The Illustrated Wavelet Transform Handbook

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The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance

The Illustrated Wavelet Transform Handbook Introductory Theory and Applications in Science, Engineering, Medicine and Finance

Paul S Addison Napier University, Edinburgh, UK

loP

Institute of Physics Publishing Bristol and Philadelphia

© lOP Publishing Ltd 2002 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with Universities UK (UUK). British Library Cataloguing-in-Publication Data

A catalogue record of this book is available from the British Library. ISBN 0 7503 06920 Library of Congress Cataloging-in-Publication Data are available

Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Frederique Swist Marketing: Nicola Newey and Verity Cooke Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS 1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 929, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset by Academic + Technical, Bristol Printed in the UK by MPG Books Ltd, Bodmin, Cornwall

For Hannah, Stephen, Anthony and Michael

Contents

Preface 1

2

3

Getting started 1.1 Introduction 1.2 The wavelet transform 1.3 Reading the book

xi

1 1 2 3

The continuous wavelet transform 2.1 Introduction 2.2 The wavelet 2.3 Requirements for the wavelet 2.4 The energy spectrum of the wavelet 2.5 The wavelet transform 2.6 Identification of coherent structures 2.7 Edge detection 2.8 The inverse wavelet transform 2.9 The signal energy: wavelet-based energy and power spectra 2.10 The wavelet transform in terms of the Fourier transform 2.11 Complex wavelets: the Morlet wavelet 2.12 The wavelet transform, short time Fourier transform and Heisenberg boxes 2.13 Adaptive transforms: matching pursuits 2.14 Wavelets in two or more dimensions 2.15 The CWT: computation, boundary effects and viewing 2.16 Endnotes 2.16.1 Chapter keywords and phrases 2.16.2 Further resources

6 6 6 9 9 11 14 21 25 28 33 35 45 51 55 56 63 63 63

The discrete wavelet transform 3.1 Introduction 3.2 Frames and orthogonal wavelet bases 3.2.1 Frames

65 65 65 65 Vll

Contents

Vlll

Dyadic grid scaling and orthonormal wavelet transforms The scaling function and the multiresolution representation The scaling equation, scaling coefficients and associated wavelet equation 3.2.5 The Haar wavelet 3.2.6 Coefficients from coefficients: the fast wavelet transform Discrete input signals of finite length 3.3.1 Approximations and details 3.3.2 The multiresolution algorithm-an example 3.3.3 Wavelet energy 3.3.4 Alternative indexing of dyadic grid coefficients 3.3.5 A simple worked example: the Haar wavelet transform Everything discrete 3.4.1 Discrete experimental input signals 3.4.2 Smoothing, thresholding and denoising Daubechies wavelets 3.5.1 Filtering 3.5.2 Symmlets and coiflets Translation invariance Biorthogonal wavelets Two-dimensional wavelet transforms Adaptive transforms: wavelet packets Endnotes 3.10.1 Chapter keywords and phrases 3.10.2 Further resources 3.2.2 3.2.3 3.2.4

3.3

3.4

3.5

3.6 3.7 3.8 3.9 3.10

4

Fluids

4.1 4.2

4.3

4.4

4.5

Introduction Statistical measures 4.2.1 Moments, energy and power spectra 4.2.2 Intermittency and correlation 4.2.3 Wavelet thresholding 4.2.4 Wavelet selection using entropy measures Engineering flows 4.3.1 Jets, wakes, turbulence and coherent structures 4.3.2 Fluid-structure interaction 4.3.3 Two-dimensional flow fields Geophysical flows 4.4.1 Atmospheric processes 4.4.2 Ocean processes Other applications in fluids and further resources

5 Engineering testing, monitoring and characterization 5.1 5.2 5.3

Introduction Machining processes: control, chatter, wear and breakage Rotating machinery

67 69 72 73 75 77 77 81 83 85 87 91 91 96 104 112 115 117 119 121 133 141 141 141 144 144 145 145 152 153 159 160 160 171 174 178 178 186 187 189 189 189 195

Contents

5.4 5.5 5.6 5.7 5.8

6

7

5.3.1 Gears 5.3.2 Shafts, bearings and blades Dynamics Chaos Non-destructive testing Surface characterization Other applications in engineering and further resources 5.8.1 Impacting 5.8.2 Data compression 5.8.3 Engines 5.8.4 Miscellaneous

IX

195 199 202 208 211 221 224 224 225 228 229

Medicine 6.1 Introduction 6.2 The electrocardiogram 6.2.1 ECG timing, distortions and noise 6.2.2 Detection of abnormalities 6.2.3 Heart rate variability 6.2.4 Cardiac arrhythmias 6.2.5 ECG data compression 6.3 Neuroe1ectric waveforms 6.3.1 Evoked potentials and event-related potentials 6.3.2 Epileptic seizures and epileptogenic foci 6.3.3 Classification of the EEG using artificial neural networks 6.4 Pathological sounds, ultrasounds and vibrations 6.4.1 Blood flow sounds 6.4.2 Heart sounds and heart rates 6.4.3 Lung sounds 6.4.4 Acoustic response 6.5 Blood flow and blood pressure 6.6 Medical imaging 6.6.1 Ultrasonic images 6.6.2 Magnetic resonance imaging, computed tomography and other radiographic images 6.6.3 Optical imaging 6.7 Other applications in medicine 6.7.1 Electromyographic signals 6.7.2 Sleep apnoea 6.7.3 DNA 6.7.4 Miscellaneous 6.7.5 Further resources

230 230 230 231 234 236 239 248 248 249 252 255 258 259 260 263 264 267 270 270

Fractals, finance, geophysics and other areas 7.1 Introduction 7.2 Fractals 7.2.1 Exactly self-similar fractals

278 278 278 279

270 273 275 275 275 276 276 277

Contents

x

7.2.2 Stochastic fractals 7.2.3 Multifractals 7.3 Finance 7.4 Geophysics 7.4.1 Properties of subsurface media 7.4.2 Surface feature analysis 7.4.3 Climate, clouds, rainfall and river levels 7.5 Other areas 7.5.1 Astronomy 7.5.2 Chemistry and chemical engineering 7.5.3 Plasmas 7.5.4 Electrical systems 7.5.5 Sound and speech 7.5.6 Miscellaneous Appendix Useful books, papers and websites 1 Useful books and papers 2 Useful websites

282 292 294 298 299 305 307 309 309 310 311 311

312 313

314 314 315

References

317

Index

351

Preface

Over the past decade or so wavelet transform analysis has emerged as a major new time-frequency decomposition tool for data analysis. This book is intended to provide the reader with an overview of the theory and practical application of wavelet transform methods. It is designed specifically for the 'applied' reader, whether he or she be a scientist, engineer, medic, financier or other. The book is split into two parts: theory and application. After a brief first chapter which introduces the main text, the book tackles the theory of the continuous wavelet transform in chapter 2 and the discrete wavelet transform in chapter 3. The rest of the book provides an overview of a variety of applications. Chapter 4 covers fluid flows. Chapter 5 tackles engineering testing, monitoring and characterization. Chapter 6 deals with a wide variety of medical research topics. The final chapter, chapter 7, covers a number of subject areas. In this chapter, three main topics are considered first-fractals, finance and geophysics-and these are followed by a general discussion which includes many of other areas not covered in the rest of the book. The theory chapters (2 and 3) are written at an advanced undergraduate level. In these chapters I have used italics for both mathematical symbols and key words and phrases. The key words and phrases are listed at the end of each chapter and the reader new to the subject might find it useful to jot down the meaning of each key word or phrase to test his or her understanding of them. The applications chapters (4 to 7) are at the same level, although a considerable amount of useful information can be gained without an in-depth knowledge of the theory in chapters 2 and 3, especially in providing an overview of the application of the theory. It is envisaged that the book will be of use both to those new to the subject, who want somewhere to begin learning about the topic, and also those who have been working in a particular area for some time and would like to broaden their perspective. It can be used as a handbook, or 'handy book', which can be referred to when appropriate for information. The book is very much 'figure driven' as I believe that figures are extremely useful for illustrating the mathematics and conveying the concepts. The application chapters of the book aim to make the reader aware of the similarities that exist in the usage of wavelet transform analysis across disciplines. In addition, and perhaps more importantly, it is intended to make the reader aware of wavelet-based methods in use in unfamiliar disciplines which may be transferred to Xl

xu

Preface

his or her own area-thus promoting an interchange of ideas across discipline boundanes. The application chapters are essentially a whistle-stop tour of work by a large number of researchers around the globe. Some examples of this work are discussed in more detail than others and, in addition, a large number of illustrations have been used which have been taken (with permission) from a variety of published material. The examples and illustrations used have been chosen to provide an appropriate range to best illustrate the wavelet-based work being carried out in each subject area. It is not intended to delve deeply into each subject but rather provide a brief overview. It is then left to the reader to follow up the relevant references cited in the text for themselves in order to delve more deeply into each particular topic as he or she requires. I refer to over seven hundred scientific papers in this book which I have collected and read over the past three or so years. I have made every effort to describe the work of others as concisely and accurately as possible. However, if I have misquoted, misrepresented, misinterpreted, or simply missed out something I apologise in advance. Of course, all comments are welcome-e-mail address below. The book stems from my own interest in wavelet transform analysis over the past few years. This interest has led to a number of research projects concerning the wavelet-based analysis of both engineering and medical signals including: non-destructive testing signals, vortex shedding signals in turbulent fluid flows, digitized spatial profiles of structural cracks, river bed sediment surface data sets, phonocardiographic signals, pulse oximetry traces (photoplethysmograms) and the electrocardiogram (ECG), the latter leading to patent applications and a university spin-off company, Cardiodigital Ltd. Quite a mixed bag, at first appearance, but with a common thread of wavelet analysis running throughout. I have featured some of this work in the appropriate chapters. However, I have tried not to swamp the application chapters with my own work-although the temptation was high for a number of reasons including knowledge of the work, ease of reproduction etc. I hope I have struck the correct balance. All books reflect, to some extent, the interests and opinions of the author and, although I have tried to cover as broad a range of examples as possible, this one is no exception. Coverage is weighted to those areas in which I have more interest: fluids, engineering, medicine and fractal geometry. Geophysics and finance are given less space and other areas (e.g. astronomy, chemistry, physics, non-medical biology, power systems analysis) are detailed briefly in the final chapter. There are some idiosyncrasies in the text which are worth pointing out. I am anf person not an w person: I prefer Hertz to radians per second. I can tap my fingers at approximately 5 Hz, or 1 Hz, I know what 50 Hz means (mains hum in the UK) and so on: w, I have to convert. Hence the frequencies in the text are in the form of l/time either in Hertz or non-dimensionalized. The small downside is that the mathematics, in general, contain a few more terms-mostly 2s and 'iTs. I have devoted a whole chapter to the continuous wavelet transform. It is noticeable that many current wavelet texts on the market deal only with the discrete wavelet transform, or give the continuous wavelet transform a brief mention en route to the theory of the discrete

Preface

Xlll

wavelet transform. I believe that the continuous wavelet transform has much to offer a wide variety of data analysis tasks and I attempt, through this text, to redress the balance somewhat. (Actually, the proportion of published papers which concern the continuous wavelet transform as opposed to the discrete wavelet transform is much higher than that represented by the currently available wavelet text books.) The book is focused on the wavelet transform and makes only passing reference in the application chapters to some of the other time-frequency methods now available. However, I have added sections on the short time Fourier transform and matching pursuits towards the end of chapter 2 and on wavelet packets at the end of chapter 3 respectively. Finally, note that I have developed the discrete wavelet transform theory in chapter 3 in terms of scale rather than resolution, although the relationship between the alternative notations is explained. I would like to thank the following people for taking the time to comment on various drafts of the manuscript: Andrew Chan of Birmingham University, Gareth Clegg of Edinburgh University (formerly at The Royal Infirmary of Edinburgh), Maria Haase of Stuttgart University and Alexander Droujinine of Heriot-Watt University. I would like to thank Jamie Watson of CardioDigital Ltd for his comments on the draft manuscript and for his close collaboration over the years (and various bits of computer code!). I would also like to thank all those authors and publishers who gave their consent to reproduce their figures within this text. I am grateful to those funding bodies who have supported my research in wavelet analysis and other areas over the years, including the Engineering and Physical Science Research Council (EPSRC), the Medical Research Council (MRC) and the Leverhulme Trust. And to those other colleagues and collaborators with whom my wavelet research is conducted and who make it so interesting, thanks. Special thanks to my wife, Stephanie, who has supported and encouraged me during the writing of this book. Special thanks also to my parents for their support and great interest in what I do. Although it has been a long hard task, I have enjoyed putting this book together. I have certainly got a lot out of it. I hope you find it useful. Paul S Addison January 2002 [email protected]

Chapter 1

Getting started

1.1

Introduction

The wavelet transform (WT) has been found to be particularly useful for analysing signals which can best be described as aperiodic, noisy, intermittent, transient and so on. Its ability to examine the signal simultaneously in both time and frequency in a distinctly different way from the traditional short time Fourier transform (STFT) has spawned a number of sophisticated wavelet-based methods for signal manipulation and interrogation. Wavelet transform analysis has now been applied in the investigation of a multitude of diverse physical phenomena, from climate analysis to the analysis of financial indices, from heart monitoring to the condition monitoring of rotating machinery, from seismic signal denoising to the denoising of astronomical images, from crack surface characterization to the characterization of turbulent intermittency, from video image compression to the compression of medical signal records, and so on. Many of the ideas behind wavelet transforms have been in existence for a long time. However, wavelet transform analysis as we now know it really began in the mid-1980s where they were developed to interrogate seismic signals. Interest in wavelet analysis remained within a small, mainly mathematical community during the rest of the 1980s with only a handful of scientific papers coming out each year. The application of wavelet transform analysis in science and engineering really began to take off at the beginning of the 1990s, with a rapid growth in the numbers of researchers turning their attention to wavelet analysis during that decade. The past few years have each seen the publication of over one thousand refereed journal papers concerning the wavelet transform, covering all numerate disciplines. Figure 1.1 shows this rapid increase in wavelet-based scientific papers published in recent years. The wavelet transform is a mathematical tool which is now common in many data analysts' toolboxes. This book aims to provide the reader both with an introduction to the theory of wavelet transforms and an overview of its use in practice. The two remaining sections of this short chapter contain, respectively, a brief non-mathematical description of the wavelet transform and a guide to subsequent chapters of the book.

1

2

Getfing started 1400

~

1200

tOOO

~

400 200

o

/

/

./

./ ~

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 ye"

Figure 1.1. Yearly count of scientific papers concerning wavelets 1990-2001. The plot contains the number of papers with 'wavelet' or 'wavelets' in the title, keywords or abstract of refereed journal papers. Source: Web of Science, httpllwos.mimas.ac.uk/ (Nole that a handful of these papers do not concern wavelet transforms, but rather refer to physical phenomena where the term wavelet has been used to describe a small localized wave.)

1.2

The wavelet transform

Wavelet transform analysis uses little wavelike functions known as wavelets. Actually. 'local' wavelike function is a more accurate description of a wavelet. Figure 1.2(a) shows a few examples of wavelets commonly used in practice. Wavelets are used to transform the signal under investigation into another representation which presents the signal information in a more useful form. This transformation of the signal is known as the wavelel trallSform. Mathematically speaking, the wavelet transform is a convolution of the wavelet function with the signal and we will see exactly how this is done in chapters 2 and 3. Here we stick to schematics. The wavelet can be manipulated in two ways: it can be moved to various locations

on the signal (figure 1.2(b)) and it can be stretched or squeezed (figure 1.2(c)). Figure 1.3 shows a schematic of the wavelet transform which basically quantifies the local matching of the wavelet with the signal. ff the wavelet matches the shape of the signal well at a specific scale and location, as it happens to do in the top plot of figure 1.3, then a large transform value is obtained. If, however, the wavelet and the signal do not correlate well, a low value of the transform is obtained. The transform value is then located in the two-dimensional transform plane shown at the bottom of figure 1.3 (indicated by the black dot). The transform is computed at various locations of the signal and for various scales of the wavelet, thus filling up the transform plane: this is done in a smooth continuous fashion for the cOllfinuol/s wal'eleI tram/arm (CWT) or in discrete steps for the discrete wavelet trans/arm (DWT).

Reading the book

3

(a)

(b)

(c)

/\

Figure 1.2. The little wave. (a) Some wavelets. (b) Location. (c) Scale.

Plotting the wavelet transform allows a picture to be built up of the correlation between the wavelet-at various scales and locations-and the signal. In subsequent chapters we will cover the wavelet transform in more mathematical detail.

1.3

Reading the book

The purpose of the book is both to introduce the wavelet transform and to convey its multidisciplinary nature. This is done in the subsequent chapters of the book by first providing an elementary introduction to wavelet transform theory and then presenting a wide range of examples of its application. It will quickly become apparent that

4

Getting started

local matching of wavelet and signal leads to a large transform value

Wavelet Transform

scale , - - - - - - - - - - - - - - - - - - - - - - - - - ,

current wavelet -- ----------------------------. scale

current wavelet location

location

Figure 1.3. The wavelet, the signal and the transform.

very often the same wavelet methods are used to interrogate signals from very different subject areas, where quite unrelated phenomena are under investigation. The book is split into two distinct parts: the first part, chapters 2 and 3, deals respectively with continuous wavelet transform theory and discrete transform theory; while the second part, chapters 4, 5, 6 and 7, presents examples of their application in science, engineering, medicine and finance. There are a number of ways to read this book: from the linear (beginning to end) via the targeted (employing the index) to the random (flicking through) approach. The reader unfamiliar with wavelet theory should read chapters 2 and 3 before moving on to the sections of particular relevance to his or her area of interest. The reader is also advised to look outside his or her own area to see how wavelets are being employed elsewhere. Details of further resources concerning the theory and applications of wavelet analysis are provided at the end of each chapter. The appendix lists a selection of useful books, papers and websites. The book contents are outlined in more detail as follows: Chapter 2: This chapter presents the basic theory of the continuous wavelet transform. It outlines what constitutes a wavelet and how it is used in the transformation of a signal. In the latter part of the chapter the continuous wavelet transform is

Reading the book

5

compared both with the short time Fourier transform and the matching pursuit method. Chapter 3: The discrete wavelet transform is described in this chapter. Orthonormal discrete wavelet transforms are considered in detail, in particular those of Haar and Daubechies. These wavelets fit into a multiresolution analysis framework where a discrete input signal can be represented at successive approximations by a combination of a smoothed signal component plus a sum of detailed wavelet components. The chapter ends by looking briefly at wavelet packets, a generalization of the discrete wavelet transform which allows for adaptive partitioning of the timefrequency plane. Chapter 4: This chapter deals with fluid mechanics, a subject that is always hungry for new mathematical techniques. The time-frequency localization properties of the wavelet transform have been employed extensively in the study of a wide variety of fluid phenomena including the intermittent nature of fluid turbulence, the characteristics of turbulent jets, the nature of fluid-structure interactions and the behaviour of large scale geophysical flows. Chapter 4 also contains the mathematics for discrete wavelet statistics and power spectra following on from some of the basic theory given in chapter 3. Chapter 5: In this chapter a close look is taken at the application of wavelet transforms to a variety of pertinent problems in engineering. These applications include the assessment of machine processes behaviour, condition monitoring of rotating machinery, the analysis of nonlinear and transient oscillations, the characterization of repeated impacting on structural elements, the interrogation of non-destructive testing signals, and the characterization of rough surfaces. Chapter 6: Medical applications of wavelet transform analysis are covered in this chapter. Wavelet transform methods have been used to characterize a wide variety of medical signals. Many of these are reviewed in this chapter, including the ECG, EEG, EMG, pathological sounds (lung sounds, heart sounds and arterial sounds), blood flows, blood pressures, DNA sequences and medical images (optical, x-ray, NMR, ultrasound etc.). Chapter 7: This final chapter covers a variety of areas of application. Most of the chapter is devoted to three main subjects-fractal geometry, finance and geophysics-with a separate section devoted to each of them. The final part of the chapter provides a brief account of the role wavelet transform analysis has played in a number of other areas including astronomy, plasma physics, electrical power systems, chemical analysis and more. Appendix: The appendix contains a list of useful papers, books and websites concerning wavelet transform theory and application. These have been chosen by the author for their extensive content and/or clarity of presentation.

Chapter 2

The continuous wavelet transform

2.1

Introduction

This chapter covers the basic theory of the continuous wavelet transform (CWT). We will first determine what constitutes a wavelet, how it is used in the transformation of a signal and what it can tell us about the signal. We then consider the inverse wavelet transform and the reconstruction of the original signal. We will look at the energy-preserving features of the wavelet transform and how it may be used to produce wavelet power spectra. Finally, we will compare the wavelet transform both with the short time Fourier transform (STFT) and matching pursuit (MP) method.

2.2

The wavelet

The wavelet transform is a method of converting a function (or signal) into another form which either makes certain features of the original signal more amenable to study or enables the original data set to be described more succinctly. To perform a wavelet transform we need a wavelet which, as the name suggests, is a localized waveform. In fact, a wavelet is a function 1jJ(t) which satisfies certain mathematical criteria. As we saw briefly in the previous chapter, these functions are manipulated through a process of translation (i.e. movements along the time axis) and dilation (i.e. spreading out of the wavelet) to transform the signal into another form which 'unfolds' it in time and scale. Note that in this chapter and the next we assume that the signal to be analysed is a temporal signal, i.e. some function of time such as a velocity trace from a fluid, vibration data from an engine casing, an ECG signal and so on. However, many of the applications discussed later in the book concern wavelet analysis of spatial signals, such as well logged geophysical data, crack surface profiles, etc. In these cases, the independent variable is space rather than time; however, the wavelet analysis is performed in exactly the same way. A selection of wavelets commonly used in practice is shown in figure 2.1. We will consider some of them in more detail as we proceed through the text. As we can see from the figure they have the form of a small wave, localized on the time axis. There 6

The wavelet ,/(1) , - - - - - - - - ,

7

,/(1) , - - - - - - - - - - - ,

v (,) ' - - - - - - - - - - - " t

(b) L -

_____='

,/(1) , - - - - - - - - - - ,

(e) ' -

J

(d) L -

-----'

t

Figure 2.1. Four wavelets. (a) Gaussian wave (first derivative of a Gaussian), (b) Mexican hat (second derivative of a Gaussian). (c) Haar. (d) Morlel (real part).

are, in fact, a large number of wavelets to choose from for use in the analysis of our data. The best one for a particular application depends on both the nature of the signal and what we require from the analysis (Le. what physical phenomena or process we are looking to interrogate, or how we are trying to manipulate the signal). We will begin this chapter by concentrating on a specific wavelet, the Mexican hat, which is very good at illustrating many of the properties of continuous wavelet transform analysis. The Mexican hat wavelet is shown in figure 2.I(b) and in more detail in figure 2.2(a). The Mexican hat wavelet is defined as

(2.1 ) The wavelet described by equation (2.1) is known as the mother wavelet or analysing wavelet. This is the basic form of the wavelet from which dilated and translated versions are derived and used in the wavelet transform. The Mexican hat is, in fact, t2j2 the second derivative of the Gaussian distribution function e- : that is, with unit variance but without the usual 1/ J2n normalization factor. The Mexican hat normally used in practice, Le. that given by equation (2.1) and shown in figure 2.2(a), is actually the negative of the second derivative of the Gaussian function. Al! derivatives of the Gaussian function may be employed as a wavelet. Which is the most appropriate one to use depends on the application. Both the first and second derivatives of the Gaussian are shown in figures I(a) and I(b). These are the two that are most often used in practice. Higher-order derivatives are less commonplace.

The continuous wavelet transform

8

1,....----------""7"1',.....----------..., 0.5

....

~

'-'

-7

0 -0.5

-1

-4

-3

-1

-2

1

0

4

t

I

a

I

(a)

3

2

~

I

a.

b

4

o -1.0

-0.8

-0.6

-0.4

-0.2

0.2

0.4

0.6

0.8

1.0

I Ip Ie a

I I I

.... I

,I

(b)

Peak frequency - I p

= Vi 2:rt

Bandpass frequency - Ie =

V5i2 2:rt

V2

and also - I p =- 2:rt

Figure 2.2. The Mexican hat mother wavelet and its associated energy spectrum. (a) The Mexican hat mother wavelet (named for an obvious reason!). Notice that, for the Mexican hat, the dilation parameter a is the distance from the centre of the wavelet to where it crosses the horizontal axis. (b) The energy spectrum of the Mexican hat shown in (a). Note that as it is a real wavelet its Fourier spectrum is mirrored around the zero axis. (a is the standard deviation of the spectrum around the vertical axis.)

The energy spectrum of the wavelet

2.3

9

Requirements for the wavelet

In order to be classified as a wavelet, a function must satisfy certain mathematical criteria. These are: 1. A wavelet must have finite energy: (2.2)

where E is the energy of a function equal to the integral of its squared magnitude and the vertical brackets II represent the modulus operator which gives the magnitude of 1jJ(t). If 1jJ(t) is a complex function the magnitude must be found using both its real and complex parts. 2. If ~(f) is the Fourier tran~form of 1jJ(t), i.e.

~(f)

=

f~oo

1jJ( t) e -i(27rf)t dt

(2.3)

then the following condition must hold: (2.4)

This implies that the wavelet has no zero frequency component, ~(O) = 0 or, to put it another way, the wavelet 1jJ(t) must have a zero mean. Equation (2.4) is known as the admissibility condition and Cg is called the admissibility constant. The value of Cg depends on the chosen wavelet and is equal to Jr for the Mexican hat wavelet given in equation (2.1). 3. An additional criterion that must hold for complex wavelets is that the Fourier transform must both be real and vanish for negative frequencies. We shall consider complex wavelets towards the end of this chapter when we will take a close look at the Morlet wavelet.

2.4

The energy spectrum of the wavelet

Wavelets satisfying the admissibility condition (equation (2.4)) are in fact bandpass filters. This means in simple terms that they let through only those signal components within a finite range of frequencies (the passband) and in proportions characterized by the energy spectrum of the wavelet. A plot of the squared magnitude of the Fourier transform against frequency for the wavelet gives its energy spectrum. For example, the Fourier energy spectrum of the Mexican hat wavelet is given by

(2.5) where the subscript F is used to denote the Fourier spectrum as distinct from the wavelet-based spectrum defined later in section 2.9. A plot of the energy spectrum

l a T h e continuous wavelet transform

of the Mexican hat wavelet is shown in figure 2(b). Note that, as the Mexican hat wavelet is a real function, its Fourier spectrum is symmetric about zero. We will see later that complex wavelets do not have negative frequency components (requirement 3 above). The peak of the energy spectrum occurs at a dominant frequency of f p = ±J2/27!". The second moment of area of the energy spectrum is used to define the passband centre of the energy spectrum, fo as follows:

50 f21~(f) 2 df 2 50 I~(f) 4f 00

1

(2.6)

00

1

where fe is simply the standard deviation of the energy spectrum about the vertical axis. For the Mexican hat mother wavelet, j~ is equal to V572/27!" or 0.251 Hz. In practice we require a characteristic frequency of the mother wavelet, such as ./'p, fe or some other, in order to relate the frequency spectra obtained using Fourier transforms to those obtained using wavelet transforms. Later we will see how these characteristic frequencies change as the mother wavelet is stretched and squeezed through its dilation parameter. When performing wavelet transform analysis it is important that the energy spectrum of the wavelet is considered, as it indicates the range and character of the frequencies making up the wavelet. From equations (2.1) and (2.2) we see that the total energy of the Mexican hat wavelet is finite and given by E

=

.f~oo 'IjJ(t)2 dt = .f~oo [(1 -

2 t2 2 t ) e- / ]2 dt

=

~ J7f

(2.7)

The energy of a function is also given by the area under its energy spectrum. For the Mexican hat wavelet this gives us

(2.8) Hence

(2.9) This is a result we would expect for any function from Parseval's theorem. Often, in practice, the wavelet function is normalized so that it has unit energy. To do this for the Mexican hat we modify its definition given in equation (2.1). From equation (2.7) we see that it is normalized to have unit energy by dividing it by (3J7f/4) 1/2. This gives (2.10) Both equation (2.1) and equation (2.10) are commonplace in the literature. The only alteration necessary when employing the normalized Mexican hat of equation (2.10) rather than that defined in equation (2.1) is in the value of the admissibility constant C g , which must be changed from 7r to 4J7f/3. In the rest of this chapter we will stick to our original definition of the Mexican hat given by equation (2.1).

The wavelet transform

2.5

11

The wavelet transform

Now we have chosen a mother wavelet, how do we put it to good use in a signal analysis capacity? First we require our wavelet to be more flexible than that defined earlier, i.e. 'IjJ(t). We can perform two basic manipulations to make our wavelet more flexible: we can stretch and squeeze it (dilation) or we can move it (translation). Figure 2.3(a) shows the Mexican hat wavelet stretched and squeezed to respectively 1.0

0.5 ,.-...

-(:l

0

b

?

-0.5

-1.0

L..---'----'----'--------1I-+-+------1f---------'--------'--------'-.....

(a)

1.0,....-------.r<"----....",....------------.r<"--------,

0.5 ,.-...

..t:> I

..... ?

'-'

0

-0.5

(b)

Figure 2.3. Dilation and translation of a wavelet. (a) Stretching and squeezing a wavelet: dilation (a] = a2/2; a3 = a2 x 2). (b) Moving a wavelet: translation.

12

The continuous wavelet transform

double and half its original width on the time axis. This dilation and contraction of the wavelet is governed by the dilation parameter a which, for the Mexican hat wavelet, is (helpfully) the distance between the centre of the wavelet and its crossing of the time axis. The movement of the wavelet along the time axis is governed by the translation parameter b. Figure 2.3(b) shows the movement of a wavelet along the time axis from b l via b2 to b 3 . We can include the dilation parameter, a, and the location parameter, b, within our definition of a wavelet given by equation (2.1). These shifted and dilated versions of the mother wavelet are denoted '1;0 [(t - b) / a]. For example, in this form the Mexican hat wavelet becomes (2.11 ) The original mother wavelet 'I;0(t) given by equation (2.1) simply had a = 1 and b = O. In the form of equation (2.11) we can now transform a signal, x(t), using a range of a's and b's. The wavelet transform of a continuous signal with respect to the wavelet function is defined as

T(a, b)

=

J~CXJ

w(a)

x(t)'I;0* (t

~ b) dt

(2.12)

where w(a) is a weighting function. The asterisk indicates that the complex conjugate of the wavelet function is used in the transform. We need not consider this when using the Mexican hat wavelet as it is a real function, but we do need to take this into account when using complex wavelets later in the chapter. The wavelet transform can be thought of as the cross-correlation of a signal with a set of wavelets of various 'widths'. Typically w(a) is set to 1/ for reasons of energy conservation (i.e. it ensures that the wavelets at every scale all have the same energy) and we will use this value for the rest of this chapter. However, w(a) = l/a is sometimes used and there is nothing to stop the user defining a function more appropriate to the application, e.g. for the visual enhancement of the transform plot (see section 2.15). From here on we will use w(a) = 1/ Thus the wavelet transform is written

va

va.

T(a, b)

=

~ .f~CXJ x(t)'I;0* (t ~ b) dt

(2.13)

This is the continuous wavelet transform or CWT. Take a closer look at this equation. It contains both the dilated and translated wavelet 'I;0((t - b)/a) and the signal x(t), where x(t) could be a beating heart, an audio signal, gearbox vibration, a financial index or perhaps even a spatial signal such as a crack profile or land surface heights. In the equation, the product of the wavelet and the signal are integrated over the signal range. In mathematical terms this is called a convolution. The normalized wavelet function is often written more compactly as

'l;0a,b(t)

=

va'11;0 (t-b) -a-

(2.14)

The where the normalization is integral may be written as

In

II l allelet

transform

13

the sense of wavelet energy. Hence, the transform

(2.15) This is the nomenclature we will use in this chapter and we will refer to 'l/Ja.b(t) simply as the wavelet. We can express the wavelet transform in even more compact form as an inner prod uct:

T(a,b)

= (x,'IjJ".bl

(2.16)

Figure 2.4 shows the effect that the dilation of a Mexican hat wavelet 'l/JlI.b(t) has on its corresponding energy spectrum. As the wavelet expands, its corresponding energy (,) 1.5

....

1.0

I:

0.5 1jJa.b(t)

-

0

" YJ':..

-1.0

-10

-8

-6

' -4

'::.XX:"\"''-" /' ~

,: ..

-.......-../

-0.5

\

I : · \ ... "/'.:

···... ·

/'"""

, -2

o

2

6

4

8

10

t

(b)

10 r----,--,-------,----,----,

5

(\ I \

I I \ 0.4

0.6

0.8

1.0

J Figure 2.4. The Fourier energy spectra of dilating wa\·elets. (a) Three Mexican hat wavelets

.,pu,b(/) = (1/Ja).,p[(t - b)/a] at three dilations, a = 0.5,1.0,2.0 and all located at b = O. (b) Energy spectra corresponding to the wavelets in (a) all have identical energy and hence the same area under their curves. Note that only the positive part of the energy spectrum is shown.

14

The continuous wavelet transform

spectrum contracts. This is an obvious consequence, as expansion in the time domain must involve the lengthening of time periods and a corresponding lowering of associated frequencies. The wavelet a scale is therefore inversely proportional to all its characteristic frequencies, including its passband centre frequency, peak frequency, central frequency (for complex wavelets) and so on. We will come back to this relationship in more detail later in this chapter when we consider wavelet power spectra in section 2.9. The wavelet transform has been called a mathematical microscope, where b is the location on the time series being 'viewed' and a is associated with the magnification at location b. Now we have defined the wavelet and its transform, we are ready to see how the transform is used as a signal analysis tool.

2.6

Identification of coherent structures

Figure 2.5(a) attempts to visualize the mechanics of the wavelet transform given by equation (2.15). In the figure, a wavelet of scale a centred at location b on the time axis is shown superimposed on top of an arbitrary signal. The time segments where the wavelet and the signal are both positive result in a positive contribution to the integral of equation (2.15), e.g. region A in the figure. Similarly, the time segments where the wavelet and the signal are both negative result in a positive contribution to the integral (region B). Regions where the signal and wavelet are of opposite sign result in negative contributions to the integral, e.g. regions C, D and E in the figure. Figure 2.5(b) shows a wavelet of fixed dilation at four locations on a signal. At the first location (bd, the wavelet is located on a segment of the signal in which the positive and negative parts of the signal are reasonably coincidental with those of the wavelet. This results in a relatively large positive value of T (a, b) given by equation (2.15). At location b2 , the positive and negative contributions to the integral act to cancel each other out, resulting in a value near zero returned from equation (2.15). At location b 3 , the signal and the wavelet are essentially out of phase which results in a large negative value returned for T(a,b). At location b4 the wavelet and the signal are again out of phase, similar to location b3 . This time, however, the signal portion in the vicinity of the wavelet contains a large local mean component. It is easy to see that the mean component contributes equal positive and negative values to T(a, b). Thus, only the local signal feature is highlighted by the wavelet at this location and the mean is disregarded. It is through this process that the wavelet transform picks out 'coherent structures' in a time signal at various scales. By moving the wavelet along the signal (increasing b) coherent structures relating to a specific a scale in the signal are identified. This process is repeated over a range of a scales until all the coherent structures within the signal, from the largest to the smallest, can be distinguished. Let us look at another simple example. Figure 2.6 shows a simple sinusoidal waveform 'interrogated' at various locations by Mexican hat wavelets of various dilations. The value of the transform convolution (equation (2.15)) depends upon both the location and dilation of the wavelet. Figure 2.6(a) shows a wavelet of similar

Identification of coherent structures negative positive contribution contribution

15

posItIve

COnlributiOn nogativo

x(t)

'4'a,b(t)

negative contribution

\i

/

:

D

B

E

contribution

••

/

analysing wavelet

(a)

x(t)

'4'a,b(t)

b

local signal mean at b 4

\

(b)

Figure 2.5. The wavelet interrogation of the signal. (a) The wavelet of specific dilation and location on the signal. The regions which give positive and negative contributions to the integral are delineated in the sketch and marked with '+' and '-' respectively. (b) A wavelet of fixed dilation at four distinct locations on the signal. A large positive value of T( a, b) is returned at location b] . A near-zero value of T(a, b) is returned at b2 and a large negative value of T(a, b) is returned at b3 . At b4 a local minimum in the signal corresponds with the positive part of the wavelet and relatively higher parts of the signal correspond with the negative parts of the wavelet. This combines to give a large negative value of T(a, b).

'periodicity' to the signal waveform superimposed on the signal at location b, which produces a reasonable local matching of the wavelet and signal. From the figure, it can be seen that there is obviously a high correlation between the signal and wavelet at this a scale and b location. The integral of the product of the signal with the wavelet here produces a large positive value of T(a, b). Figure 2.6(b) shows the wavelet moved to a new location where the wavelet and signal seem to be out of phase. In this case the convolution produces a large negative value of T(a, b). In between these two extremes, the value of the transform reduces from a maximum (figure 2.6(a)) to a

16

The continuous wavelet transform sinusoidal waveform P i

1

,/

0.5

\

1

0.5

a

a

-0.5 -1

-0.5 -1 (a)

b

(d)

1

1

0.5

0.5

a

a

-0.5

-0.5

b

-1

-1 (b)

(c)

b

(e)

b

b

Figure 2.6. A wavelet interrogating a sinusoidal waveform. (a) The wavelet is in phase with the waveform giving good positive correlation. (b) The wavelet is now out of phase with the waveform giving good negative correlation. (c) The wavelet is again out of phase with the waveform, this time giving zero correlation. (d) The squeezed wavelet does not match the waveform locally. (e) The stretched wavelet does not match the waveform locally.

minimum (figure 2.6(b». Figure 2.6(c) shows the point at which the wavelet and signal produce a zero value of T(a,b). Figures 2.6(a), (b) and (c) consider a wavelet which matches locally the signal, i.e. it has approximately the same 'shape' and 'size' of the signal in the vicinity of b. Figure 2.6(d) shows the effect that using a smaller a scale has on the transform. From the plot we see that the positive and negative parts of the wavelet are all convolved by roughly the same part of the signal, producing a value of T(a, b) near zero. Hence, T(a, b) tends to zero as the dilation a tends to zero width. T(a, b) also tends to zero as a becomes very large (figure 2.6(e», as now the wavelet covers many positive and negatively repeating parts of the signal, again producing a near-zero value of T(a,b) in the integral of equation (2.15). Thus when the wavelet function is either very small or very large compared with the signal features, the transform gives near-zero values. Continuous wavelet transforms are not usually computed at arbitrary dilations and isolated locations but rather over a continuous range of a and b. A plot of T(a, b) versus a and b for a sinusoidal signal is shown in figure 2.7, where the Mexican hat wavelet has been used. This plot of T(a, b) against a and b is known as a wavelet transform plot. Two methods are employed to present the resulting transformed

Idellti}icatioll of cohel'enr structures

17

_P-..

(.)

[\7\7\7\7\] (b)

O.05p O,lOp O.25p a

I.CIOp 2.50p b

(c)

a

Figure 2.7. \Val'elellransform plots of a sinusoidal wal'eform. (a) Five cycles of a sinusoid of period p. (b) Contour plot of T(a,b) for Ihe sinusoid in (a). (Note the logarithmic scaling of the a axis. Note also that a greyscale is used where while corresponds to transform maxima and black 10 minima. This is the formal used in all subsequent transform plots unless otherwise stated.) (c) Isometric surface plol of T(a,b). Viewed with the smallest a scales 10 the fore.

signal in figure 2.7: as a contour plot (figure 2.7(b» and as a surface plot (figure 2.7(c». The contour plot is more commonly used in practice. The near-zero values of T{a, b) are evident in the plot at both large and small values of a, However, at intermediate values of a we can see large undulations in T(a, b) corresponding to the sinu· soidal form of the signal. We can explain these large undulations by referring back to figures 2.6(a)-(c), where wavelets of a 'size' comparable with the waveform move in

18

The camilluotls wavelet transform

and out of phase with the signal. For the Mexican hat wavelet, the a scale is required to be roughly one quarter of the period, P, of the sine wave for this to occur. (This is covered in more detail in section 2.9.) In figure 2.7(b) we can see that the maxima and minima of the transform plot actually do occur at an a scale of approximately 0.25p, indicating maximum correlation between the wavelet and signal at this scale. In the figure, the a axis has logarithmic scaling. This is the most common form used in practice; however, note that linear scales are sometimes used. All wavelet transform plots

(a)

(b)

b

(e)

a

Figure 2.8. Wa\'elet Iransform plOIS of Iwo combined sinusoids. (a) A signal composed of a combination of two sinusoids of period PI and P2- where P2 = Pl/5. (b) Contouf plot of T(a,b) fOf the waveform in (a). (c) Isometric surface pial of T(a,b). Viewed with the smallest a scales 10 the fore.

Identification of coherent structures

19

in this chapter have logarithmic a scales, with the exception of figure 2.12. In addition, in chapters 2 and 3 we stick to the convention of showing transform plots with the smallest a scales at the top, although there is nothing to stop the reader plotting them the other way up. The signal shown in figure 2.8 is composed of two sinusoidal waveforms, one with a period (pd five times the other (P2)' The transform plot below shows up very well the two periodic waveforms in the signal at a scales of one quarter of each of the periods. This figure clearly shows the ability of the transform to decompose the signal into its separate components. The transform has unfolded the signal to show its two constituent waveforms. Figure 2.9 contains a segment of a chirp signal which has the form x(t) = sinx2 . The chirp begins just into the signal window and finishes just before the end of the window. The increase in freq uency of the oscilla tion can be seen in the signal. The transform plot of the transformed chirp is shown below the signal. The transform plot shows the oscillation as peaks at decreasing a scales from left to right. Figure 2.1 O(a) contains a signal with a number of isolated features: four identical wavegroups containing three sinusoidal oscillations of the same periodicity, a group of three bumps, a small negative constant region (a block pulse) and a further sinusoidal wavegroup at a higher frequency. Three representations of the wavelet transform are plotted directly below the signal in figures 2.IO(b), (c) and (d): these are,

small a scales ...- - - - - - - - - - - - - -......

,

large a scales L

~

b

Figure 2.9. Segment of a chirp signal with associated transform plot-Mexican hat wa\'eJet.

20

The camilluotls wavelet transform

Ca)

higher frequency oscillation '\.

(b)

OC>l;'~+--II~)flJ~

ii

associated with wavegmup

",.•'1\-,

,').'

"

, ('''' -- ... /

J

\ ..

_ ./

.../ "

-

(\

-'-

'---..

'.

-

periodicity of

Ce)

wavegroup

(d)

spectral r - - . - - - - - - - - - - - - - , energy

spectral energy

(e)

frequency

frequency

Figure 2.10. The wlnelet transform of an intermittent signal. (a) Intermittent signal. (b) Filled transform plot. (c) The unfilled transform plot using two contours. (d) The unfilled transform plot using twelve contours. (The arrows point to the edges of the constant discontinuity.) (e) Fomier power spectrum of signal. (Arbitrary axis units. Left-hand plol has both axes with linear scales, righthand plot has a linear horizontal scale and a logarithmic vertical scale.)

Edge detection

21

respectively, a filled plot, a contour plot using two equally spaced contours and a contour plot using twelve equally spaced contours. The three different representations are shown to illustrate their advantages and disadvantages as a method for viewing the signal in the timescale plane ('wavelet domain' or 'wavelet space'). We can see from all three transform plots that the four identical wavegroups all have the same morphology in the wavelet domain. In addition, the periodicity of the wavegroups can be easily differentiated from the periodicity of the waves within the group, found from the associated wavelet scale as indicated in figure 2.10(c). Wavelet transforms are particularly good at picking out recognizable signal features in this way, where the features occur intermittently. The wavegroup with the higher frequency oscillation appears at a smaller a scale towards the top of the plot as we would expect. The group of three bumps appears to have a similar form to the oscillations in the filled plot, the difference being more apparent in the two-contour plot. The edges of the block pulse are more apparent in the twelve-contour plot which points to these discontinuities. These are located by the two arrows at the top of the plot. Figure 2.10(e) contains the Fourier energy spectrum of the signal. The energy spectrum on the left-hand side is plotted with linear scaling axes. The same energy spectrum is plotted on the right-hand side with a logarithmic vertical axis which shows up the rich structure in the Fourier domain. However, we can see that the Fourier representation does not provide us with any useful information regarding the coherent (obvious even) nature of the localized features within the signal.

2.7

Edge detection

Another useful property of the wavelet transform is its ability to identify abrupt discontinuities ('edges') in the signal. A simple example of a discontinuity is shown in figure 2.11(a), where a constant signal, x(t) = 1, suddenly drops to a constant negative value, x(t) = -1. To see how the wavelet picks out such a discontinuity we follow a wavelet of arbitrary dilation a as it traverses the signal discontinuity. The effect of wavelet location, b, on the transform T(a, b) is discussed for each of five locations on the signal, A, B, C, D and E. Location A: At locations much earlier than the discontinuity, for example at location A, the wavelet and the (constant) signal combine to give near-zero values of the integral for T(a, b). As it is a localized function, the wavelet becomes approximately zero at relatively short distances from its centre. Hence the wavelet transform (equation (2.15» effectively becomes a convolution of the wavelet with a constant valued signal producing a zero value. Location B: At this location the wavelet is just beginning to traverse the discontinuity. The left-hand lobe of the wavelet produces a negative contribution to the integral, the right-hand lobe of the wavelet produces an equal positive contribution, leaving the central bump of the wavelet to produce a significant positive value for the integral at this location. Location C: When the signal discontinuity coincides with the wavelet centre, b, the right and left halves of the wavelet contribute to a zero value of the integral.

22

The continuous wavelet transform x(t) 'lj!«t-b)/a)

signal

i

wavelet

E

BCD

A

(a)

discontinuity

t

(b)

C

B

T(a,b)

D

'width' controlled bya

\'

V

~

local undulation in T(a, b)

b

(c)

Figure 2.11. A schematic illustration of the wavelet interrogation of a signal discontinuity. (a) A schematic diagram of the wavelet interrogation of a signal discontinuity. The regions which give positive and negative contributions to the wavelet transform integral are indicated. (b) A blow-up of the wavelet as it traverses the discontinuity. (c) A plot of T(a, b) against location b at a specific a scale.

Note that, as the wavelet has zero mean by definition, we can see that the four regions of the wavelet in the figure all have the same area. Location D: This is similar to location B. As the wavelet traverses the discontinuity further, the left-hand lobe of the signal produces a negative contribution to the integral, the right-hand portion of the wavelet produces an equal positive contribution, as with location B. This time, however, the central portion of the wavelet

Edge detection

(a)

23

original signal

a=1

a=2

a=6

(b)

a=16

a=1

a

(0)

a =32 b

Figure 2.12. The w3\'elel decomposition of a signal discontinuily. (a) A signal with a step discontinuity. (b) Plots of T(a,b) ag"inst b at four arbitrary a sc"les for the discontinuity: a = 1,2,6 and 16. (Total window length = 64 units.) (c) The transform plot for the discontinuity in (a). Light greys correspond to large positive values of T(a, b) and dark greys to large negative values of T(a,b). Note that here a linear a scale axis is used. The plots in (b) represent vertical slices taken through the transform surface T(a,b) plotted in (c).

24

The camilluotls wavelet tralls/ol'm

coincides with the negative constant signal and hence the integral produces a significant negative value at this location. Location E: At locations far greater than C, the wavelet and signal combine to give near-zero values of the integral. Hence, as the wavelet traverses the discontinuity there are first positive then negative values returned by the transform integral. These values are localized in the vicinity of the discontinuity. This is illustrated in figure 2.1 I(c), where a schematic diagram is given of T(a, b) plotted against the b location on the time axis. From the figure, we see an undulation in T(a,b) centred at the signal discontinuity. The width of this ripple in T(a, b) is controlled by the width orthe wavelet, a. In ract, it is directly proportional to it. This is illustrated in figure 2.12, which plots the wavelet transform of a signal discontinuity for various a scales. Below these the wavelet transform plot is shown. Notice how the ripple in the transform plot becomes more localized as the dilation parameter reduces. This has the effect of making the transform plot 'point' to the location of discontinuities in the signal. Think for yoursclfwhat the result would be of an antisymmetric wavelet (e.g. first derivative of the Gaussian shown in figure 2.1 (a)) passing across the edge shown in figures 2.11 and 2.12. Can you see that T(a, b) plotted against b would have the shape of a single bump with width proportional to the wavelet a scale? Actually if we were using the first derivative of the Gaussian function as a wa velet, the bump would be Gaussian in shape. Figure 2.13 contains another example of a

(,)

,

I\. (b)

small a scales

,

,

a

(log scale)

large a scales _

b Figure 2.13. Pointing 10 an exponential discontinuity. (a) A sudden spike with an exponential tail. (b) The transform plot for the discontinuity. (Note that in the transform plot a logarithmic {j scale is used and small (j scales are located
The inverse

II l avelet

transform

25

signal discontinuity-a sudden spike in the signal half way along its length followed by a smooth exponential decay. As the transform plot has been oriented with the smallest a scales at the top, it 'points' to the signal discontinuity in the signal above.

2.8

The inverse wavelet transform

As with its Fourier counterpart, there is an iI/verse wavelet transform, defined as

x(t)

1 Joo = -c

Joo

g-ooO

da db

T(a,b)1/J",b(t)-,a

(2.17)

This allows the original signal to be recovered from its wavelet transform by integrating over all scales and locations, a and b. Note that for the inverse transform, the original wavelet function is used, rather than its conjugate which is used in the forward transformation. If we limit the integration over a range of a scales rather than all a scales, we can perform a basic filtering of the original signal. Figures 2.14 and 2.15 illustrate this on a segment of signal constructed from two sinusoidal waveforms, one with a period one quarter of the other, plus a local burst of noise. Figures 2.14(a)-(c) show the three component waveforms which are added together to make the composite signal shown in figure 2.14(d). The transform plot of the composite signal (figure 2.14(e)) shows up the two constituent waveforms at scales

(a)

(d)

(b)

(e)

Figure 2.14. A composite signal and its transform plot. (a) Sinusoidal waveform. (b) Sinusoidal waveform with a period one quarter of that in (a). (c) A burst of high frequency noise. (d) Composite signal obtained by combining (a), (b) and (c). (e) The transform plot of the composite signal (d).

26

The camilluotls wavelet transform

(a)

(0)

(b)

(d)

Figure 2.15. Wa\'elet filtering of the composite signal. (a) Small scale (i.e. high frequency) components above the line indicated are removed frolll the transform plot in figure 2. I4(e). (b) Reconstructed signal using the transform plot in (a). (c) Small scale components above the line indicated are removed from the transform plot in figure 2. I4(e). (d) Reconstructed signal using the transform plot in (c).

al and a2' In addition, the high frequency (i.e. small a scale) burst of noise is shown up

as a patch within the top left-hand quadrant of the transform plot. Figure 2.15 shows two reconstructions of the signal where the components in the transform plot, T(a, b), are set to zero above the white line indicated. In effect we are reconstructing the signal uS1l1g

X(I) =

1 [00 Joo -c . g.-ooa

da db

T(a,b),p",b(l) - , a

(2,18)

i.e. over a range of scales a* < a < 00. The lower integral limit, a*, is the cut-ofT scale indicated by the white lines in the figures. The reduction in the high frequency noise components in the reconstructed signal is evident as the cut-off a scale value increases. This simple noise reduction method is known as scale-dependent thresholding. Figures 2, 16(a) and (b) show, in a very simple fashion, the ability of the wavelet transform to perform a manipulation of the signal which is localized in both time and scale. Only those T(a,b) values in the region contained within the box in the transform plot arc set to zero. In this way the burst of noise can be dealt with locally in the signal, thus the denoising does not affect other parts of the signal. Figures 2. 16(c)-(e) show the effect of using a global cut-off at a much higher a scale. Here the transform plot components are restricted to those associated mainly with the low frequency waveform. The inverse transform (figure 2.16(d)) shows a sinusoidallike waveform. Figure 2.16(e) plots the reconstructed signal of figure 2.16(d) at a greater vertical scale and compares it with the original waveform component. We can see that a reasonably good match is obtained. However, as the spectral information in the transform components is smeared across scales, perfect reconstruction of the individual sinusoidal components is not achievable (it would be for the Fourier

The inverse II l alleler transform

(,)

27

(oj

(b)

(0)

Figure 2.16. Further wal'elet filtering. (a) Small scale (i.e. high frequency) components are removed from the transform plOI in figure 2.14(e) in the location within the box shown. (b) Reconstructed signal using the transform plol in (
transform for the specific case of a signal composed of sinusoidal components). The denoising strategies shown in figures 2.15 and 2.16 are very simple in nature and are shown here as an illustration of the inverse transform. A better way to separate pertinent signal features from unwanted noise, or other larger scale artefacts, using the continuous wavelet transform is by using a wavelet transform modI/Ius maxima method. Figure 2.17 shows a composite signal together with its transform plot and corresponding modulus maxima lines. The modulus maxima lines are the loci of the local maxima and minima of the transform plot, with respect to b, traced over wavelet scales. Various signal features are identified within the modulus maxima plot. Modulus maxima plots allow the salient information within the transfonn plot to be expressed in a much more compact form. Following maxima lines down from large to small a scales allows the high frequency information corresponding to large features within the signal to be differentiated from high frequency noise components. This lends itself to novel methods of filtering out noise from coherent signal features. We will come across examples of the use of modulus maxima filtering methods as we proceed through the rest of the book: in the context of the analysis of turbulent fluid measurements in

28

The camilluotls wavelet transform

I~

~

~

~

~,

V

(0)

irregular maxima of __ noise

I

'... ., ~

irregular -- - maxima of noise extending to larger scaJes

,

/. .:> \

\

.~'" . regu Iar max.lma ;.' of sinusoid

.... \ /

/ (c)

discontinuity

end of sinusoid

. edge effect

Figure 2.17. Modulus maxima of a composite signal. (a) A composite signal (bottom right) constructed from the noise, curtailed sinusoid and exponential decay signals shown. (b) Wavelet transform plot of the composite signal (Mexican hat wavelet). (c) Modulus maxima plot corresponding to the transform plol in (b).

chapter 4; the filtering of non-destructive testing data in chapter 5, section 5.6 (see figures 5.26-5.28); the filtering of ECG signals, Doppler ultrasound traces, DNA sequences and medical images in chapter 6 (see figures 6.20 and 6.21) and the analysis of multi fractal signals in chapter 7.

2.9

The signal energy: wavelet-based energy and power spectra

The total energy contained in a signal, x(t), is defined as its integrated squared magnitude

E

=

roo IX(I)I'

dl

= Ilx(I)II'

(2.19)

The signal energy: wavelet-based energy and power spectra

29

For this equation to be useful the signal must contain finite energy. We have already come across this expression in equation (2.7), where we found the energy in the Mexican hat function (i.e. substitute x(t) for 7jJ(t)). The relative contribution of the signal energy contained at a specific a scale and b location is given by the twodimensional wavelet energy density function: E(a,b)

=

IT(a,b)1 2

(2.20)

A plot of E(a, b) is known as a scalogram (analogous to the spectrogram, the energy density surface of the short time Fourier transform-see section 2.12). In practice, all functions which differ from T(a, b) 2 by only a constant multiplicative factor are also called scalograms, e.g. IT(a,b)1 2 /Cg , IT(a,b)1 2 /Cg !;" etc. The scalogram can be integrated across a and b to recover the total energy in the signal using the admissibility constant, Cg , as follows: 1

E = _1 Cg

1

Joo Joo °

1

-00

T(a, b)

2 1

d~ db

(2.21)

a

Figures 2.18(a)-(c) show an experimental signal, x(t), with associated wavelet transform plot, T(a, b), and scalogram, E(a, b). A Mexican hat wavelet was used in the signal transformation. The scalogram (figure 2.18(c)) is very similar in form to the wavelet transform plot. This is to be expected when using real wavelets as the scalogram is simply the squared magnitude of the wavelet transform values. For complex wavelets (see later) we can view the modulus, phase, real and complex parts separately. The scalogram surface highlights the location and scale of dominant energetic features within the signal. The relative contribution to the total energy contained within the signal at a specific a scale is given by the scale dependent energy distribution: E(a) =_1 Cg

foo •

IT(a,b)1 2 db

(2.22)

-00

Peaks in E(a) highlight the dominant energetic scales within the signal. Figure 2.18(d) plots E(a) against a for the signal segment in figure 2.18(a). The plot shows that two dominant scales exist within the signal which are linked to the dominant oscillatory regime of the original experimental signal. We may convert the scale dependent wavelet energy spectrum of the signal, E(a), to a frequency dependent wavelet energy spectrum in order to compare directly with the Fourier energy spectrum of the signal EF(J'). To do this, we must convert from the wavelet a scale (which can be interpreted as a representative temporal, or spatial, period for physical data) to a characteristic frequency of the wavelet. One of the most commonly used characteristic frequencies used in practice is the passband centre of the wavelet's power spectrum. We will use this here, but note that another representative frequency of the mother wavelet such as either the spectral peak frequency, !~, or the central frequency, fo, could be chosen and would be equally valid in the following discussion. We saw in section 2.5, figure 2.4, that the spectral components are inversely proportional to the dilation, i.e. f ex 1/ a, and in section 2.4 we defined the passband centre frequency of the mother wavelet (i.e. a = 1) as!;,. Hence, using this passband frequency, the characteristic frequency

Ewen

30

The comiflliOUS wavelet transform

0 3~ 0.2

£(0)

0.1

o o

2

4

6

8

o

(a)

0.5

1.0

(d)

o (b)

1.5

2.0

o

2

(e)

4

6

8

10

f

om <0

«) Figure 2.18. Wavelet energy density and power spectra. (a) Experimental signal X(I): a velocity trace taken within a vortex shedding regime in a fluid. (b) Transform plot T(a, b) using a Mexican hat wavelet. (Negative values are shown dark grey to black. Positive values are shown light grey to white.) (c) The wavelet scalogram 11'((1, h)1 2 . (Note all values are positive as the square of the modulus is plotted. Hence all contours enclose peaks.) (d) Wavelet energy distribution £(a). (e) Wavelet power spectral density PwU). The horizontal axis is related to that in (d) throughf = 0.251/(1 (Mexican hat wavelet). (r) Power spectral densities (logarithmic plot). Fourier spectrum PF(J), continuous line. Wavelet spectrum Pw(f), circles. (Note that times are in seconds, frequencies are in Hertz and the scale (I = I corresponds to I second.)

associated with a wavelet of arbitrary a scale is given by

jJ

a

(2.23)

where the passband centre of the mother wavelet,fn becomes a scaling constant and! is the representative or characteristic frequency for the wavelet at scale a. We saw in section 2.4 that.fc is equal to .j5/2/27r or 0.251 for the Mexican hat mother wavelet.

The signal energy: wavelet-based energy and power spectra

31

Hence, for this specific wavelet we havef = 0.251ja and this is why the peaks in the transform plot of the sinusoid in figure 2.7 occurred at around 0.25 of its period p (= l/f). Using equation (2.23), we can now associate the scale dependent energy, E(a), to the passband frequency of our wavelet. We can also see from equations (2.21) and (2.22) that the total energy in the signal is given by

Jo E(a) da CXJ

E

=

(2.24)

a2

We can rewrite this equation in terms of passband frequency by making the change of variable f = f j a. The relationship between the derivatives is daja2 = -dfIfe and, after modifying then swapping the integral limits to get rid of the negative sign, we get E =

foo

(2.25)

EwCI') df

,0

where we define EwCI') = E(a)jj~ for f = j~ja, and the subscript W corresponds to 'wavelet' to differentiate it from its Fourier counterpart. A plot of the wavelet energy, Ew(f) against f (the wavelet energy spectrum) has an area underneath it equal to the total signal energy and may be compared directly with the Fourier energy spectrum EF(f) of the signal. (Remember that EF(f) is defined as the squared magnitude of the Fourier transform of the signal. We have already come across it in equation (2.5) where the energy spectrum of the Mexican hat wavelet function was given.) From equation (2.22), we see that the total energy in the signal is given by

=~. J Joo0 CgJ e CXJ

E

2

(2.26)

IT(f, b)1 df db

-00

where we define T(f, b) = T(a,b) forf =fcla. We can see also that the energy density surface in the time-frequency plane, defined by E(f, b) = (IT(f,b)12)j(CgfJ, contains a volume equal to the total energy of the signal, i.e. E

=

.f~oo .f~ E(f, b) df db

(2.27)

This energy density surface can be compared directly with the energy density surface of the short time Fourier transform (the spectrogram). Note that the timescale representations of the scalogram, E(a, b), and scale dependent energy distribution, E(a), do not enclose, respectively, volumes and areas proportional to the energy of the signal, whereas their time-frequency counterparts, E(f, b) and Ew(f), do. In fact, the way we have defined E(f, b) and Ew(1') above means that they enclose a volume and, respectively, an area exactly equal to the energy of the signal. However, the peaks in E (a, b) and E (a) do correspond to the most energetic parts of the signal as do the peaks in E(f, b) and E(f). We can, therefore, use both the scalogram and the scale dependent energy distribution to determine the energy distribution relative to wavelet scale. Scalograms are normally plotted with a logarithmic a scale axis. As f = f j a and hence log(f) = 10g(fJ - log( a), the plot of T (f, b) 2 using a logarithmic frequency scale is simply a shifted, inverted plot of IT(a,b)1 2 using a logarithmic a scale. For example, figure 2.18(c) containing T(a, b) 2 with logarithmically 1

1

1

1

32

The continuous wavelet transform

decreasing a scales towards the top of the plot can also be interpreted as a plot of T(f, b) 2 with logarithmically increasing frequencies towards the top of the plot. In the literature both representations are commonplace. If the signal in figure 2.18(a) were infinitely long, we can see that its energy would be infinitely large. However, in practice, experimental signals (see chapters 4 to 7) are of finite length-usually long enough for the pertinent statistics of the signal to settle down sufficiently for analysis. Hence, in practice, power spectra are more often used to characterize experimental signals of finite length. The power spectrum is simply the energy spectrum divided by the time period of the signal under investigation. Hence, the area under the power spectrum gives the average energy per unit time (i.e. the power) of the signal. For example, for a signal of length T, the Fourier and wavelet power spectra are, respectively, 1

1

(2.28) Pw(f)

=!T Ewe!) = ~ JT T.JcCg 0

2

IT(f,b)1 db

(2.29)

Figure 2.l8(e) plots the wavelet power spectrum, PwU'), for the experimental signal shown in figure 2.18(a). The wavelet power spectral density plot contains the same two peaks as those of the scale energy distribution plot of figure 2.l8(d), but in reverse order as the horizontal frequency axis is the rescaled inverse of the scale axis. The area underneath the P w (f) plot is equal to the power of the signal. Figure 2.l8(f) again contains the wavelet power spectrum; this time logarithmic axes are used and the corresponding Fourier power spectrum is also drawn for comparison. Such logarithmic power spectral plots are commonly used in practice (e.g. see the fluid turbulence spectra of chapter 4) where, for example, some form of power-law scaling is expected or when the pertinent spectral components span quite different orders of magnitude. Due to the frequency distribution within each wavelet, the resulting wavelet power spectrum is smeared compared with the Fourier spectrum. However, the wavelet spectrum is more than simply a smeared version of the Fourier spectrum as the shape of the wavelet itself is an important parameter in the analysis of the signal. Some wavelets will correlate better with specific signal features than others, so accentuating these features in the resulting spectra. Note also that we are using the passband centre of the wavelet as its representative frequency in our discussion; if we had used another characteristic frequency of the wavelet then this would affect the resulting wavelet power spectrum of the signal (and energy density plots), either squashing it or stretching it while retaining the same overall shape and, of course, the same power (respectively, energy). Finally, it is worth noting that the wavelet variance, defined for the continuous wavelet transform as ci(a)

=!T JT0

2

]T(a,b)1 db

(2.30)

is often used in practice to determine dominant scales in the signal. Again we assume that T is of sufficient length to gain a reasonable estimate of (i (a). We can see that this expression is very similar to both the scale dependent energy distribution of

The wavelet transform in terms of the Fourier transform

33

equation (2.22) and the power spectral density function of equation (2.29), differing from both equations only by constant multiplicative factors. We will come across many examples of the use of wavelet and Fourier spectra (and wavelet variance) as we proceed through the application chapters of this book.

2.10

The wavelet transform in terms of the Fourier transform

As we saw in equation (2.15), the wavelet transform is the convolution of the signal with the wavelet function. Hence we can employ the convolution theorem to express the wavelet transform in terms of products of the Fourier transforms of the signal, x(f), and wavelet, (/Ja,b(f), as follows: T(a, b)

=

J~oo x(f){/J~,b(f) qf

(2.31)

where we note that the conjugate of the wavelet function is used. The Fourier transform of the dilated and translated wavelet is (/Ja,b(f) Making the substitution t'

=

J~oo ~ 1fi( t ~ b )e-i(27ff)t dt

(t - b)/a (hence dt

=

(/Ja,b(f)

=

~ J:oo

=

(2.32a)

adt') we obtain

1fi(t')e- i(27ff)(at'+b)adt'

(2.32b)

Separating out the constant part of the exponential function and dropping the prime from t' we get (2.32c) The integral expression in the above equation is simply the Fourier transform of the wavelet at rescaled frequency af Hence we can write equation (2.32b) as (2.33) The Fourier transform of the wavelet function conjugate is then simply (/J~,b(f) = va{/J* (af) ei (27ff)b

(2.34)

Hence equation (2.30) can be written in expanded form as T(a, b)

=

va

J~CXJ x(f)'~*(af) ei (27ff)b df

(2.35)

which we can see has the form of an inverse Fourier transform. This is a particularly useful result when using discretized approximations of the continuous wavelet transform in practice with large signal data sets, as the fast Fourier transform (FFT) algorithm may be employed to facilitate rapid calculation of the wavelet transform and its inverse. In addition, the Fourier transform of the wavelet function, '~a,b(f), is usually known in analytic form and hence need not be computed using an FFT. Only an FFT

34

The continuous wavelet transform

of the original signal, x(f), is required. Then, to get T(a,b), we take the inverse FFT of the product of the signal Fourier transform and the wavelet Fourier transform for The equivalence between the time each required a scale and multiply the result by convolution and Fourier integrals for determining T (a, b) is depicted in figure 2.19.

va.

'i'a,b(t) x(t)

x(t)

x (t)1jJ a,b(t)

/

I

t

Near zero values of the wavelet here lead to near zero contributions to the transform in regions far fromb

Features in the signal are highlighted by the wavelet in the vincinity of b

(a)

x(f}lr(af)

\lJ (af) x(f)

¥af)

x(f)

/ Frequencies in this region are accentuated by the filtering process

Frequencies in this region are diminished by the filtering process

(b)

Figure 2.19. Schematic representation of the wavelet transform in its time and frequency representations. (a) The convolution of the wavelet with the signal. (b) The convolution in (a) expressed in the Fourier domain involves a product of the signal Fourier transform and the wavelet Fourier transform.

Complex wavelets: the M orlet wavelet

35

The bandpass nature of the wavelet is evident from figure 2.19(b). The inverse transform (equation (2.17» can similarly be written in terms of an inverse Fourier function.

2.11

Complex wavelets: the Morlet wavelet

So far we have used the Mexican hat wavelet to illustrate many of the features of the wavelet transform. In this section we consider wavelets which have both real and imaginary parts. Complex or analytic wavelets have Fourier transforms which are zero for negative frequencies (requirement 3 in section 2.3). By using such complex wavelets we can separate the phase and amplitude components within the signal. Actually, we can easily make a complex version of the Mexican hat wavelet by taking its Fourier transform, setting the negative frequency components in the Fourier domain to zero and then performing an inverse Fourier transform to get the complex wavelet. However, in this section, we focus on the most commonly used complex wavelet, the Morlet wavelet, which is defined as (2.36) where.fa is the central frequency of the mother wavelet. The second term in the brackets is known as the correction term, as it corrects for the non-zero mean of the complex sinusoid of the first term. In practice it becomes negligible for values of .fa » 0 and can be ignored, in which case, the Morlet wavelet can be written in a simpler form as (2.37)

normalization

complex

Gaussian

factor

sinusoid

bell curve

This wavelet is simply a complex wave within a Gaussian envelope. We can see this by looking at equation (2.37) in conjunction with figure 2.20(a). The complex sinusoidal waveform is 2contained in the term ei2Trfot (= cos(2Jr.fat) + i sin(2-n:-fot)). The Gaussian envelope e- t /2 has unit standard deviation and 'confines' the complex sinusoidal waveform. Figure 2.20(a) shows the real and imaginary parts of the Morlet wavelet together with its confining Gaussian envelope. We can see that the real and imaginary sinusoids differ in phase by a quarter period. The Jrl/4 term is a normalization factor which ensures that the wavelet has unit energy. Note that the function given by equation (2.37) is not really a wavelet as it has a non-zero mean, i.e. the zero frequency term of its corresponding energy spectrum is non-zero and hence it is inadmissible according to equation (2.4). However, it can be used in practice with.fa » 0 with minimal error. The Fourier transform of the Morlet wavelet is given by (2.38)

36

The continuous wavelet transform

real component

imaginary component

energy density

Ig(f)1 2 4 .-------,---,.....,...--.,..-----,

·_---iGaussian envelope

-1

L.------'-_----L_---'--_--L.-_...L..------'

-6

-4

-2

0

2

4

2

OL-----J.L---+---l.-~-....I....-----J

o

6

0.5

(b)

(a)

i 1.0

!

1.5 2.0 frequency

fa l.-----,-----r---,---,...---..,..----,

-1

L.------'-_----L_---'--_--L.-_-'-------'

-6 -4

-2

0

2

4

-1 '-----------'-----'------'------'------'---------'

-6 -4

6

(c)

f

1,-------,-----,-----,-----,--,--------,

-2

0

2

4

6

(d)

o -1

L.-----'-_----'-_---'-_--'---_-'--------'

-6 -4

(e)

-2

0

2

4

-1

6

L.------'-_----'--_---'--_--'---_..L.------'

-6 -4

-2

0

2

4

6

(f)

Figure 2.20. Morlet wavelets. (a) The Morlet wavelet a = 1 andjo = 0.894. (b) Energy spectrum of the MOrlet wavelet. (c)jo = 0.318 (a = 1). (d)jo = 1.909 (a = 1). (e) a = 0.5 (fo = 0.894). (f) a = 2 (fo = 0.894).

which has the form of a Gaussian function displaced along the frequency axis by fo. Note that the central frequency of the Gaussian spectrum is generally chosen to be the characteristic frequency of the analytic Morlet wavelet rather than the passband frequency, which we used previously for the Mexican hat wavelet. The energy spectrum (the squared magnitude of the Fourier transform) is given by (2.39) The integral of this gives the energy of the Morlet wavelet, which is equal to unity according to our definition given by equation (2.37). The energy spectrum of the Morlet wavelet is shown in figure 2.20(b). The central frequency fa is the frequency of the complex sinusoid and its value determines the number of 'effective' or 'significant' sinusoidal waveforms contained within the envelope, i.e. those which are not very close to zero amplitude. The value of 0.849 (=vlj(2In2)) is often used in practice. It produces a decay where the magnitude of the two peaks in the real waveform adjacent to central peak are half its amplitude. (In the literature the

Complex wavelets: the Morlet wavelet

37

angular frequency Wo = 2rr/o of the wavelet is often quoted, hencejo = 0.849 becomes Wo = 5.336 = "J2/ln 2. Values of Wo equal to 5 and 6 (and in between) are also commonly used in practice. For values of Wo less than 5 (fa < 0.8) the full or 'complete' Morlet wavelet of equation (2.36) should be used as the simplified wavelet function of equation (2.37) contains a significant non-zero mean.) Figures 2.20(c) and (d) show Morlet wavelets with], equal to 0.318 and 1.909 respectively. The figure shows that the number of effective oscillations contained within the Gaussian window increases with/a. To construct the dilated and translated Morlet wavelet we replace t with (t - b)ja as we did for the Mexican hat in equation (2.11). Its form is then

1/J( t -

a

b) =

_1_ e">if.I(,-hl/.,] e-j[(,-b)/.,I'

(2.40)

rr 1/ 4

Figures 2.20(e) and (f) show Morlet wavelets with a scales of 0.5 and 2 respectively. We can see directly from equation (2.40) that the standard deviation of the Gaussian

(al

(h)

(e)

(0)

(f)

Figure 2.21. Morlel wal'elet analysis of a Iwo-component sinusoidal waveform. (a) Original signal. (b) The real part of the wavelet transform Re(T(a, b)) (positive maxima in white, negative minima in black). (c) The real part of the wavelet transform Re(T(a,b)) (same plot as (b) but with contours added and a coarser shading used). (d) The imaginary part of the wavelet transform Im(T(a,b)). (e) The phase of the wavelet transform. ¢(a,b) = tan- l {[lm(T(a,b))IJ[Re(T(a,b))} (-1T phase in black, 1T phase in white, zero phase in mid grey tone). (f) The modulus of the wavelet transform. T(a,b) = J[Re(T(a,b))]2 + [Im(T(a,b))j2 (maximum values in white. zero values in black).

38

The comiflliOUS wavelet transform

envelope on the time axis is, in fact, simply equal to a. Figures 2.20(e) and (f) show the stretching and squeezing of the wavelet with a scale. Figure 2.21 illustrates the use of the Morlet wavelet (Jo ~ 0.849) in analysing a two-component sinusoidal waveform. Performing the wavelet transform on the signal using the complex Morlet wavelet results in complex transform values T(a,b) which we may view in a number of ways. The real part ofT(a,b) is shown in figure 2.2I(b). As expected, the two wavelet components are picked up and displayed as ripples in the transform plot at two distinct scales. Figure 2.21 (c) also contains the real part of T(a,b). This time a coarser shading is used and contour lines are added to enhance visualization of the periodic structure of the transform plot. Figure 2.2I(d) contains the imaginary part of T(a,b). Notice the similarity between this transform plot and that of figure 2.2I(c). In fact, the imaginary plot is a phase-shifted version of the real plot. The reason for this is apparent if we consider the form of Morlet wavelet as shown in figure 2.20(a). In the figure we see that, for the Morlet wavelet defined above, the imaginary part of the wavelet comprises a sinusoidal waveform within a Gaussian envelope which leads the real part by one quarter of a cycle. In other words it is phase shifted by one quarter of a cycle from the real part. However, as we use the complex conjugate in the transform, the imaginary part is

(0)

Figure 2.22. Phase shifted sinusoid. (a) Original signal. (b) Phase. (e) Modulus.

Complex wavelets: the Morlet wavelet

39

inverted, leading to an imaginary waveform which lags behinds by one quarter of a cycle from the real part. Hence, the imaginary part of the Morlet best matches one quarter of a cycle later than the real part. Therefore, the maxima of the transform plot for the imaginary part are phase shifted forward by one quarter of a cycle. This we see when comparing figures 2.21(c) and (d). As T(a,b) is a complex number, i.e. T(a, b) = Re(T(a,b)) + Im(T(A,b)), we can write T(a,b) in terms of its phase 1J(a,b) and modulus IT(a,b)l. The phase of the Morlet transform plot is shown in figure 2.21(e). The phase varies cyclically between -7i and 7i over the duration of the component waveforms. Zero phase corresponds to the real part of the Morlet wavelet centred at the maximum amplitude of the sinusoidal waveforms. Hence, zero phase corresponds to the peaks on the real transform plot of figure 2.21(c). A phase of 7i (and -7i) corresponds to the minima of the real transform plot. Figure 2.21(f) contains the modulus of the transform plot where we can see that the periodic sinusoidal waveforms manifest themselves as continuous bands across the modulus plot. Figures 2.22 to 2.28 show various simple signals together with their associated Morlet wavelet transform plots. Figure 2.22 shows a sinusoidal waveform which is

,..

-3 -2

(a)

-t

0

2

3

(h)

(d)

(d)

Figure 2.23. Morlel decomposilion of signal wilh abrupl change in periodicily. (a) Original signal. (b) Morlet wavelet 10 = 0.849 (period = lifo = 1.117) at scale (I = I. (c) Phase or the transrorm. (d) Real part or the transrorm. (e) Modulus or the transrorm. (r) Imaginary part or the transrorm.

40

The camilluotls wavelet transform

shifted by half a cycle in the middle. Hence, the resulting signal contains both a discontinuity and a phase shift. Both transform plots indicated the location of the discontinuity. The phase plot shows up the location of the phase shift in the waveform. Figure 2.23 shows the Morlet wavelet decomposition of a signal which contains a change in periodicity midway along its length. The change in periodicity is clearly shown in all four transform plots. Figure 2.24 shows the effect of interrogating the same signal as figure 2.23 using different Morlet wavelets (i.e. different value of fo). On the left-hand side of the figure, the signal is decomposed using a Morlet wavelet with fo = 0.318. We see that the real part of this wavelet is more like a Mexican hat in form. (As mentioned above, we should not really use such a low frequency fo for a Morlet wavelet in practice, as its power spectrum is significantly non-zero at the origin. rnstead, its complete form, given by equation (2.36), should be employed. An example of the use of the complete Morlet wavelet can be found in section 5.6 and figure 5.29 of chapter 5.) Comparing the transform plots with those of the previous figure we see some differences. The phase plot for thefo = 0.318 wavelet is much smoother at the transition point. This is because this wavelet has essen-

<.

,

., "

"

,'

,,

,.

(a)

(h)

(c)

Figure 2.24.. The effect off0 on the Morlct decomposition of signal with abrupt change in periodicity. (a) MorICI wavelets ffj = 0.318 (left) and Wfj = 1.909 (right). (b) Phase plols corresponding 10 10 = 0.318 (left) and Wo = 1.909 (right). (c) Modulus plots corresponding 10 fo = 0.318 (left) and Wo = 1.909 (right).

Complex wavelets: the Morlet wavelet

41

-

-

-

Figure 2.25. Comparison of Morlet power spectra for three values of 10' (a) Power spectrum for 10 = 1.909 (arbitrary axis units). (b) Power spectrum for 10 = 0.849 (arbitrary axis units). (c) Power spectrum for 10 = 0.318 (arbitrary axis units). (Note that the Fourier spikes are shown with a finite height and width as they were calculated numerically using a discrete Fourier algorithm. In theory they are Dirac delta functions.)

tially only a single positive bump which matches the signal smoothly over the transi· tion region, whereas the/a = 0.849 wavelet has five distinct peaks within the window which correspond to the five ridges converging at small scales in the phase plot in figure 2.23(c). The right-hand side of figure 2.24 shows the Morlet wavelet decomposition of the signal using/a = 1.909. The power spectra for the three Morlet wavelet decompositions given in figures 2.23 and 2.24 are given in figure 2.25. We can see the greater degree of frequency localization as fo increases (and hence the number of cycles within the Gaussian envelope increases). This is also evident from the transform plots in figures 2.23 and 2.24 where we can see that a narrower band in the modulus transform plot is associated with higher values of 10' (Remember from section 2.9 that we can interpret the logarithmic vertical axis of the transform plot in terms of characteristic wavelet frequency rather than scale.) However, this greater degree of frequency localization with increasing fo is associated with much poorer

42

The camilluotls wavelet transform

temporal localization, as can be seen when comparing both the modulus and phase plots in figures 2.23 and 2.24. Figure 2.26 shows two signals containing repeating features-bumps and spikes. The transform plots are plotted below the signals. We can see that even for these

(0)

(d)

(b)

(e)

(0) j ........

(

\

(g)

Figure 2.26. Morlel decomposition of bump sigmtls. (a) Single bump- positive part of sinusoid. (b) Wavelet phase of signal (a). (c) Wavelet modulus of signal (a). (d) Bumps and spikes. (e) Wavelet phase of signal (d). (f) Wavelet modulus of signal (d). (g) The Morlet wavelet at a periodicity correlating best with the bump signal. This value of Morlet wavelet results in the dominant band of (c). (h) The Morlet wavelet with half the period oftha! in (g). This value of Morlet wavelet results in the second band in (c) at an a scale half that of (g).

Complex wavelets: the Morlet wavelet

43

relatively simple repeating signals the phase plots are already exhibiting a significant degree of complexity. The modulus plot of the bump and spike signal points to the discontinuous spikes in the signal (compare with the ECG signal shown in chapter 6, figure 6.12). Figure 2.26(g) shows a schematic of the wavelet at approximately the same periodicity as the bump signal of figure 2.26(a). At this scale a maximum correlation is produced which shows up as the dominant (white) band in the modulus plot of figure 2.26(c). The next most dominant (grey) band in the modulus plot is generated when the Morlet wavelet correlates with the signal as shown in figure 2.26(h). Figure 2.27 contains the same chirp signal as that shown earlier in

(,)

(b)

•

(d)

-Ridge

a R ~-----:::;:;"",,._.~

.----

(e)

Figure 2.27. Segment of a chirp signal with associated transform plots-Morlet wal'elel. (a) Chirp signal segment. (b) Real part of Morlet wavelet transform. (c) Phase. (d) Modulus. (e) A schematic of the ridge found from the maxima of the rescaled scalogram IT(a,b)1 2 /a. The instantaneous frequency at time b R can be found from aR' We can see the relation between maxima in the rescaled scalogram and instantaneous frequency by substituting a complex sinusoid as the signal X(I) in the wavelet transform integral given by equation (2.13), and using a Morlet wavelet. Then, using a change of variable t' = (f - b)/a, it can be shown that maxima in the rescaled scalogram correspond to the instantaneous frequencies through their associated scales.

44

The camilluotls wavelet transform

figure 2.9. The Morlet wavelet withJO = 0.849 was used to transform the signal. The real part of the transformed signal is plotted in figure 2.27(b) and has similarities with the Mexican hat transform plot in figure 2.9. The discontinuities at the beginning and end of the chirp segment are picked up well in the phase plot of figure 2.27(c). These are located using arrows at the top of the phase plot. The continuous increase in instantaneous frequency associated with the chirp is highlighted in the modulus plot of figure 2.27(d). The instantaneous frequency associated with a signal can be found from its wavelet t,.all~lol'lI1 ridges. These are the maxima found in the rescaled wavelet transform scalogram, IT(a 1 b)1 2 ja associated with the instantaneous frequency of the signal. The ridge associated with the chirp signal is shown schematically in figure 2.27(e), where the instantaneous scale aR at time b R can be used to find the instantaneous frequency fR (=fojaR)' The instantaneous amplitude and phase can also be found from the ridge. Further, if we plot the rescaled scalogram in terms ofa characteristic wavelet freq uency 1T(j, b) 1'1 a where.r ~ fal a (refer back to equations (2.26) and (2.27)), then the instantaneous frequency can be read directly off this plot. We do not go into the derivation of ridges here, but rather refer the reader to section 2.16 at the end of the chapter. A final test signal for the Morlct wavelet is shown in figure 2.28. It is composed of a dominant sinusoid of unit amplitude which has three features superimposed on it.

Ca)

(b)

o

Cd)

(e)

Figure 2.28. Wa\'elet decomposition of a sinusoidal wa\'eform with added small scale features. (a) Signal. (b) Phase. (c) Real part. (d) Modulus. (e) Unfilled contour plot or modulus.

The wavelet transform, short time Fourier transform and Heisenberg boxes

45

At the second peak a small spike has been added. The fourth oscillation has noise added to it. A high frequency oscillatory component has been added to the sixth oscillation of the dominant sinusoid. This has a frequency ten times that of the large waveform. The location of the spike can be easily identified in the phase plot, as are the regions of random noise and the high frequency component. The real transform plot locates the dominant peaks and troughs of the signal together with superimposed signal features. The modulus is plotted as a filled plot in figure 2.28(d) and as a contour plot in figure 2.28(e). The dominant waveform is obvious in both figures. The high frequency oscillatory feature in the signal is made much more obvious in the contour plot. In general, as the signal becomes more complex in nature the phase and modulus information quickly becomes more difficult to interpret.

2.12

The wavelet transform, short time Fourier transform and Heisenberg boxes

In this section we take a brief look at the time-frequency characteristics of the wavelet

transform and compare it with the short time Fourier transform (STFT). We will consider the specific cases of the Morlet wavelet transform and the Gabor STFT, both of which employ a Gaussian window. As we will see below, the Morlet wavelet has a form very similar to the analysing function used for the short time Fourier transform within a Gaussian window. The important difference is that, for the Morlet wavelet transform, we scale the window and enclosed sinusoid together, whereas for the STFT we keep the window length constant and scale only the enclosed sinusoid. The wavelet can therefore localize itself in time for short duration, i.e. high frequency, fluctuations. There is, however, an associated spreading of the frequency distribution associated with wavelets of short duration. Conversely, there is a spreading in temporal resolution at low frequencies. This is illustrated in figure 2.29(a). The middle of the figure contains a schematic of a Morlet wavelet (real part only) shown at three a scales. The energy densities of the wavelets are plotted below in both the time and frequency domains, i.e. respectively [?);a,h(t) 2 and I~a,b(f) 2 . We can see from the figure that, as the wavelet contracts in time, it becomes composed of higher frequencies with a wider spread. The spread of I?);a,b(t) 2 and I~a,b(f) 2 can be quantified using at and ar respectively-the standard deviations around their respective means. We can represent the spread of the wavelets in the time-frequency plane by drawing boxes of side lengths 2at by 2ar. These are shown at the top of figure 2.29(a). These boxes are known as Heisenberg boxes after the Heisenberg uncertainty principle, which tells us the minimum area that these boxes can have. Specifically, the product atar must be greater than or equal to 1/47T, thus the area of the Heisenberg box is 1/7T. In fact, for the Gaussian windowed functions used in the Morlet wavelet transform and STFT considered here, atar is exactly equal to 1147T as the Gaussian distribution is the optimal window shape. (You can verify this for the Morlet wavelet given in the time domain by equation (2.37) and frequency domain by equation (2.38); integrating their squared modulus leads to at = I/V2 and ar = V2/47T.) The Heisenberg uncertainty principle actually addresses the problem of the simultaneous resolution 1

1

1

1

46

The continuous wavelet transform (a)

Heisenberg box in the time-frequency plane

f

J;.+

····················································1+..

Of

i

f,----------------lEB--

'\j! a b(t) (real part)

\

decrease in wavelet width associated with increase in spread of spectral components

/

liPa,b(/)1 2

f Figure 2.29. Heisenberg boxes in the time-frequency plane. (a) Heisenberg boxes in the timefrequency plane for a wavelet at various scales. Do not confuse Of with./;, the passband frequency, given earlier in section 2.4: J, is the standard deviation of the spectrum around the origin; CJt is the standard deviation of the spectrum around the mean spectral components II ,12 and 13 shown in the figure.

The wavelet transform, short time Fourier transform and Heisenberg boxes (b)

47

f

lighfo

-_m-mll+ Of

t+_+1

modiumfo ---

------8

lowfo

lowfo

tjJ a,b(t) (real part)

mediumfo

,

fr

...,'lRl.,.II\I'\O_--hi-.g-h--O - - - - - - - + - .

Figure 2.29 (continued). (b) Heisenberg boxes in the time-frequency plane for a Morlet mother wavelet with three different central frequencies set to a low, medium and high value. The confining Gaussian windows are all of the same dimensions. Notice that altering the central frequency of the mother wavelet simply shifts the associated 'mother' Heisenberg box up and down the timefrequency plane without altering the box dimensions. This mother Heisenberg box then defines the relative shapes of all the others in the time-frequency plane associated with each wavelet, i.e. the pattern shown in (a) is simply shifted up or down the plane.

in time and frequency that can be attained when measuring a signal. To get a good idea of the frequency composition we need to sample a long period of the signal. If instead we pinpoint a small region of the signal to measure it with accuracy, then it becomes very difficult to determine the frequency makeup of the signal in that region. That is, the more accurate the temporal measurement (smaller O"t) the less accurate the spectral measurement (larger O"r) and vice versa. Note that the Morlet central frequency fo sets the location of the Heisenberg box in the time-frequency

48

The continuous wavelet transform (c)

f 1; +..

+

;

+.

Of

h

t

+

L

+ .. ,..+

hf,b(t) (real part)

Figure 2.29 (continued). (c) Heisenberg boxes in the time-frequency plane for the STFT. Other window lengths will produce longer and thinner or shorter and fatter boxes. However, once the window length is fixed, all Heisenberg boxes in the time-frequency plane corresponding to the STFT will have the same dimensions.

plane for the mother wavelet, and hence the relative locations for all dilated wavelets. This is shown in figure 2.29(b). Thus, when comparing Heisenberg boxes centred at the same location in the time-frequency plane, lower values of fo correspond to Heisenberg boxes that are wider in frequency and narrower in time than boxes corresponding to higher ios. Thus Morlet wavelets with lower ios correspond to time-frequency decompositions that are 'more temporal than spectral' than their higher central frequency counterparts. (See the example in chapter 5, figure 5.29.) The Fourier transform of a signal x(t) is defined as

x(f)

=

.f~oo

x(t) e- i(21ff)t dt

(2.41)

The wavelet transform, short time Fourier transform and Heisenberg boxes

49

We can modify the Fourier transform to allow localized features in the signal to be interrogated. This short-time Fourier transform (STFT) employs a window function to localize the complex sinusoid. It is defined as Fer, b) =

J~oo

x(t)h(t - b) e-

i27r/t dt

(2.42)

where h(t - b) is the window function which confines the complex sinusoid e-i2K}t. The STFT is also commonly known as the windowed Fourier transform. There are many shapes of window available, for example Hanning, Hamming, cosine, Kaiser and Gaussian. We will consider the Gaussian windowed STFT, known as the Gabor transform, which has the form

1

h(t) =

VC7IT 1/ 4

1(2/ 2) e-2: t (T

(2.43a)

where (]" is a fixed parameter (the standard deviation) which sets the width of the Gaussian window on the time axis. The combined window plus complex sinusoid is known as a windowed Fourier atom or more generally as a time-frequency atom, denoted (2.43b) Convolving the complex conjugate of this atom with the signal x(t) results in its timefrequency decomposition: the short time Fourier transform. We can see that, when the Gaussian window is employed within the STFT integral of equation (2.42), we obtain F(f, b)

=

J~oo

x(t)h;,b(t) dt

=

Joo

x(t)

-00

1

VC7IT

1 4 /

e-~[(t-b)2/(T2] e- i2K/t dt

(2.44)

This has a very similar form to the Morlet wavelet transform. We can see this by combining equations (2.13) and (2.40) and rearranging the terms as follows: T(a, b)

= =

J~oo x(t)?/J~,b(t) dt

foo . x(t) . -oc

1

y!aIT 1/ 4

e-~[(t-b)2p] e-i2KUO/a)(t-b) dt

(2.45)

The main difference between the Gabor STFT and the Morlet WT is now obvious: the internal frequency, j; is allowed to vary within a Gaussian window of fixed width (given by (]") in the former, whereas the latter employs an internal frequency f (=fo/ a) which is linked to the window width (given by a). There is another less significant difference in that the wavelet's complex sinusoid is centred at b on the time axis whereas the complex sinusoid contained within the Gabor atom is 'centred' at the origin (t = 0). To put it another way, in the STFT the sinusoid remains fixed in relation to the origin and the window slides across it, whereas for the wavelet

The camilluotls wavelet transform

50

transform, the origin of the complex sinusoid is at b and hence the sinusoid and window move together. The atom used for the Gabor STFT is shown in figure 2.29(c) for three internal frequencies within the fixed width window. This atom also has a constant width in the

(.j

time

time

(bj

i

.

i time

time

(oj

Figure 2.30. 51'FT and WT time-frequency plots. (a) Sinusoid containing a spike at the location indicated by the arrow. (b) Morlel wavelet transforms of the signal in (a). The wavelet transform on the left has been generated llsing a Morlel wavelet with a central frequency half that used to produce the transform plot on the right. ee) Gabor STFTs of the signal in (a). The left-hand plol corresponds to a Gaussian window which is half the width of the one used to generate the righthand pial. Note that the horizontal axis is time and the vertical axis is frequency. In addition, high to low energies correspond with white to black in the grey-scale used.

Adaptive transforms: matching pursuits

51

frequency domain. This leads to boxes in the time-frequency plane of equal shape, regardless of the internal frequency. The dimensions of the Heisenberg boxes shown in the time-frequency plane at the top of figure 2.29(c) are determined by the preselected window width (J. Longer and thinner or shorter and fatter Heisenberg boxes can be obtained by changing the window width. However, whatever length of window is used, once fixed, the corresponding Heisenberg boxes associated with the STFT all have exactly the same dimensions in the time-frequency plane. This is true for all window shapes used in the STFT-not just Gaussians. As with the Morlet wavelet transform, the Gaussian windowed complex sinusoids used in the Gabor STFT have the smallest areas of Heisenberg boxes in the time-frequency planethat is, they have optimal time-frequency energy distributions. An example of the difference between the WT and STFT is shown in figure 2.30 which contains a test signal together with corresponding Morlet-based wavelet transforms and Gabor STFTs. The signal comprises a sinusoid plus a single spike located by the arrow in figure 2.30(a). We can see that the sinusoid manifests itself as a dominant horizontal ridge in both the WT and STFT plots. The spike, however, is localized in time at high frequencies by the wavelet transform, whereas it corresponds to the vertical ridge of constant width in the STFT due to the constant width of its Heisenberg boxes. Hence, the wavelet transform can discern individual high frequency features located close to each other in the signal, whereas the STFT smears such high frequency information occurring within its fixed width window (a good example of this effect is shown in figure 6.17 of chapter 6). We can also see from figure 2.30(b) that the Morlet wavelet transform corresponding to the lower central frequency (left-hand plot) produces a sharper resolution in time but correspondingly poorer resolution in frequency than the transform based on the higher central frequency (right-hand plot). Similarly, comparing the plots in figure 2.30(c) we can see that the STFT corresponding to the shorter window width produces a sharper resolution in time but correspondingly poorer resolution in frequency than that generated using the longer window width.

2.13

Adaptive transforms: matching pursuits

From the previous section we saw that the wavelet transform becomes more localized as it interrogates smaller scales, whereas the STFT has a fixed window length for all scales. We will now take a brief look at an adaptive transform called the matching pursuit (MP) which offers an alternative, more flexible way of providing timefrequency information. Although more flexible it does not provide a regular repeatable or even full coverage of the time-frequency plane, but rather one that adapts to each signal and is hence signal dependent. The matching pursuit method involves the decomposition of the signal piece by piece using a dictionary of analysing functions. At each stage in the decomposition an analysing function is chosen which 'best' represents part or all of the remaining signal. After a number of decompositions the original signal, x(t), can be represented to some arbitrary resolution by a series of expansion coefficients, M i , i = 1,2, ... ,n, where n is the number of iterations of the decomposition algorithm. The signal

52

The camilluotls wavelet tralls/ol'm

approximation reconstructed from these expansion coefficients is given by 1/-1

X,,(I)

=L

(2.46)

M,h,(I)

;=0

where h;(t) are the functions used in the decomposition. The essential differences between wavelet transform analysis and matching pursuit analysis are in the way the transform coefficients are selected and the flexibility in the choice of analysing function used in the signal decomposition. The signal is first examined using each of the analysing functions contained within a preselected dictionary of functions, and the one which takes the most energy from the signal is chosen to decompose the signal. The residual signal is constructed and then examined to find the next function from the dictionary which takes the most energy from this new signal. The process is repeated until the residual signal falls below some predetermined cut-off. The procedure is illustrated in figure 2.31. In the figure we can see the original signal with the first analysing function-a time-frequency atom-used to decompose it shown directly below. The bottom plot in the figure contains the first residual signal. The example in figure 2.31 uses a Gabor atom, one of the most commonly employed analysing functions in matching pursuit analysis. We have already come across the Gabor atom used in the STFT. For the matching pursuit method the atom is defined as h"'/(I)

.,

= va ~h( r-a b)e'(2'1)'

(2.47)

where the scale, a, location, b, and frequency,.f; can all be varied independently. Thus it has an increased flexibility over the Gabor STFT atom. We will define the Gaussian

:1(a)Si~nal : y~=: .1 0

20

40

60

80

100

120

140

:. : .1 160

180

2f

:f~)at~m ~ : : : : ..

0 . 11 l~tdUal 20

o

•

-1 0

20

= =_--==__ ::::

4O~80 .J;~~~ I~ I~ j~Q r

40

60

80 100 120 time (samples)

140

"f_ _

160

180

200

Figure 2.31. The matching pursuil method. (a) A damped sinusoidal signal. (b) The first Gabor atom chosen to represent the signal in (a). (c) The residual signal. (Note that this example comes from a paper by Goodwin and Vetterli (1999), which goes on to show the difficulties in using symmetric window functions in representing transient signals.) After Goodwin and Vetlerli (1999) IEEE Transactions 011 Signal Processing, 47(7) 1890-1902. © IEEE 1999.

Adaptive transforms: matching pursuits

53

window as (2.48)

Note that this is different in form from that used for the STFT in equation (2.43). Both have unit energy and are equally valid, however, equation (2.48) is prevalent in the literature for the matching pursuit method, hence we use it here. The timefrequency atom defined by equations (2.47) and (2.48) has a similar form to both those used in the Morlet wavelet transform and the short time Fourier transform of equations (2.44) and (2.45). However, the Gabor atom used in the matching pursuit method is more flexible in that its scale, location and internal frequency may all be varied independently. In order to get a decomposition with real expansion coefficients and real residuals, which is often required in practice, real-only atoms are used of the form h

. .. (t) Uj7 iJi,cPi

=

2 1/ 4

K· I

va; e-

7T

2

[(t-b i)/Ui] cos(27rf:t J

I

+ A•. ) '+'1

(2.49)

where ai and hi are the scale and location factors for the Gaussian envelope,f; and cPi are respectively the frequency and phase of the real sinusoid within the Gaussian envelope and K i is a normalization factor used to maintain unit energy for hUi,bi,fi,cPi (t). The subscript i relates to the specific set of parameters a, h, f and cP used for the ith decomposition to get the Mi' The expansion coefficients M i are determined in turn by examining the signal with the analysing atom and selecting the parameter set (ai, h i ,/;, cPi)' which provides the largest value of IMi l2 where (2.50)

and where we have used hi as a more compact representation of hUi,biJi,cPi' In addition to ai, hi,.I; and cPi, we need to retain K i and, of course, M i at each iteration of the matching pursuit method in order to perform the reconstruction. The matching pursuit method is very flexible and the dictionary of analysing functions can include functions other than the Gabor function hi, such as sinusoids within another shape of window, continuous sinusoids (i.e. a Fourier function), wavelet functions, Dirac delta functions, and so on. All these functions are kept within a preselected dictionary and each are used in turn to determine which gives the maximum value IMi l2 and, hence, takes the most energy from the signal. The matching pursuit method is an iterative method. At the first iteration the signal is decomposed into two orthogonal components x(t) =

XI

(t)

+ R 1x(t)

(2.51 )

which can also be written as (2.52)

where R1x(t) is the residual vector after approximating the original signal xo(t) (now subscripted with 0) in the direction of ho(t). R 1x(t) and ho(t) are orthogonal to each

54

The continuous wavelet transform

other, hence the energy of the signal can be expressed as (2.53) where IIxl1 2 = f Ixl 2 dx. In addition, as ho(t) has unit energy by definition and M o is a constant, then we can write (2.54) The first coefficient M o is found by applying each of the dictionary functions to the signal in turn and choosing the one which maximizes the value of M o, or conversely, the one which minimizes the energy of the residuaIIIR 1x(t)11 2 . This residual signal is then examined in the same way to find the second coefficient M 1 and the new signal residual R 2 x(t), and so on. At any stage in the decomposition, the original signal can be partitioned into two components: the reconstructed part using all the expansion coefficients (given by equation (2.46», and the residual component after n iterations, R l1 x(t). This can be written as 11-1

xo(t) =

L

Mihi(t)

+R

l1

x(t) = x l1 (t)

+R

l1

x(t)

(2.55)

i=O

where x l1 (t) is the signal approximation and R l1 x(t) is the residual signal left after the nth iteration of the matching pursuit algorithm. In addition, the energy at the nth decomposition is given by 11-1

Il x o(t)11

2

=

L

2

IMi l

+ IIR

l1

x(t)W

(2.56)

i=O

In practice the matching pursuit algorithm is terminated either when the residual energy is below a preset cut-off level E, defined by (2.57) or, alternatively, after a predetermined number of iterations, n. Figure 2.32 shows an example of a synthetic signal decomposed using the matching pursuit method selecting from a dictionary containing three basic elements: Gabor atoms, Dirac delta functions (spikes in the time domain) and sinusoidal Fourier functions (spikes in the frequency domain and continuous sinusoids in the time domain). We can see from the figure that the MP algorithm leads to an intermittent, patchy covering of the time-frequency plane as only those coefficients obtained when the time-frequency atom is matched to the signal at locations of maximum energy removal are retained. This contrasts both with the wavelet transform and short time Fourier transform where the time-frequency plane is covered evenly. The time-frequency plot used to represent the energy distribution of the matching pursuit decomposition is based on a Wigner distribution. We do not go into the details of this representation here, but instead the reader is referred elsewhere at the end of this chapter. Modified versions of the basic matching pursuit algorithm are now available and, in later chapters in this book, some examples are

WavelelS in Iwo or more dimensions

55

-. IV ~ ... -

'.'f;

--

•

time

'~'v"'l-'_-'--

1lI U

I

Figure 2.32. The time-frequency decomposition of a test signal using the matching pursuit method. The figure contains the time-frequency plane with the energy distributions associated with the MP decomposition of the signal IV. Leiters mark signal components and corresponding atoms or groups of atoms: A, B=single transients; C. D=sinusoids modulated by a Gaussian function; E=single sharp transient; F=sinllsoid. Signal IV is a sum of I, II and III. The MP analysis time~freqllency plot for signal IV is shown above the signals. The darkness of the plot elements is proportional to the logarithm of the energy distribution in the time-frequency plane according to a Wigner-Ville distribution. With kind permission of P.J. Durka.

given of the use of matching pursuit methods in medicine, geophysics, finance and englneenng.

2.14

Wavelets in two or more dimensions

The two-dimensional Mexican hat specified on a

11112

coordinate plane is given by

,p(I) ~ (2 - III') e- Itl '/'

(2.58)

JIT

where t is the coordinate vector (l111z) and It1= +/~. Figure 2.33 shows a couple of two-dimensional Mexican hats on the plane. The two-dimensional wavelet transform is given by

T(a, b)

(I-b) x(l) dl a

= -'fOO ,po a

-00

(2.59)

where b is the coordinate vector (b 1 , bz). Note that the weighting function w(a) has been set to I/a required to conserve energy across scales for the two-dimensional wavelets. Note also that the coordinate vector t = (/1,/2) is likely to specify two

56

The comiflliOUS wavelet transform

/

20 Mexican hat on the plane

Figure 2.33.

Two~dimensional

reduced a scale

side on view

Mexican hat.

spatial (rather than temporal) coordinates in practice, where the function x(t) could for example be surface heights (e.g. fracture surfaces, topographic features) or image greyscales (e.g. medical images, fluid visualization studies). The corresponding inverse wavelet transform is

IfOOJOOI(t-b) da -g - - T(a,b),db

x(t)=-C

g-oooa

a

a

(2.60)

Wavelet transforms in higher dimensions, D, are also possible simply by extending the length of the vectors t and b to D components. To preserve energy in the D-dimensional transformation, the weighting function becomes l/a D/ 2 . Hence, the D-dimensional wavelet is defined as

,p".,(I)

I (t-b) = av/2,p a

(2.61)

The transform in D dimensions becomes

T(a, b)

= roo

,p;..(t)x(t) dt

(2.62)

with inverse I x(t) =-C

foo [00 ,pa,b(t)T(a,b)

g-oo.o

da V+l db a

(2.63)

and the energy of the signal may be found from

I roo fOO IT(a,b)1 2D+T da db

E=-C

g.-ooO

2.15

a

(2.64)

The CWT: computation, boundary effects and viewing

As with all mathematical tools used to investigate physical phenomena, a number of practical issues must be taken into consideration. This is no less the case when using

The CWT: computation, boundary effects and viewing

57

the wavelet transform. The results obtained by the investigator must be viewed in terms of the limitations in the data analysis method used. These limitations stem from a number of sources, including the discrete nature of the data, the finite resolution of the data, the finite extent of the data, the wavelet used, the discretization and numerical computation of the transform, and so on. To compute the continuous wavelet transform we could simply perform a naive discretization of the transform integral given by equation (2.13), replacing the integral with a discrete summation involving the sampling interval of the measured time series 6.t together with a suitable discretization for the a and b parametersusually logarithmic for a and linear steps of 6.t for b. This is a very cumbersome way to compute the CWT integral and a much better approach is to use the FFT method described in section 2.10 and given by equation (2.35). The FFT provides a much faster algorithm for the computation of the transform integral. In fact, this approach has been used in all the transform plots presented in this chapter. We will consider again the discretization of the wavelet transform in the next chapter when we tackle the discrete wavelet transform (DWT). However, note that the DWT is fundamentally different from the discretized CWT (see chapter 3, section 3.2.2) and in the scientific literature the term 'continuous wavelet transform' generally includes all discretizations using continuous wavelets such as the Mexican hat, Morlet, etc. In practice, experimental data sets are finite in extent and often the investigator wants to analyse the whole of the available data. However, an obvious consequence of wavelet analysis of a finite data set is that, as the wavelet gets closer to the edge of the data, parts of it begin 'spill over' the edge as illustrated in figure 2.34(a). This creates a boundary effect, where transform values close to the boundary of the signal are tainted by the discontinuous nature of the signal edge. The affected region increases in extent as the dilation of the analysing wavelet increases. An example of boundary effects is shown in the transform plot of figure 2.34(c), which is generated for the simple signal shown in figure 2.34(b) using a Mexican hat wavelet. The transform plot points to the location of the signal discontinuity as expected, but also treats the edges of the signal as additional discontinuities. Hence, large T(a, b) values are realized close to the edge of the transform plot, which increase in extent as a increases. This region affected by a discontinuity is known as the cone of influence. The extent of the cone increases linearly with a, i.e. it is proportional to the temporal support (or 'width') of the wavelet. (We have already seen the cone of influence corresponding to a signal discontinuity in figure 2.12.) However, when plotting T(a, b) using a logarithmic a scale (e.g. figure 2.34(c)) the cone contours become curved. The cone boundaries at either ends of the signal define the region which is significantly influenced by the signal edges. It is very much up to the investigator where to take the boundary of the cone by choosing a limiting value of the distance from the wavelet centre-a multiple of the wavelet a scale-at which point pertinent information contained in the transform plot is not considered masked by edge effects. A number of methods have been developed to cope with the boundaries of signals of finite extent. A variety of these are illustrated in figure 2.35 and include: adding a line of zero values at either end of the signal (zero padding,

58

The continuous wavelet transform x(t)

....

'ljJ«t-b)/a)

(a)

End of signal

Part of wavelet not within signal segment

(b)

Edge effects

Signal discontinuity

\

(c)

Figure 2.34. The manifestation of boundary effects in the scalogram. (a) Schematic diagram of the wavelet encountering the signal boundary. (b) A simple signal with a discontinuity halfway along its length. (c) The scalogram of the signal in (b).

figure 2.35(a)); adding a line of constant values at either end of the signal equal to the last value of the signal (value padding, figure 2.35(b)); adding some form of decay to zero for the last value at each end of the signal (decay padding, figure 2.35(c)); continuing the signal on from the last point back to the first point (periodization, figure 2.35(d)); reflecting the signal at the edges (reflection, figure 2.35(e)); convolving the signal with a window function which reduces the edge values of the signal to zero (windowing, figure 2.35(f)); using a polynomial extrapolation of the

The CWT: computation, boundary effects and viewing

59

x(t)

Signal set to zero here

Signal set to zero here

\

j ~

t

r

(a)

Beginning of signal segment

End of signal segment

x(t)

Signal set to initial value here

Signal set to final value here ~ ~

t

(b)

Beginning of signal segment

r End of signal segment

x(t)

Signal decays to zero

Signal decays to zero ~

t

(c)

Beginning of signal segment

End of signal segment

Figure 2.35. Schemes to deal with signal boundaries. (a) Zero padding. (b) Value padding. (c) Decay padding.

signal at either end (polynomial fitting, figure 2.35(g)); and, if we are focusing on a small segment of a much larger signal available to us, we may simply use the data points outside the segment under consideration (signal following, figure 2.35(h)). In addition, there is the wraparound method which uses only the length of signal available but wraps the parts of the wavelet which fall off each end of the signal back to the other end. Wraparound is similar to periodization but gives wavelet transform values only within the extent of the signal. Wraparound is used extensively in the next chapter. Whichever method is employed to cope with the signal edges, we must be aware that features appearing near the edges of the transform plot will contain information (synthetic or real) from outside the region of the signal segment under consideration. In addition, this effect will increase in

60

The continuous wavelet transform x(t)

Repeated segment

Beginning of signal segment

(d)

End of signal segment

x(t)

Reflected segment

Reflected segment

I

End of signal segment

Beginning of signal segment

(e)

x(t)

Smoothing window

Original signal

Signal set to zero here

\

.......".,.,../ Modified signal ,/

f"\_>··

Signal set to zero here

j

r (f)

Beginning of signal segment

End of signal segment

Figure 2.35 (continued). (d) Periodization involves the repetition of the signal along the time axis. (e) Reflection. (f) Smoothing window.

extent as the wavelet a scale increases and the wavelet extends further beyond the edge of the signal when close to it. Finally, when plotting the wavelet transform, we want to be able to discern features associated with the signal at specific a scales. It is often the case that features

The CWT: computation, boundary effects and viewing

61

x(t)

Polynomial fit

(g)

Beginning of signal segment

End of signal segment

x(t)

Signal outwith segment

(h)

Beginning of signal segment

r

End of signal segment

Figure 2.35 (continued). (g) Polynomial fitting. (h) Signal following.

at certain scales dominate the transform plot, obscuring the detail at other scales. In order to accentuate this hidden detail we can change the weighting parameter w(a) in value to a value which is more suitable at the transform from the usual 1/ highlighting features at a specific a scale of interest. This is shown in figure 2.36 for a sinusoidal signal containing a burst of noise. The transform plot with w( a) set to 1/ shows up the sinusoidal waveform well, but this dominates and the noise is not highlighted particularly well. By changing the weighting to w(a) = a-1.5 we can enhance the noise within the plot at lower a scales. When w(a) is set to a- 2.S we can see that the noise now dominates the transform plot. It is sometimes useful to vary w(a) in this way to accentuate features at different scales in the transform plot which might otherwise have been missed. Another way to weight the transform plot to show up small and large amplitude features is to use a logarithmic scale for T(a, b). However, two problems arise with this approach. The first is that we cannot take the logarithm of the negative values of T(a, b) and the second is that taking the logarithm of near-zero values of T(a, b) produces very large negative

va

va

62

The comiflliOUS wavelet transform

ty\/V\JI (aJ

(b)

II"

(0)

(d)

(eJ Figure 2.36. Accentuating features in the wavelet transform plot. (a) Original signal. (b) Transform plot with \I'(a) = lj.fii. (c) Transform plot with \I'(a) = a-l.5. (d) Transform plot with \I'(a) = a- 2.5 . (e) Logarithmic plot of the modulus of the Mexican hat wavelet transform IT(a,b)l. Near-zero values of IT(a, b)1 produce large negative logarithmic values. These are omilled using a minimum cut-ofT value, where aIlIT(a,b)1 values less than the cut-off are set to this value before computing logarithms.

numbers. We can get around these two problems, respectively, by using the modulus of the transform IT(a , b)1 to avoid negatives and setting a cut-off, or floor, value of IT(a,b)1 in order to both avoid zero and near-zero values and limit the extent of the logarithmic scale. This is illustrated in figure 2.36(e). Logarithmic plotting is particularly good at highlighting simultaneously features in the signal occurring at very different orders of magnitude, see for example the logarithmic plots of the ECG signals in figures 6.12 to 6.15 of chapter 6.

Endnotes

2.16

Endnotes

2.16.1

Chapter keywords and phrases

63

(You may find it helpful to jot down your understanding of each of them.) wavelet mother/analysing wavelet Mexican hat wavelet Fourier transform admissibility condition admissibility constant energy spectrum bandpass filter

2.16.2

passband centre dilation parameter location parameter wavelet transform inverse wavelet transform wavelet transform modulus maxima scalogram spectrogram

wavelet variance M orlet wavelet wavelet transform ridges short time Fourier transform windowed Fourier atom time-frequency atom matching pursuit boundary effects

Further resources

There are a number of sources of information at an introductory level concerning the continuous wavelet transform. Gade and Gram-Hansen (1997) compare wavelet transforms with short time Fourier transforms, presenting time-frequency plots of a variety of signals including speech and engine vibrations. Sarkar and Su (1998) detail the properties of the continuous wavelet transform from the perspective of electrical engineers. The relationship between Fourier and wavelet spectra is described by Perrier et al (1995). Wong and Chen (2001) provide a number of illustrative examples of phase and modulus plots in their Morlet wavelet-based study of nonlinear structural oscillations. See also chapter 1 of the book by Holschneider (1995) which contains many informative modulus and phase plots for a variety of wavelets and various simple signal features. Holshneider gives details of many other wavelets including Bessel, chirp, Cauchy, Poisson and Marr. Many websites contain good introductory material on the continuous wavelet transform. The appendix lists some websites from which to begin a search. Concise treatments of the continuous wavelet transform which provide a little more of the mathematical background at a reasonable mathematical level are to be found in chapter 3 of the book by Kaiser (1994), chapter 3 of the book by Blatter (1998), chapter 5 of the book by Vetterli and Kovacevic (1995) and the paper by Koornwinder in the book edited by the same author (Koornwinder, 1993). Heisenberg's uncertainty principle is covered by many authors; see for example Mallat (1998) or Kaiser (1994) for some extra information. Ridges have been used by Staszewski (1997, 1998a) in a new procedure for nonlinear system identification. (More details are given in chapter 5, section 5.4.) See also the early paper by Delprat et al (1992) and the algorithms for ridge detection of noisy signals by Carmona et al (1997, 1999). On a related topic, the relationship between standard wavelet scalograms and Fourier wavelengths is discussed by Meyers et al (1993), who provide brief mathematical detail in the appendix. See also Torrence and Compo (1998) who provide the information on how to find the Fourier wavelengths for the Morlet, Paul and all the derivatives of Gaussian wavelets (including the Mexican

64

The continuous wavelet transform

hat). More information on modulus maxima and signal reconstruction can be found in the papers by Mallat and Hwang (1992) and Mallat and Zhong (1992). Examples of the use of modulus maxima and ridge methods are to be found in chapters 4 to 7 of this book (e.g. figures 5.27 and 5.28 in chapter 5 and figures 6.20 and 6.21 in chapter 6). Further information on the application of ridges, modulus maxima, the complete Morlet wavelet and complex Mexican hat can also be found in Addison et al (2002a). In this chapter we considered the two-dimensional Mexican hat wavelet. More information concerning the two-dimensional continuous wavelet transform can be found in the book by Antoine (1999). A two-dimensional Morlet wavelet is given by Kumar and Foufoula-Georgiou (1994, p 29) and Peyrin and Zaim (1996). Koornwinder (1993) provides a brief outline of the mathematics of multidimensional wavelet transforms. See also the discussion by Daubechies (1992, pp 33-34) concerning the continuous wavelet in higher dimensions where rotations can be introduced into the definition of higher-dimensional wavelets. Numerous, diverse applications of the continuous wavelet transform to real signals are given in chapters 4 to 7 of this book, all of which require a discretized version of the transform. We dealt briefly with this issue in section 2.15. For more information on the discretization of the continuous wavelet transform see for example Jones and Baraniuk (1991), Sadowsky (1994) and Jordan et al (1997). The latter authors discuss many of the implementation issues that arise from the discretization itself. In particular they show how to convert from the non-dimensional times and frequencies of the mother wavelet to physical measures. They use the Morlet wavelet to illustrate their discussion, applying it to the velocity fluctuations measured in a subsonic wake undergoing transition to turbulence. Note that they use angular frequency and a Morlet wavelet which is not normalized to have unit energy. In this chapter we have not dealt directly with the issue of the physical meaning of the a scale parameter. It is non-dimensional. However, in practice it has to be linked to the timescales under investigation. The easiest thing to do is relate a = 1 to an appropriate unit of time, e.g. 1 second, 1 day, 1 year, etc. This has been done in this chapter when necessary: see for example figure 2.18 containing the vortex shedding time series where a = 1 corresponded to 1 second. It can also be linked to the sampling interval ~t or some multiple of it. We do not even have to use the a scale parameter to link to a temporal scale but can employ another measure of spread such as the standard deviation of the wavelet energy density in time; this has been done for the Morlet wavelet by Jordan et al (1997). A good account of the matching pursuit method is given in the original paper by Mallat and Zhang (1993). The method has been applied in many areas of science, technology and finance. We will come across some of these applications in subsequent chapters of this book. However, the paper by Zygierewicz et al (1999), which investigates sleep patterns in the EEG, gives a concise account in its appendix of the timefrequency (Wigner) representation of energy distribution used for the matching pursuit (something we have not gone into detail with here). The matching pursuit is a member of a larger family of adaptive approximation techniques. Jaggi et al (1998) give a brief account of related methods and introduce a high resolution pursuit which overcomes some of the shortcomings of the matching pursuit. Goodwin and Vetterli (1999) used a matching pursuit method which incorporates damped sinusoids as the analysing functions.

Chapter 3

The discrete wavelet transform

3.1

Introduction

In this chapter we consider the discrete wavelet transform (DWT). We will see that when certain criteria are met it is possible to completely reconstruct the original signal using infinite summations of discrete wavelet coefficients rather than continuous integrals (as required for the CWT). This leads to a fast wavelet transform for the rapid computation of the discrete wavelet transform and its inverse. We will then see how to perform a discrete wavelet transform on discrete input signals of finite length: the kind of signal we might be presented with in practice. We will also consider briefly biorthogonal wavelets, which come in pairs, and some space is devoted to two-dimensional discrete wavelet transforms. The chapter ends with wavelet packets: a generalization of the discrete wavelet transform which allows for adaptive partitioning of the time-frequency plane.

3.2 3.2.1

Frames and orthogonal wavelet bases Frames

In chapter 2, the wavelet function was defined at scale a and location b as

VJa,b(t)

=

va1 VJ (t-b) -a-

(3.1 )

In this section the wavelet transform of a continuous time signal, x(t), is considered where discrete values of the dilation and translation parameters, a and b, are used. A natural way to sample the parameters a and b is to use a logarithmic discretization of the a scale and link this, in turn, to the size of steps taken between b locations. To link b to a we move in discrete steps to each location b which are proportional to the a scale. This kind of discretization of the wavelet has the form (3.2)

65

66

The discrete wavelet transform

where the integers m and n control the wavelet dilation and translation respectively; ao is a specified fixed dilation step parameter set at a value greater than 1, and bo is the location parameter which must be greater than zero. The control parameters m and n are contained in the set of all integers, both positive and negative. It can be seen from the above equation that the size of the translation steps, !:1b = bodf/, is directly proportional to the wavelet scale, a3 The wavelet transform of a continuous signal, x( t), using discrete wavelets of the form of equation (3.2) is then 1

•

Tm,11

=

foo-00 x(t) a1~/2 'I/J(ao mt -

nb o) dt

(3.3a)

o

which can also be expressed as the inner product ~11,11 = (x, 'l/Jm,l1)

(3.3b)

where T ml1, are the discrete wavelet transform values given on a scale-location grid of index m, n. For the discrete wavelet transform, the values Tm,11 are known as wavelet coefficients or detail coefficients. These two terms are used interchangeably in this chapter as they are in the general wavelet literature. To determine how 'good' the representation of the signal is in wavelet space using this decomposition, we can resort to the theory of wavelet frames which provides a general framework for studying the properties of discrete wavelets. Wavelet frames are constructed by discretely sampling the time and scale parameters of a continuous wavelet transform as we have done above. The family of wavelet functions that constitute a frame are such that the energy of the resulting wavelet coefficients lies within a certain bounded range of the energy of the original signal, i.e. 00

AE:::;

00

L 11 L T m,111 m =-00 I

2

:::;

BE

(3.4)

= -00

where T m ,11 are the discrete wavelet coefficients, A and B are the frame bounds, and E is the energy of the original signal given by equation (2.19) in chapter 2: E = f~oo Ix(t)1 2 dt = Ilx(t)11 2 , where our signal, x(t), is defined to have finite energy. The values of the frame bounds A and B depend upon both the parameters ao and bo chosen for the analysis and the wavelet function used. (For details of how to determine A and B see Daubechies (1992).) If A = B the frame is known as 'tight'. Such tightframes have a simple reconstruction formula given by the infinite series 1 00 00 (3.5) x(t) = Tm,I1'I/Jm,I1(t) m=-oo 11=-00

ALL

A tight frame with A (= B) > 1 is redundant, with A being a measure of the redundancy. However, when A = B = 1 the wavelet family defined by the frame forms an orthonormal basis. If A is not equal to B a reconstruction formula can still be written: I 2 ~ x(t)= A+B L

~ Tm,I1'I/Jm,11 (t) L m=-oo 11=-00

(3.6)

Frames and orthogonal wavelet bases

67

O.5a

scale m a

= (V2)m

b

width

xV2

O.5a

b

Figure 3.1. The nearly tight Mexican hat wavelet frame with ao = 2'/2 and bo = 0.5. Three consecutive locations of the Mexican hat wavelet for scale indices m (top) and m + 1 (lower) and location indices n, n + I, n + 2. That is, a = 2111 and a = 2111 + 1 respectively, and three consecutive b locations separated by aj2.

where x' (t) is the reconstruction which differs from the original signal x( t) by an error which depends on the values of the frame bounds. The error becomes acceptably small for practical purposes when the ratio B/A is near unity. It has been shown, for the case of the Mexican hat wavelet, that if we use ao = 2 1/ v , where v ~ 2 and bo ::; 0.5, the frame is nearly tight or 'snug' and for practical purposes it may be considered tight. (This fractional discretization, v, of the power-of-two scale is known as a voice.) For example, setting ao = 2 1/ 2 and bo = 0.5 for the Mexican hat leads to A = 13.639 and B = 13.673 and the ratio B/A equals 1.002. The closer this ratio is to unity, the tighter the frame. Thus discretizing a Mexican hat wavelet transform using these scale and location parameters results in a highly redundant representation of the signal but with very little difference between x(t) and x'(t). The nearly tight Mexican hat wavelet frame with these parameters (ao = 2 1/ 2 and b o = 0.5) is shown in figure 3.1 for two consecutive scales m and m + 1 and at three consecutive locations, n = 0, 1 and 2. 3.2.2

Dyadic grid scaling and orthonormal wavelet transforms

Common choices for discrete wavelet parameters ao and b o are 2 and 1 respectively. This power-of-two logarithmic scaling of both the dilation and translation steps is known as the dyadic grid arrangement. The dyadic grid is perhaps the simplest and most efficient discretization for practical purposes and lends itself to the construction

68

The discrete wavelet transform

of an orthonormal wavelet basis. Substituting ao = 2 and b o = 1 into equation (3.2), we see that the dyadic grid wavelet can be written as (3.7a) or, more compactly, as (3.7b) Note that this has the same notation as the general discrete wavelet given by equation (3.2). From here on in this chapter we will use 1/Jm,n(t) to denote only dyadic grid scaling with ao = 2 and b o = 1. Discrete dyadic grid wavelets are commonly chosen to be orthonormal. These wavelets are both orthogonal to each other and normalized to have unit energy. This is expressed as

oo f

1/Jm,n(t)1/Jm',n,(t) dt

• -00

=

{I

0

if m = m' and n . otherwIse

=

n'

(3.8)

That is to say, the product of each wavelet with all others in the same dyadic system (i.e. those which are translated and/or dilated versions of each other) are zero. This means that the information stored in a wavelet coefficient T m ,n is not repeated elsewhere and allows for the complete regeneration of the original signal without redundancy. In addition to being orthogonal, orthonormal wavelets are normalized to have unit energy. This can be seen from equation (3.8) as, when m = m' and n = n', the integral gives the energy of the wavelet function equal to unity. Orthonormal wavelets have frame bounds A = B = 1 and the corresponding wavelet family is an orthonormal basis. (A basis is a set of vectors, a combination of which can completely define the signal, x(t). An orthonormal basis has component vectors which, in addition to being able to completely define the signal, are perpendicular to each other.) The discrete dyadic grid wavelet lends itself to a fast computer algorithm, as we shall see later. Using the dyadic grid wavelet of equation (3.7a), the discrete wavelet transform (DWT) can be written as:

T,n,n =

J:oo x(t)1/Jm,n(t) dt

(3.9)

By choosing an orthonormal wavelet basis, 1/Jm,n(t), we can reconstruct the original signal in terms of the wavelet coefficients, T,n ,11' using the inverse discrete wavelet transform as follows: 00

x(t)

=

00

L L

Tm,n1/Jm,n(t)

(3.10a)

m=-oo 11=-00

requiring the summation over all integers m and n. This is actually equation (3.5), with A = 1 due to the orthonormality of the chosen wavelet. Equation (3. lOa) is often seen

Frames and orthogonal wavelet bases

69

written in terms of the inner product 00

00

x(t)

L L

=

(x,7/Jm,n)7/Jm,n(t)

(3.lOb)

m=-oo n=-oo

where the combined decomposition and reconstruction processes are clearly seen: going from x(t) to Tm,n via the inner product (x, 7/Jm,n) then back to x(t) via the infinite summations. In addition, as A = B and A = 1, we can see from equation (3.4) that the energy of the signal may be expressed as (3.11 )

Before continuing it is important to make clear the distinct difference between the DWT and the discretized approximations of the CWT covered in chapter 2. The discretization of the continuous wavelet transform, required for its practical implementation, involves a discrete approximation of the transform integral (i.e. a summation) computed on a discrete grid of a scales and b locations. The inverse continuous wavelet transform is also computed as a discrete approximation. How close an approximation to the original signal is recovered depends mainly on the resolution of the discretization used and, with care, usually a very good approximation can be recovered. On the other hand, for the DWT, as defined in equation (3.9), the transform integral remains continuous but is determined only on a discretized grid of a scales and b locations. We can then sum the DWT coefficients (equation (3. lOa)) to get the original signal back exactly. We will see later in this chapter how, given an initial discrete input signal, which we treat as an initial approximation to the underlying continuous signal, we can compute the wavelet transform and inverse transform discretely, quickly and without loss of signal information. 3.2.3

The scaling function and the multiresolution representation

Orthonormal dyadic discrete wavelets are associated with scaling functions and their dilation equations. The scaling function is associated with the smoothing of the signal and has the same form as the wavelet, given by --I, ~m~

(t)

=

2- m / 2 ~ --I'(2- m t - n )

(3.12)

They have the property

J:oo ¢o,o(t) dt

=

1

(3.13)

where ¢o,o(t) = ¢(t) is sometimes referred to as the father scaling function or father wavelet (cf. mother wavelet). (Remember from chapter 2 that the integral of a wavelet function is zero.) The scaling function is orthogonal to translations of itself, but not to dilations of itself The scaling function can be convolved with the signal to produce approximation coefficients as follows:

Sm,n

=

J:oo x(t)¢m"Jt) dt

(3.14)

70

The discrete wavelet transform

From the last three equations, we can see that the approximation coefficients are simply weighted averages of the continuous signal factored by 2111 / 2. The approximation coefficients at a specific scale m are collectively known as the discrete approximation of the signal at that scale. A continuous approximation of the signal at scale m can be generated by summing a sequence of scaling functions at this scale factored by the approximation coefficients as follows: 00

x l11 (t)

L

=

SI11,I1¢I11,I1(t)

(3.15)

11=-00

where x l11 (t) is a smooth, scaling-function-dependent, version of the signal x(t) at scale index m. This continuous approximation approaches x(t) at small scales, i.e. as m --+ -00. Figure 3.2(a) shows a simple scaling function, a block pulse, at scale index 0 and location index 0: ¢o,o(t) = ¢(t)-the father function-together with two of its corresponding dilations at that location. It is easy to see that the convolution of the block pulse with a signal (equation (3.14» results in a local weighted averaging of the signal over the nonzero portion of the pulse. Figure 3.2(b) shows one period of a sine wave, x(t) contained within a window. Figure 3.2(c) shows various approximations of the sine wave generated using equations (3.14) and (3.15) with the scaling function set to a range of widths, 2° to 27 . These widths are indicated by the vertical lines and arrows in each plot. Equation (3.14) computes the approximation coefficients Sml1, which are, as mentioned above for this simple block scaling function, the weighted average of the signal over the pulse width. The approximation coefficients are then used in equation (3.15) to produce an approximation of the signal which is simply a sequence of scaling functions placed side by side, each factored by their corresponding approximation coefficient. This is obvious from the blocky nature of the signal approximations. The approximation at the scale of the window (=2 7 ) is simply the average over the whole sine wave which is zero. As the scale decreases, the approximation is seen to approach the original waveform. This simple block pulse scaling function used in this example here is associated with the Haar wavelet, which we will come to shortly. We can represent a signal x(t) using a combined series expansion using both the approximation coefficients and the wavelet (detail) coefficients as follows: 00 mo 00 (3.16) x(t) = SI110,n¢1110)t) + ~n.I11/JI11,I1(t)

L

L L

11=-00

111=-00 11=-00

We can see from this equation that the original continuous signal is expressed as a combination of an approximation of itself, at arbitrary scale index mo, added to a succession of signal details from scales mo down to negative infinity. The signal detail at scale m is defined as 00

dm(t)

L

=

~11.I11/Jm,I1(t)

(3.17)

11=-00

hence we can write equation (3.16) as

1110

x(t)

=

xmo(t)

+

L

111=-00

dm(t)

(3.18)

71

Frames and orthogonal wavelet bases
~

/

/

/

(a)

I -1

o

1

2

3

4

t

....

ts:J

(b)

~~ ~~ ~--1 ~ d 1 1 m=O

m=1

m=2

m=3

m=4

m=5

I.

m=6

.\

1"1

1 I·

m=7

·1

(c)

Figure 3.2. Smooth approximation of a sine wave using a block pulse scaling function. (a) Simple block scaling function shown at scale I (scale index 111 = 0) and location n = 0, i.e. cPa.a(t) (shown bold), together with its dilations at that location. (b) Sine wave of one period. (c) Selected smooth approximations, x l11 (t), of the sine wave at increasing scales. The width of one of the scaling functions cPl11.n(t) at scale index 111 is depicted in each figure by the arrows. Note that the example has been set up so that one

From this equation it is easy to show that X I11 - ]

(t)

=

xm(t)

+ dm(t)

(3.19)

which tells us that if we add the signal detail at an arbitrary scale (index m) to the approximation at that scale we get the signal approximation at an increased resolution (i.e. at a smaller scale, index m - 1). This is called a multiresolution representation.

72

3.2.4

The discrete wavelet transform

The scaling equation, scaling coefficients and associated wavelet equation

The scaling equation (or dilation equation) describes the scaling function ¢( t) in terms of contracted and shifted versions of itself as follows: ¢(t) = ~ ck¢(2t - k)

(3.20)

k

where ¢(2t - k) is a contracted version of ¢(t) shifted along the time axis by an integer step k and factored by an associated scaling coefficient, c/" (Take note of the similar but different terminology-scaling equation and scaling function.) Equation (3.20) basically tells us that we can build a scaling function at one scale from a number of scaling equations at the previous scale. The solution to this two-scale difference equation gives the scaling function ¢(t). For the sake of simplicity in the rest of the chapter we concern ourselves only with wavelets of compact support. These have sequences of nonzero scaling coefficients which are of finite length. Integrating both sides of the above equation, we can show that the scaling coefficients must satisfy the following constraint: (3.21 ) In addition, in order to create an orthogonal system we require that

~

C Ck k+2k' =

2 if k' = 0 { 0 otherwise

(3.22)

This also tells us that the sum of the squares of the scaling coefficients is equal to 2. The same coefficients are used in reverse with alternate signs to produce the differencing of the associated wavelet equation, i.e. 7jJ(t) = ~

(-I)k c1

_k¢(2t - k)

(3.23)

k

This construction ensures that the wavelets and their corresponding scaling functions are orthogonal. This wavelet equation is commonly seen in practice. In this chapter, however, we will consider only wavelets of compact support which have a finite number of scaling coefficients, N k . For this case we can define the wavelet function as 7jJ(t) = ~(-llcNk-l-k¢(2t-k)

(3.24)

k

This ordering of scaling coefficients used in the wavelet equation allows for our wavelets and their corresponding scaling equations to have support over the same interval [0, N k - 1]. (The ordering of equation (3.23) leads to wavelet and scaling functions displaced from each other, except for the Haar wavelet where N k = 2.) Note that, if the number of scaling coefficients is not finite, we cannot use this reordering and must revert back to an ordering of the type given by equation (3.23). We will stick to the ordering specified by equation (3.24) in this text. Often the reconfigured coefficients used for the wavelet function are written more compactly as (3.25)

Frames and orthogonal wavelet bases

73

where the sum of all the coefficients bk is zero. Using this reordering of the coefficients, equation (3.24) can be written as Nk-I

L

?jJ(t) =

bk¢(2t - k)

(3.26)

k=O

From equations (3.12) and (3.20) and examining the wavelet at scale index m + 1, we can see that for arbitrary integer values of m the following is true:

2-(I11+I)/2A-.(_t__ n ) =2- 111 / 22- 1/ 2 ~ c A -.( 2t -2n-k) '+' 2111 + 1 ~ k'+' 2 x 2111

(3.27a)

k

which may be written more compactly as

¢111+ I,n(t)

1

J2 ~

=

Ck¢I11,2n+k(t)

(3.27b)

That is, the scaling function at an arbitrary scale is composed of a seq uence of shifted scaling functions at the next smaller scale each factored by their respective scaling coefficients. Similarly, for the wavelet function we obtain

?jJ111+ I,n(t) =

3.2.5

1

J2 ~ bk¢I11,2n+k(t)

(3.28)

The Haar wavelet

The Haar wavelet is the simplest example of an orthonormal wavelet. Its scaling equation contains only two nonzero scaling coefficients and is given by

¢(t)

=

¢(2t)

+ ¢(2t -

1)

(3.29)

that is, its scaling coefficients are Co = CI = 1. We get these coefficient values by solving equations (3.21) and (3.22) simultaneously. (From equation (3.21) we see that Co + C1 = 2 and from equation (3.22) COCo + C1 C1 = 2.) The solution of the Haar scaling equation is the single block pulse shown in figure 3.3(a) and defined as

1 O
(3.30)

This is, in fact, the scaling function used earlier in figure 3.2 to generate the signal approximations of the sine wave. Reordering the coefficient sequence according to equation (3.25) we can see that the corresponding Haar wavelet equation is

?jJ(t)

=

¢(2t) - ¢(2t - 1)

(3.31)

The Haar wavelet is shown in figure 3.3(b) and defined as 1 O
(3.32)

74

The discrete wavelet transform

<j>(2t)

<j>(t)

SCALING FUNCTION

<j>(2t-l)

1

0

0

1

1/2

+

0

1/2

1

(a)

<j>(2t)

'ljJ (t)

WAVELET FUNCTION

+ -<j>(2t-l)

(b)

1-------1~...------+ 1-~r------+ I I I

(c)

(d)

I I

(e)

Figure 3.3. Discrete orthonormal wavelets. (a) The Haar scaling function in terms of shifted and dilated versions of itself. (b) The Haar wavelet in terms of shifted and dilated versions of its scaling function. (c) Three consecutive scales shown for the Haar wavelet family specified on a dyadic grid, e.g. from top to bottom: '1)/1/,I1(t), '1fJ/1/+ 1,11(t) and '!/J/1/+2,I1(t). (d) Three Haar wavelets at three consecutive scales on a dyadic grid. (e) Three Haar wavelets at different scales. This time the Haar wavelets are not defined on a dyadic grid and are hence not orthogonal to each other.

The mother wavelet for the Haar wavelet system, 'l/J(t) = 'l/Jo,o(t), is formed from two dilated unit block pulses sitting next to each other on the time axis, with one of them inverted. From the mother wavelet we can construct the Haar system of wavelets on a dyadic grid, 'l/Jm.n(t). This is illustrated in figure 3.3(c) for three consecutive scales. The

Frames and orthogonal wavelet bases

75

1

o

Ol---~,

-1 '--_-'-(f)

.1....-_ _----'_-----'

o

-5

-1

5

'-----'------.1....-------'------'

-5

o

5

Figure 3.3 (continued). (f) A Meyer wavelet and associated scaling function (right).

orthogonal nature of the family of Haar wavelets in a dyadic grid system is obvious from figure 3.3(d), where it can be seen that the positive and negative parts of the Haar wavelet at any scale coincide with a constant (positive or negative) part of the Haar wavelet at the next larger scale (and all subsequent larger scales). In addition, Haar wavelets at the same scale index m on a dyadic grid do not overlap. Hence, it is obvious that the convolution of the Haar wavelet with any others in the same dyadic grid gives zero. Figure 3.3(e) shows three Haar wavelets which are not specified on a dyadic grid. The non-orthogonal nature of the Haar wavelets across scales, when specified in this way, is obvious from the plot. In addition, if these wavelets overlap each other along each scale this also destroys orthogonality. (Although this is not the case for other orthonormal wavelets-see later.) Finally, note that the Haar wavelet is of finite width on the time axis; that is, it has compact support. As was stated in the last section, wavelets which have compact support have a finite number of scaling coefficients and these are the type of wavelet we concentrate on in this chapter. Not all orthonormal wavelets have compact support. Figure 3.3(f), for example, shows a Meyer wavelet which is orthonormal with infinite support, although, as with all wavelets, it is localized, decaying relatively quickly from its central peak. 3.2.6

Coefficients from coefficients: the fast wavelet transform

From equation (3.14) we can see that the approximation coefficients at scale index m + 1 are given by

SI11+ [,11 = J:CXJ x(t)¢111 + [,I1(t) dt

(3.33)

Using equation (3.27b) this can be written as

Sm+l,,,

~ [~ x(t) [~~;>k4>m,2"+k(t)]dt

(3.34)

We can rewrite this as SI11+1,11 =

~L

y2

k

Ck

[Joo-CXJ x(t)¢111,211+k(t) dt]

(3.35)

76

The discrete wavelet transform

The integral in brackets gives the approximation coefficients Sm,211+k for each k. We can therefore write this equation as I Sm+I,11 =

!2 L

VL

Ck S m,211+k =

k

I

!2 L

VL

Ck-2I1 S m,k

(3.36)

k

Hence, using this equation, we can generate the approximation coefficients at scale index m + I using the scaling coefficients at the previous scale. Similarly the wavelet coefficients can be found from the approximation coefficients at the previous scale using the reordered scaling coefficients bk as follows: (3.37) We can see now that, if we know the approximation coefficients SmO,ll at a specific scale mo then, through the repeated application of equations (3.36) and (3.37), we can generate the approximation and detail wavelet coefficients at all scales larger than mo. Notice that, to do this, we do not even need to know exactly what the underlying continuous signal x(t) is, only SmO,ll. Equations (3.36) and (3.37) represent the multiresolution decomposition algorithm. The decomposition algorithm is the first half of the fast wavelet transform which allows us to compute the wavelet coefficients in this way, rather than computing them laboriously from the convolution of equation (3.9). Iterating equations (3.36) and (3.37) performs respectively a highpass and lowpass filtering of the input (i.e. the coefficients Sm211+k) to get the outputs (Sm+ 1,11 and ~n+l,n). The vectors containing the sequences (l/V2)Ck and (1/V2)b k represent the filters: (l/V2)ck is the lowpassjilter, letting through low signal frequencies and hence a smoothed version of the signal, and (I / V2)b k is the highpass jilter, letting through the high frequencies corresponding to the signal details. We will come back to the filtering process in more detail in later sections of this chapter. We can go in the opposite direction and reconstruct Sm,11 from Sm+ I,ll and T m + I,n. We know already from equation (3.17) that X I11 -1 (t) = xm(t) + dl11 (t); we can expand this as (3.38) 11

11

Using equations (3.27b) and (3.28) we can expand this equation in terms of the scaling function at the previous scale as follows:

(3.39) Rearranging the summation indices, we get

(3.40)

Discrete input signals offinite length

77

We also know that we can expand X I11 -1 (t) in terms of the approximation coefficients at scale m - 1, i.e. X I11 -1

(t)

=

L

SI11-1,n¢I11-I,n(t)

(3.41 )

n

Equating the coefficients in equation (3.41) with equation (3.40) we note that the index k at scale index m relates to the location index n at scale index m - 1. In addition, location index n in equation (3.40) is not equivalent to location index n in equation (3.41), as the former corresponds to scale index m, with associated discrete location spacings 2111 , and the latter to scale index m - 1, with discrete location spacings 2111 - 1. Hence the n indices are twice as dense in the latter expression. The simplest way to proceed before equating the two expressions is to swap the indices k and n in equation (3.40) which, after some algebra, produces the reconstruction algorithm: (3.42) where we have reused k as the location index of the transform coefficients at scale index m to differentiate it from n, the location index at scale m - 1. Hence, at the smaller scale, m - 1, the approximation coefficients can be found in terms of a combination of approximation and detail coefficients at the next scale, m. Note that if there are only a finite number of nonzero scaling coefficients (=NK ), then Cn -2k has nonzero values only when in the range 0 to N k - 1. The reconstruction algorithm is the second half of the fast wavelet transform (FWT). Note that in the literature the fast wavelet transform, discrete wavelet transform, decomposition/reconstruction algorithms, fast orthogonal wavelet transform, multiresolution algorithm, pyramid algorithm, tree algorithm and so on, are all used to mean the same thing. It becomes even more confusing when other discretizations of the continuous wavelet transform are referred to as the discrete wavelet transform. Take care!

3.3 3.3.1

Discrete input signals of finite length Approximations and details

So far we have considered the discrete orthonormal wavelet transform of a continuous function x(t), where it was shown how the continuous function could be represented as a series expansion of wavelet functions at all scales and locations (equation (3.10a)) or a combined series expansion involving the scaling and wavelet functions (equation (3.16)). In this section, and from here on, we will consider discrete input signals specified at integer spacings. To fit into a wavelet multiresolution framework, the discrete signal input into the multiresolution algorithm should be the signal approximation coefficients at scale index m = 0, defined by SO,n =

J::)() x(t)¢(t -

n) dt

(3.43)

78

The discrete wavelet tramlol'm

which, as we now know from equations (3.36) and (3.37), will allow us to generate all subsequent approximation and detail coefficients, 511/./1 and TII/,/I' at scale indices greater than 111 = O. In this section we will assume that we have been given 50 ,11' Section 3.4 considers further the question of discrete input data which may not be 5 0 ./1' In practice our discrete input signal 5 0 /1 is of finite length N, which is an integer power of 2: N = 2M . Thus the range of ~cales we can investigate is 0 < m < M. Substituting both m = 0 and 111 = M into equation (3.16), and noting that we have a finite range of 11 which halves at each scale, we can see that the signal approximation scale 111 = 0 (the input signal) can be written as the smooth signal at scale M plus a combination of detailed signals as follows: M

5 0.II ¢o,lI(t)

= 5 M ,II¢M.II(t) + L m=l

2,11

III _

L

I

(3.44)

11=0

This is the form we use to describe our finite length discrete signal in terms of its discrete wavelet expansion. The covering of a finite length time segment with wa velets is illustrated in figure 3.4 for Daubechies D4 wavelets at two successive scales. The lower scale covers the time window using eight wavelets, and the larger scale uses four wavelets. One of the wavelets in each plot is shown bold for clarity. The wavelets shown which spill over the end of the window have been wrapped around back to the

(,j

Figure 3.4. CO\'ering the lime axis with dyadic grid wa\'elels. (a) Eight Daubechies 04 wavelets covering the time axis at scale m. (b) Four Oaubechies 04 wavelets covering the time axis at scale III + I. These wavelets are twice the width of those in (a).

Discrete input signals offinite length

79

beginning. Known as wraparound, it is the simplest and one of the most common treatments of the boundary for a finite length signal and we will employ it throughout the rest of this chapter. However, note that, by employing this method, we assume that the signal segment under investigation represents one period of a periodic signal and we are in effect pasting the end of the signal back on to the beginning. Obviously, if the signal is not periodic, and in practice it usually is not, then we create artificial singularities at the boundary which results in large detail coefficients generated near to the boundary. We can rewrite equation (3.44) as M

xo(t)

=

XM(t)

+L

dm(t)

(3.45)

m= I

where the mean signal approximation at scale M is

XM(t) = SM,n¢M,n(t)

(3.46)

As the approximation coefficients are simply factored, weighted averages of the signal then, when wraparound is employed to deal with the boundaries, the single approximation component S M n is related to the mean of the input signal through the relationship SO,n = SM,nlV2M, where the overbar denotes the mean of the sequence So ,17" In addition, when wraparound has been used to deal with the boundaries, the mean signal approximation at the largest scale, XM(t), is a constant valued function equal to the input signal mean. (We will see why this is so later in section 3.5.1.) The term on the far right of equation (3.45) represents the series expansion of the fluctuating components of the finite length signal at various scales in terms of its detail coefficients. The detail signal approximation corresponding to scale index m is defined for a finite length signal as 2M

dm(t)

=

-

III

L

-I

Tm,n'l/Jm,n(t)

(3.47)

n=O As we saw above (equation (3.45)), adding the approximation of the signal at scale index M to the sum of all detail signal components across scales 0 < m < M gives the approximation of the original signal at scale index o. Figure 3.5(a) shows the details of a chirp signal with a short burst of noise added to the middle of it. A Daubechies D20 wavelet was used in the decomposition (see an example of this wavelet later in figure 3.l5(e)). The original signal is shown at the top of the plot. Below the signal the details for ten wavelet scales, d l (t) to d lO (t), are shown. The bottom trace is the remaining signal approximation XIO(t). Adding together all these details plus the remaining approximation (which is the signal mean) returns the original signal. Two things are noticeable from the plot. First, there is a shift to the left of the large amplitude details with increasing scale, as we would expect as the chirp oscillation increases in frequency from left to right. The second thing to notice is that the high frequency burst of noise is captured at the smallest scales, again as we would expect. We saw in equation (3.19) that the signal approximation at a specific scale was a combination of the approximation and detail at the next lower scale. Ifwe rewrite this

80

The discrete wavelet transform (a)

14 xo(t) = original signal

12 d 1(t) d 2 (t)

10

dlt) dit) dlt) dit) dit) d 8(t) d 9(t) d lO (t) x lO (t) = signal mean

8 6 4 2 00

(b)

200

400

600

800

1000

14 xo(t) = original signal

12 10

x 1(t)

8

xi t ) x 3(t) x4 (t) - xs(t)

6

xit) xi t ) x 8(t) x 9(t) x lO (t)

4 2 0

200

0

400

600

800

1000

Figure 3.5. Multiresolution decomposition of a chirp signal containing a short burst of noise. (a) Signal details dl1l (t). (The signals have been displaced from each other on the vertical axis to aid clarity.) (b) Signal approximations x l1l (t).

equation as (3.48)

and begin at scale m - 1 = 0, that of the input signal, we can see that at scale index m = 1, the signal approximation is given by

x[ (t) = xo(t) - d[ (t) at the next scale (m

=

(3.49a)

2) the signal approximation is given by

X2(t)

=

xo(t) - d[ (t) - d2(t)

(3.49b)

and at the next scale by (3.49c)

Discrete input signals affinite length

81

and so on, corresponding to the successive stripping of high frequency information (contained within the dm (t)) from the original signal. Figure 3. 5(b) contains successive approximations xm(t) of the chirp signal. The top trace is the original signal xo(t). Subsequent smoothing of the signal takes place throughout the traces from the top to the bottom of the figure. As we saw in equation (3.48), the difference between each of the approximation traces Xm-l (t) and xm(t) is the detail component dm(t). We can view these differences in the detail component traces of figure 3.5(a). We have glossed over much of the mathematical detail of multiresolution analysis here. Most mathematical accounts of the subject begin with a discussion of orthogonal nested subspaces and the signal approximations and details, xm(t) and dm(t), as projections on to these spaces. This tack has not been followed here; see for example Mallat (1998), Blatter (1998), Sarkar et al (1998) or Williams and Armatunga (1994). In this chapter we concentrate on the mechanics, rather than the mathematics, of multiresolution analysis. 3.3.2

The multiresolution algorithm-an example

Once we have our discrete input signal SO,lD we can compute Sm,1l and Tm ,1l using the decomposition algorithm given by equations (3.36) and (3.37). This can be done for scale indices m > 0, up to a maximum scale determined by the length of the input signal. To do this, we use an iterative procedure as follows. First we compute St,1l and T l ,1l from the input coefficients SO,11' i.e.

1 SI,1l =

"2

VL

L

('k S O,21l+k

(3.50a)

k

and (3.50b) In the same way, we can then find SliD , i.e.

S21l ,

and T 21l, from the approximation coefficients

(3.51a) and

1 T 2 ,11

=

"2

VL

L

bk S 1,21l+k

(3.51 b)

k

Next, we can find S3,11 and T 3,11 from the approximation coefficients S2,11' and so on, up to those coefficients at scale index M, where only one approximation and one detail coefficient is computed: SM,O and T M,O' At scale index M we have performed a full decomposition of the finite-length, discrete input signal. We are left with an array of coefficients: a single approximation coefficient value, S M.O, plus the detail coefficients, T mll , corresponding to discrete wavelets of scale a = 2m and location b = 2m n. The' finite time series is of length N = 2M . This gives the ranges of m and M n for the detail coefficients as respectively 1 < m < M and 0 < n < 2 -m - 1.

82

The discrete wavelet transform

Notice that the range of n successively halves at each iteration as it is a function of scale index m for a finite length signal. At the smallest wavelet scale, index m = 1, 2M /2 1 = N /2 coefficients are computed, at the next scale 2M /2 2 = N /4 are computed and so on, at larger and larger scales, until the largest scale (m = M) where only one (=2 M /2 M ) coefficient is computed. The total number of detail coefficients for a discrete time series of length N = 2M is then, 1 + 2 + 4 + 8 + ... + 2M - I, or 2..:~==-/ 2m = 2 M - 1 = N - 1. In addition to the detail coefficients, the single approximation coefficient S M 0 remains. This is related to the signal mean as we saw above, and is required in addition to the detail coefficients to fully represent the discrete signal. Thus a discrete input signal of length N can be broken down into exactly N components without any loss of information using discrete orthonormal wavelets. In addition, no signal information is repeated in the coefficient representation. This is known as zero redundancy. The decomposition of approximation coefficients into approximation and detail coefficients at subsequent levels can be illustrated schematically thus sO,n ---~ sI,n ---~ s2,n ---~ s3,n ---~ ---~ S M,O

"" """-

"" TI,n

"" """-

"""T 2,n

""

""

"" "-

T 3 ,n

"" "TM,o

Figure 3.6 shows an alternative schematic of the same process, illustrating the decomposition and insertion of the approximation and detail coefficients at each

.

original signal

/n~", , N-' Scale index SD,n

m=O

m=l

m=2

m=3

m=4

T l,n

m=5

T l,n

wavelet transform L.......L---'-----'-_--'---'--_----'-_ _' - -

-,t: \ ~'N~ n=O, I ,...,N/16-1

n=O,1, ..,N/4-1

--'--

----'

+--- vdector at ~~l ecomposlt1on

n~oin-I

Figure 3.6. Schematic diagram of the Haar filtering algorithm. - - - ----+ ST1LI1 used to derive Sm + 1.11; -------+ Sm.11 used to derive T m + 1.11; - - T m .11 taken to next level as Tm .", i.e. no further manipulation.

Discrete input signals affinite length

83

iteration within the wavelet transform vector for an arbitrary input signal vector containing 32 components. The wavelet transform vector after the full decomposition has the form W(M) = (SM, T M, T M -I,'" ,T/1w'" T 2 , T 1), where T m represents the sub-vector containing the coefficients 01111 at scale index m (where n is in the range o to 2M -m - 1). We can halt the transfo~mation process before the full decomposition. If we do this, say at an arbitrary level mo, the transform vector has the form W(mo) = (Smo' T mo ' T mo - I , " " T 2 , T 1), where mo can take the range 1 :::; mo :::; M - 1. In this case, the transform vector does not contain a single approximation component but rather the sequence of approximation components Smo,n' However, the transform vector always contains N = 2M components. For example, we can see from figure 3.6 that stopping the algorithm at m = 2 results in W(2) = (S2' T 2, T] ). Remember, the range of n is a function of scale index m, hence this vector contains eight S2,11 components, eight T 2,11 components and 16 T] ,11 components, matching the 32 components of the original input signal vector. The range ofn is indicated below the full decomposition vector in figure 3.6. Notice also that we can express the original input signal as the transform vector at scale index zero, i.e. W(O). An example of a wavelet decomposition using a discrete wavelet is shown in figure 3.7. The input signal is composed of a section of a sine wave, some noise and a flatline. The signal is decomposed using a Daubechies D6 wavelet. A member of this family is shown in figure 3.7(b). (We will look more closely at the Daubechies wavelet family later in this chapter.) The discrete transform plot is shown in figure 3.7(c), where the dyadic grid arrangement may be seen clearly. This plot is simply a discretized dyadic map of the detail coefficients, 011,10 where the coefficients at larger scales have correspondingly longer boxes (as the wavelets cover larger segments of the input signal). In addition to the detail coefficients, T m ,I1' the remaining approximation coefficient SM.O is added to the bottom of the plot. As we would expect it covers the whole time axis. We can see from the transform plot that the dominant oscillation is picked up at scale index m = 6 and the high frequency noise is picked up within the middle segment of the transform plot at the smaller scales. We can use the reconstruction algorithm (equation (3.42)) to get back the original input signal SO,11 from the array of detail coefficients shown in figure 3.7(c). Alternatively, as with the continuous transform, we can reconstruct a modified version of the input signal by using only selected coefficients in the reconstruction. This is shown in figures 3.7(d) and (e), where only the coefficients corresponding to scales m = 5 to 8 are kept (the others are set to zero) and the signal is reconstructed. This has removed a significant amount of the noise from the signal although the sinusoidal waveform is less smooth than the original in figure 3.7(a). We will come across more sophisticated ways to remove noise and other signal artefacts later, in section 3.4.2 of this chapter. 3.3.3

Wavelet energy

After a full decomposition, the energy contained within the coefficients at each scale is given by 2:11-111_1

Em =

L

11=0

(011,11)2

(3.52)

84

The discrete wavelet transform

JV\!&R :1 I==-_ o

200

1000

(b)

(,j

scale index m= 1

scale index

m= I m=5

Iti5a;:.--;--:~"'t.ained

coefficients

remaining approximalion coefficient (cj

o

(dj

500

1000

(ej

Figure 3.7. Discrete wavelet transform of a composite signal. (a) Original composite signal. (b) A member of the Daubechies D6 wavelet family. (c) Discrete transform plot. (Note dyadic structure-large positive coefficient values are white and large negative values black.) (d) Coefficient removal. (e) Reconstructed signal using only retained coefficients in (d). The original composite signal (a) is composed of three segments: a sinusoid, uniformly distributed noise and a flatline. The signal is decomposed using Daubechies D6 wavelets (b) to give the dyadic array of transform coefficients plotted in (c). The coefficients corresponding to scales 5 to 9 are kept (d) and used to reconstruct the signal in (e). Note that a grey scale is used to depict the coefficient values, where the maximum value is white and the minimum value is black.

A wavelet-based power spectrum of the signal may be produced using these scaledependent energies. To do so, we require a frequency measure which is a reciprocal of the wavelet dilation, e.g. the passband centre of the power spectrum of the wa velct. A wavelet power spectrum can then be produced for the signal which is directly comparable with both its Fourier and continuous wavelet counterparts (see chapter 2, section 2.9). This topic is dealt with in detail in chapter 4, section 4.2.1, in connection with the statistical measures used to analyse turbulent fluid signals. The total energy of the discrete input signal N-I

E~

L

11=0

(So,,)'

(3.53)

Discrete input signals affinite length

85

is equal to the sum of the squared detail coefficients over all scales plus the square of the remaining approximation coefficient, S M 0, as follows: 2M "'_I

M

E= (SM,O/+

L L m=

[

(Tm,n)2

(3.54)

11=0

In fact, the energy contained within the transform vector at all stages of the multiresolution decomposition remains constant. We can, therefore, write the conservation of energy more generally as N-I

E =

L

(W?11))2

(3.55)

;=0

where W/ m) are the individual components of the transform vector W(m) ordered as described in the previous section. When m = 0, this equation corresponds to the summation of the component energies of the input signal (equation (3.53» and when m = M it corresponds to the summation of the energies within the coefficients at full decomposition (equation (3.54». 3.3.4

Alternative indexing of dyadic grid coefficients

There are three main methods used in practice to index the coefficients resulting from the discrete wavelet transform. It is worth discussing these now. All three methods are popular in the scientific literature and appear often in many of the examples of the practical application of wavelet analysis in the subsequent chapters of this book. We will use the full decomposition of a 32 component input signal as an illustration. Scale indexing: The scale indexing system (m, n) corresponding to an input signal of length N = 2M is shown schematically on a dyadic grid in figure 3.8(b) for the discrete signal shown in figure 3.8(a). The lowest scale on the grid m = 1 corresponds to a spacing of2 1 = 2 on the data set. The discrete input signal is at scale index m = 0. We have already come across this grid structure in the plot of the transform coefficients in figure 3.7(c). Such plots give a good visual indication of the covering of the timescale plane by the wavelets and their relative importance in making up the signal. Sequential indexing: Again, we have already come across this form of indexing (figure 3.6). Figure 3.8(c) contains a schematic of the transform coefficients given in sequential format. We know that a signal N samples long produces N wavelet vector components. It makes sense, therefore, to find an ordering of these coefficients to fit into a vector oflength N. The discrete time series, SO,II' n = 0, I, 2, ... ,N - I, can be decomposed into N - 1 detail coefficients plus one approximation coefficient where N is an integer power of 2: N = 2 M . After a full decomposition these transform coefficients can be resequenced from the two-dimensional dyadic array T m n to the M ) have the same range as the origitransform vector W(M), where the components nal signal (i = 0, I, 2, ... , N - I). The vector component index i is found from the dyadic grid indices m and n through the relationship i = 2M -m + n. In addition, the superscript M in parentheses denotes a full decomposition of the signal over M scales. The transform vector components, W;(M)are plotted in figure 3.8(c). The last

W/

86

The discrete wavelet transform

(a) (N=32)

r

1,1 ,0

scale index 1 scale index 2 scale index 3 scale index 4 scale index 5

\ I Tz,o

r

T

I

I

Tz,1

Tz,z

I

T 3,0

I

I

I'

I

T Z,7 T 33

T 3,Z

T 3,1 T 4,0

T 4,1 T 50 (=TMO)

S5,0 _

signal mean component

(b)

sequential indexing WOo WI. W Z• W 3 • W 4 • W 5.·······················WN_3• W N_Z• W N_ I•

SM,O

---=±:.l;JkJ.c-W-,..w~Uli--.lJLLLl-.LJ---LU..L1---LLLli--.lJ~

TM,O TM-I,O TM_I,I

TI,O - T I ,15

TM-Z,O TM_Z,I TM-Z,Z T M- Z,3

(C)

corresponding scale indexing

T4,Z

T

41 ,0 "\

level index 4 level index 3 level index 2 level index 1 level index 0

\

\ I T 3,0

T

5

1'1

I

I

T 3,1

T 3,Z

TZ,O

I

I

I

I

I' T 3,7

TZ,Z

TZ,I TI,O

T Z,3 TI,I

To,o

signal mean component

level index -1 (d)

Figure 3.8. Schematic diagram of alternative indexing methods for dyadic grid wavelet transform coefficients. (a) Original signal containing 32 components (scale index m = 0). (b) Scale indexing. M T I1l ,I1' (c) Sequential indexing. W/ ). where i = 2M -11 + n (corresponding scale indexing is shown at the bottom of the plot). (d) Level indexing, T'.I1'

half of the series in the figure represents the coefficients corresponding to the smallest wavelet scale (index m = 1). The next quarter, working backwards, represents the coefficients corresponding to scale index m = 2, the next eighth to the m = 3 coefficients, and so on, back to W~M), which is the single coefficient for scale index m = M. M The remaining component ) is the single approximation coefficient (SM,O)' As

W6

Discrete input signals affinite length

87

mentioned in section 3.3.2, if we halt the decomposition algorithm at scale mo (before full decomposition) this results in an intermediate transform vector W(mo) which contains a number of approximation coefficients, Sm 0, n at its beginning. Level indexing: Level indices, I, are often used instead of scale indices, m. The level index I is simply equal to M - m. In this case, the number of wavelets used to cover the signal at each level at a specific level is i, e.g. level I = 0 corresponds to a single wavelet and the scale of the whole time series, level I = 1, corresponds to two wavelets, and the scale of one half of the time series, and so on. The number of wavelets used at a level is simply 2'. In addition, it is standard in this nomenclature to denote the remaining approximation coefficient at level I = -1. The wavelet array becomes T,,/J" Figure 3.8(d) shows the I,n indexing system. Level indexing is used when the signal is specified over the unit interval and the analysis is set up in terms of resolution rather than scale (i.e. the lowest resolution I = 0 corresponds to the length of the whole signal, whereas the smallest scale m = 0 corresponds to the distance between each signal point). 3.3.5

A simple worked example: the Haar wavelet transform

Now we will illustrate the methods described above using a Haar wavelet in the decomposition of a discrete input signal, So ,n: n = 0, 1,2, ... ,N - 1. To do so, we employ the decomposition algorithm given by equations (3.36) and (3.37). The Haar wavelet has two scaling coefficients, Co = 1 and CI = 1. Substituting these into equation (3.36) we can obtain the approximation coefficients at the next scale through the relationship Sm+l,n

=

1 J2[Sm,2n

+ Sm,2n+l]

(3.56)

Similarly, through equation (3.37) we can obtain the detail coefficients at subsequent scales using 1

'011+ I,n

=

J2 [Sm,2n

- Sm,2n+ d

(3.57)

Using these equations, we will perform the Haar wavelet decomposition of the (very) simple discrete signal, (l, 2, 3, 4). As the signal contains only four data points, we can only perform two iterations of the Haar decomposition algorithm given by equations (3.56) and (3.57). After two iterations we expect to obtain four transform coefficients: three wavelet coefficients Tm,n-two at scale index m = 1, (TI,o, TI,I), and one at scale index m = 2, (T2,o), plus a signal mean coefficient at scale index m = 2, (S2,O). This is illustrated through the iteration of the transform vector in figure 3.9 and also through a schematic of the coefficients in figure 3.10. The first iteration of the decomposition algorithm gives T IO = -

,

1

J2

[1 - 2]

1 SI,O = J2[0

+ 3]

1

= --

J2

3

=

J2

88

The discrete wavelet transform

~

scale m=O

1st iteration D

~

scale m=l

2nd iteration D

scale m=2

Figure 3.9. Iteration of the wavelet transform vector in the decomposition of a simple signal. - - - ----+ 5/11,11 used to derive 5/11 + 1,11; -------+ 5/11,11 used to derive T/11 + 1,11; - - TI1l ,11 taken to next level as T/11,,,, i.e. no further manipulation.

The transform coefficient vector, after this first iteration, is then W

(I)

=

(SI,O, SI,I, TI,o, Tl,l)

(3J2' J2' 7 - J2' 1 - J21)

=

The second iteration (only involving the remaining approximation coefficients SI,O and Su) yields T 2 ,o =

~ [ ~ - ~]

= -2

and

The transform coefficient vector after this second iteration is now Wi(2)

=

(S2,O, T2 ,o, TI,o, TI,I)

=

( I) 5, -2,I - J2' - J2

The signal mean is found from the remaining approximation coefficient, i.e. S2,O/( J2? = 2.5. We can also see that the energy of the original discrete signal (12 + 22 + 32 + 42 = 30) is equal to the energy of the transform coefficient vector after the first iteration

and the second iteration, giving the full decomposition 2 2 ( 5+(-2)+

1)2 + -J2

(

1)2 =30 -J2

which is a result we expect from the conservation of energy condition expressed in equations (3.52) to (3.55). To return from the transform coefficients to the original discrete signal we work in reverse. The inverse Haar transform can be written simply as: at even locations 2n: I

Sm,2n =

J2 [Sm + I,n + ~n + I,n]

(3.58)

89

Discrete inpll! signals offinite length 5o,oSo,1 5 O,25 0,3

~ ~ ~

SI,O SI,I TI,O T1,1

*

5

:Gill

iteration 1

c::::::)

0

-5

0

(a) discrete input signal Xi '" (1,2,3,4)

0

(h) ,-'

"

:8 o

(d) Xz(t) = signal mean

(0) <1,(')

(g) L -

x (2,5)

---.J

+

x (-0.5) x (-0.5)

~

-'o(t)

(h)

Figure 3.10. Haar decomposition and reconstruction of a simple ramp signal. The original discrete input signal is decomposed through an initial iteration to (b), the first decomposition, then further to (c), the second and final decomposition, From the coefficients at the full decomposition (c), the following can be constructed: the scale index 2 approximation (the signal mean) (d), the scale index 2 detail (e) and the scale index I detail (r), (g) The transform coefficient plot associated with the transform values given in (c). (h) Schematic diagram of the addition of the rescaled Haar wavelets to reconstruct the original signal. (Adding the rei;onstructions in (d), (e) and (f) returns the approximation of the signal at scale index 11/ = 0.)

and at odd locations 21/

+ I: SIII,2/1 + 1 =

I

J2[Sm+l,1J - Tm+l,n]

(3.59)

For the Haar wavelet transform, we can derive these reconstruction equations directly from equations (3.56) and (3.57). Alternatively, we could derive them from the reconstruction algorithm of equation (3.42).

90

The discrete wavelet transform

We will perform the reconstruction on the transform vector we have just computed: W (2) -- (S2,o, T 2,o, T 1,0' T) 1,1 --

(S , -2 , _0_1' _0_1 )

The first iteration using the reconstruction pair (equations (3.S8) and (3.S9)) yields 1 3 1 7 SI,O = 0 (S2,0 + T2,0) = 0 SI,1 = 0 (S2,0 - T2,0) = 0 which results in

1 So , I = y M (SI 0- T I 0) 2' , SO,\,-

=

=

2

1 M (SI I - T I I) = 4 y 2' ,

hence we get back the original signal (So,0,SO,I,SO,2,SO,3) = (1,2,3,4). Figure 3.10 attempts to show visually the decomposition of the signal into the Haar wavelet components. Figure 3.lO(a) contains the original signal, figure 3.lO(b) plots the coefficients after the first iteration of the decomposition algorithm and figure 3.lO(c) plots the coefficients after the second iteration of the algorithm. The coefficients contained in figure 3.lO(c) correspond in turn to a single scaling function at scale index m = 2, a wavelet at scale index m = 2 and two wavelets at scale index m = 1. These are shown respectively in figures 1O( d)-(f). The transform coefficient plot for this signal is equally simplistic, consisting of four coefficient values partitioned as shown in figure 3.l0(g). We can find the corresponding approximation and details of the signal by taking the inverse transform of the coefficients at each scale. First the approximation coefficient is used to determine the signal approximation at the largest scale: (3.60) where the scaling function rP2,0(t), for the Haar wavelet, is simply a block pulse of length 4 and amplitude 1/(V2Y =~, beginning at t = O. Thus X2(t) is simply a block pulse of length 4 and magnitude S2,0 x ~ = 2.S, which is, in fact, the signal mean. A plot of X2(t) is shown in figure 3.lO(d). Similarly, we can obtain the signal detail component at scale index 2 as follows: d2(t)

=

(3.61)

T 2,0'l/J2,0(t)

This detail component (shown in figure 3.l0(e)) is simply a single Haar wavelet spanning the data set with coefficient value T 2,0 = -2, hence magnitude _2/(0)2 =-1. Next the detail signal component at scale index I is found from I

d l (t)

=

L 11=0

T I,I1'l/JI,I1(t)

=

TI,O'l/JI,O(t)

+ TI,1 1PI,1 (t)

(3.62)

Everything discrete

91

which is simply two Haar wavelets at scale index 1, side by side, with amplitudes given by T1,o/(yfi) = (-I/yfi)/(yfi) = -0.5 and Tu/(yfi) = (-I/yfi)/(yfi) = -0.5. The detail component d] (t) is plotted in figure 3.10(f). We already know from equation (3.45) that the approximation of the signal at scale index can be found by adding together all the detail components plus the signal approximation at scale index M, i.e.

°

M

xo(t)

=

XM(t)

+L

dm(t)

(3.63a)

m=l

We have already seen an example of this partitioning of the signal into approximation and detail components in figure 3.5 for a chirp signal. For the Haar case considered here M = 2, hence (3.63b) This is simply the addition of the approximation and detail components shown in figures 3.1 O(d)-(f). This is shown schematically in figure 3.1O(h).

3.4 3.4.1

Everything discrete Discrete experimental input signals

We will now consider the case where we have a discrete experimental signal collected, say, using some piece of experimental apparatus, and we want to perform a wavelet decomposition of it using the multiresolution algorithm. In addition, we would also like to represent the approximation and detail signal components discretely at the resolution of the input signal. This discrete signal, which we will denote Xh is of finite length N, i.e. i = 0, 1, ... , N - 1. It has been acquired at discrete time intervals 6.t (the sampling interval) to give the discrete time signal x(t i ): i = 0, 1,2, ... , N - 1. The sampling of the signal provides a finite resolution to the acquired signal. This discretization of the continuous signal is then mapped on to a discrete signal Xi, where the sampling interval has been normalized to 1. In doing so we must remember 6.t and add it when required, for example in real applications when computing signal frequencies. Common practice is to input the discretely sampled experimental signal, Xi, directly as the approximation coefficients at scale m = 0, and begin the multiresolution analysis from there. However, it is not correct to use the sampled time series Xi directly in this way. We should really use the approximation coefficients So ,n obtained from the original continuous signal at scale m = 0, as defined by equation (3.43). In practice we usually do not know exactly what x(t) is. As SO,n is a weighted average of x(t) in the vicinity of n, then it is usually reasonable to input Xi as So ,n if our signal is slowly varying between samples at this scale. That is we simply set SOn , = Xn

(3.64)

92

The discrete wavelet transform

where, at scale index m = 0, both the coefficient location index n and signal discretization index i have the same range (0 to N - 1) and are equal to each other. In the rest of this chapter we will assume that the sampled experimental signal has been input directly as So ,11. The literature is full of studies of real data where this has been done and, if anything, it is the rule rather than the exception. Obviously, it does not make it any more correct! In fact, Strang and Nguyen (1996) call the use of the sampled signal directly within the transform a 'wavelet crime' and suggest various ways to preprocess the sampled signal prior to performing the analysis. In practice, continuous approximations, X I11 (t), and details, dl11 (t), of the signal are not constructed from the multiresolution analysis, especially when inputting the signal Xi as the detail coefficients at scale index m = 0. Instead, either the approximation and detail coefficients, SI11,11 and ~11,11' are displayed at their respective scales or, alternatively, they are used to construct representations of the signal at the scale of the input signal (m = 0). The latter is sometimes preferable as the scalings of the displays are visually comparable. As an example, let us consider the Haar wavelet decomposition of a simple discrete signal containing eight components: Xi = (4,5,-4,-1,0,1,2,0). The signal is shown in figure 3.11(a). We can see that the first half of the signal has a marked step from 5 to -4, whereas the second half of the signal appears much smoother. The transform vector after a full decomposition is w(3) =

(S3,0, T 3,0, T 2.0, T 2,1, T 1,0, T 1,1, T 1,2, T) 1,3

=

(2.475,0.354,7.000, -0.500, -0.707, -2.121, -0.707, 1.414)

=

(

7

1

14

1

1

3

1

2)

(J2)3'(J2)3'2'-2'- J2'- J2'- J2'J2

(You can find these values for yourself using the procedure described in section 3.3.5.) This vector is shown schematically in figure 3.11 (b). Performing the inverse transform on this vector leads us back to the original discrete input signal. Referring back to the schematic of the wavelet decomposition given in section 3.3.2, we can similarly represent the reconstruction from the transform vector, W(3), as

s3,n

---~

,

S2,n

---~

",

",

",

T 3 ,n

S 1,n

---~ ",

",

", ",

,

T 2,n

SO,n

",

", ",

,

", ",

T 1,n

Remember that the original signal Xi was input into the multiresolution algorithm as SOn. , Let us now look at what happens to the reconstructed signal when we remove the smallest-scale (i.e. highest-frequency) detail coefficients in turn (figures 3.11(c)-(e)). Removing the components, T 1,n, the modified transform vector becomes (2.475, 0.354, 7.000, -0.500, 0, 0, 0, 0), where the coefficients set to zero are shown bold. Performing the inverse transform on this vector with the coefficients at the smallest wavelet scale set to zero removes the high-frequency details at this scale and returns a smoother version of the input signal. We will denote this smooth version of the

Everything discrete

o

93

5

(b)

(a)

o1---'---'----,--,-.........................---'---1

o

5

o

5

o

5

o

5

(d)

(c)

o

5 (f)

(e)

o

5 (h)

(g)

Figure 3.11. Multiresolution decomposition as scale thresholding. (a) xO,i (= Xu + dl,J (b) Wavelet coefficients (S3,O' T 3,o, Tz,o, Tz,J, Tl,o, Tl,l' T1,2, T l ,3)' (c) Xl,i (=XZ,i + dZ,i), (=XO,i - du)o (d) dl,i' (e) Xv (=X3,i + d3,i), (=xu - dz.J (f) dv · (g) X3,i (signal mean), (=XZ,i - d 3,i)' (h) d3,i'

input signal as

Xl ,i'

This operation is shown schematically as

s

3~

---~ ",.

,

S2~

---~ ",.

",.

T 3 ,n

S 1~

---~ ",.

",.

",. ",.

,

T 2,n

x 1~.

",.

",. ",.

,

",. ",.

zeros

The reconstructed discrete signal becomes xl'; = (4.5, 4.5, -2.5, -2.5, 0.5, 0.5, 1, 1). That is, the signal is smoothed by the averaging of each pair of signal values as shown in figure 3.ll(c). Note that we use the nomenclature xl'; for the approximation signal Xl (t) expressed in terms of discrete coefficients at the scale of the original input signal (i.e. m = 0). Remember from section 3.2.3 that the approximation coefficients, Sm ,11'

94

The discrete wavelet transform

provide a discrete approximation of the signal at scale index m. By passing the coefficients through the reconstruction filter we can express the contributions of these discrete approximations at the scale of the original signal (scale index 0). In the rest of this section, we will see how to express all the discrete approximations and details at the scale of the input signal with index m = O. Now we remove the detail coefficients associated with the next smallest scales from the transform vector to get (2.475, 0.354, 0, 0, 0, 0, 0, 0). Reconstructing the signal using this modified transform vector, we get x2,i = (1, 1, 1, 1, 0.75, 0.75, 0.75, 0.75) shown in figure 3.l1(e). Again we can show this schematically as S3,n

---~

""

""

,

S2,n

---~

"" zeros

T 3 ,n

""

,

S'1,n

---~ X 2,i

"" "" zeros

,

where we have put the prime on S't.n to indicate that it has different valued components from S',n above in the full reconstruction as it is computed with the T 2,n components set to zero. We can see that the nomenclature is beginning to get a bit awkward here as x2,i is in effect sb,n: a modified version of the input signal. In x2,i' the subscript 2 relates to the largest scale of the original approximation coefficients used at the beginning of the reconstruction, and the index i reminds us that we are back at the input resolution of the original signal (m = 0) with N components. Removing the last detail coefficient, the modified transform vector becomes (2.475, 0, 0, 0, 0, 0, 0, 0) which, when used to reconstruct the signal, produces X3,i = (0.875, 0.875, 0.875, 0.875, 0.875, 0.875, 0.875, 0.875). This can be shown as S3,n

---~

"" zeros

""

,

S'2,n

---~

"" zeros

""

,

S'1,n

---~ X

,

3,i

"" "" zeros

Actually, we have now removed all signal detail and all the components of vector X3 ,i are of constant value. The signal has been progressively smoothed until all that is left is a row of constant values equal to the signal mean (figure 3.11(g)). We can by see comparing figures 3.11 (c), (e) and (g) that the high-frequency information has been successively stripped from the discrete input signal (figure 3.11(a)) by successively removing the discrete details at scale indices ma = 1, 2 and 3 in turn. These discrete signal details are shown in figures 3.11(d), (f) and (h). If we want to generate the details of the discrete signal at any scale using the multiresolution algorithm, we perform the inverse transform using only the detail coefficients at that scale (we could also subtract two successive discrete approximations). For example, to compute the detail at the lowest wavelet scale (index m = 1), which we will denote dl,i' the following multiresolution operation is required: zeros

---~

"" " zeros

,

"

zeros

---~

"" " zeros

,

"

zeros

---~

,

" ""

T 1,n

"

d 1,1.

Everything discrete

95

dl,i is in fact the contribution of the wavelets at scale index m = 1 (i.e. '1fJI,O(t), '1fJI,1 (t), '1fJI,2(t) and '1fJI,3(t)) to the original signal Xi expressed at discrete points at the scale of the input function m = 0 at locations i = 0, 1, ... ,N - 1. Similarly we can calculate the contribution d2,i using the following operation:

zeros - - -~ zeros - - -~ S'1,11 ---~ d2,1. .- ..- ..- ..- ..- ..- .zeros T 2,11 zeros

,

,

,

where, obviously, the S;,II vector has all of its components equal to zero as it comes from S3,1I and T 3,1I with elements set to zero. Then d2,i is the contribution of the wavelets at scale index m = 2 (i.e. which for this example come from the two wavelets '1fJ2,0(t) and '1fJ2,1 (t)) to the original signal Xi given at discrete points at the scale of the input function m = 0, i.e. at locations i = 0 to N - 1. Similarly, beginning with T 3,n we can calculate the contribution d3,i' In the above example we were effectively taking the information stored in the approximation and detail coefficients Sm,1I and T m,1I and expressing it at scale index m = O. The discrete approximation and detail contributions Xm,i and dm,i are related to their continuous counterparts xm(t) and dm(t) through the scaling equation at scale m = 0 as follows: 2M

N-I

xm(t)

=

L

xm/PO,i(t)

171_1

L

[ =

Sm,n¢m,11 (t)]

(3.65)

T m,n'1fJm,n(t)]

(3.66)

n=O

i=O

and 2M

N-I

dm(t)

=

L

dm,i¢o,i(t)

[ =

i=O

171_1

L

11=0

Notice that the scaling function ¢O,i(t) is not only used to compute the continuous approximation from xm,i but also the detail components from dm,i' This is because the scaling coefficient sequence for the wavelet, blo has already been used in the initial stages of the reconstruction sequence to take T ml1 to S:n- Ill' Thereafter the contributions can be expressed in terms of scaling functi~ns, i.e. fr~m S:n _ ',11 to S:n- 2,11 and so on. You can verify equations (3.65) and (3.66) using the reconstruction algorithm of equation (3.42) (noting that one of its terms will be zero) and the scaling function relationship given by equation (3.27b). Figure 3.11(c) shows xl,i' i.e. where only the detail signal dl,i corresponding to the smallest scale wavelets (m = 1) has been removed (figure 3.11(d)). Figure 3.11(e) shows X2 ,..i' where both detail signals d, , i and d2,i' corresponding to the smallest and next smallest scale wavelets, have been removed from the original discrete input signal. d2,i is shown in figure 3.11 (f). Figure 3.11 (g) shows X1- ,i' where the detail signals dm i corresponding to all the wavelet scales (m = 1, 2 and 3) have been stripped from the original signal leaving only the signal mean component. This can be written as .

3

X1- ,l. -- Xo ,I. - L..J """"" dm,1. m= I

(3.67a)

96

The discrete wavelet transform

or, in general, over M scales as M

X M ,i

=

Xo , i L - ~ d/11,i ..J /11 =

(3.67b)

1

where Xo ,i is simply the input signal Xi. This equation stems directly from its continuous counterpart of equation (3.45) using equations (3.65) and (3.66). We have now seen how we can express everything discretely. We can use a suitable discrete input signal within the multiresolution algorithm, and we can express the approximation and details of this signal at discrete points at the input resolution; and, as we will see later in section 3.5.1, even the wavelet and scaling functions are expressed discretely within the multiresolution algorithm, built up by the repeated iteration of the scaling coefficient vectors. 3.4.2

Smoothing, thresholding and denoising

Let us look again at the discrete input signal used in the previous section (figure 3.12(a)). Once we have performed the full decomposition, we are free to alter any of the coefficients in the transform vector before performing the inverse. We can set groups of components to zero, as we did in the last section (figure 3.11), or set selected individual components to zero. We can reduce the magnitudes of some components rather than set them to zero. In fact, we can manipulate the components in a variety of ways depending on what we want to achieve. Notice, however, that the transform vector contains a range of values: some large and some small. Let us see what happens if we throwaway the smallest valued coefficients in turn and perform the inverse transforms. We start with the smallest valued coefficient 0.354. Setting it to zero, the modified transform vector becomes (2.475, 0, 7.000, -0.500, -0.707, -2.121, -0.707,1.414). Performing the inverse on this vector gives the reconstructed discrete signal (3.875, 4.875, -4.125, -1.125,0.125,1.125,2.125,0.125), as shown in figure 3.12(b). There is no discernible difference between the original discrete signal and the reconstruction. We now remove the next smallest valued coefficient (smallest in an absolute sense), the -0.5 coefficient. The transform vector becomes (2.475, 0, 7.000, 0, -0.707, -2.121, -0.707, 1.414) and the reconstructed signal is (3.875, 4.875, -4.125, -1.125,0.375, 1.375, 1.875, -0.125). Again we have to look carefully at figure 3.12(c) to discern any difference between the original discrete signal and the reconstruction. Next we remove the two -0.707 components to get (2.475, 0, 7.000, 0, 0, -2.121, 0, 1.414). Reconstructing we get (4.375, 4.375, -4.125, -1.125, 0.875, 0.875, 1.875, -0.125). This reconstruction is shown in figure 3.12(d) where we can now notice obvious smoothing between both the first and second signal point and the fifth and sixth. Next setting the last transform vector component to zero, we reconstruct to get (4.375, 4.375, -4.125, -1.125,0.875,0.875,0.875,0.875). This is shown in figure 3.12(e). Removing the -2.121 component leaves the coefficient vector as (2.475, 0, 7.000, 0, 0, 0, 0, 0) and the reconstruction becomes (4.375, 4.375, -2.625, -2.625,0.875, 0.875, 0.875, 0.875) as shown in figure 3.12(f). Removing the 2.475 component (the signal mean component) leaves the coefficient vector with only a single component: (0, 0, 7.000, 0, 0, 0, 0, 0). Reconstructing using this

Everything discrete

o

5

(a) (2.475,0.354,7.000,-0.500,-0.070,-2.121,-0.707,1.414)

o

5

(e) (2.475,0,7.000,0,-0.707,-2.121,-0.707,1.414)

o

97

5

(b) (2.475,0,7.000,-0.500,-0.707,-2.121,-0.707,1.414)

o

5

(d) (2.475,0,7.000,0,0,-2.121,0,1.414)

o

I----'----'-----r---r---'----'----'---'--I

o

5

(e) (2.475,0,7.000,0,0,-2.121,0,0)

o

o

5

(f) (2.475,0,7.000,0,0,0,0,0)

5

(9) (0,0,7.000,0,0,0,0,0)

Figure 3.12. Signal reconstruction using thresholded wavelet coefficients. The coefficient vectors used in the reconstructions are given below each reconstructed signal. Note that (a) is the original signal as it uses all of the original eight wavelet coefficients.

component leaves a single (wavelet shaped) fluctuation contained within the first half of the signal (figure 3.l2(g». Can you see what has happened? The least significant components have been smoothed out first, leaving the more significant fluctuating parts of the signal intact. What we have actually done is to threshold the wavelet coefficients at increasing magnitudes. First, all the components whose magnitude was equal to or below 0.354 were removed; i.e. 0.354 was the threshold. Then 0.5 was set as the threshold, then 0.707 and so on. By doing this we removed the least significant influences on the signal first. Hence, the shape of the reconstructed signal resembles the original even with a large number of coefficients set to zero. Compare this magnitude thresholding of the coefficients to the scale-dependent smoothing of figure 3.11, which

98

The discrete wavelet transform

removed the components at each scale in turn, beginning with the smallest scales. Note that in practice we would normally deal with the signal mean coefficient separately, either retaining it regardless of magnitude or removing it at the beginning of the thresholding process. We can define the scale-dependent smoothing of the wavelet coefficients as scale Tm,1l

= {

0T

m;::: m* m,1l

m

< m*

(3.68)

where m* is the index of the threshold scale, or the transform vector. Figure 3.13(a) shows a schematic diagram of scale-dependent smoothing for sequentially indexed coefficients Wi' (Assuming a full decomposition, we have dropped the M superscript from W iM .) In this case the thresholding criterion is defined as W?cale =

{o

i> 2M

Wi

- m*

i < 2M -

m*

(3.69)

where the range of the sequential index i is from 0 to N - 1 and N is the length of the original signal; hence i = 2M - m ' is the first location index within the transform vector where the coefficients are set to zero. (Remember that the smallest scale coefficients are placed at the right-hand end of the transform vector and hence have the highest index values.) Note that in practice the coefficients at the very largest scales are sometimes also set to zero to remove drift effects from the signal. The reconstructed signal using the scale thresholded coefficient vector with scale threshold m* is simply the smooth approximation of the signal at scale m* expressed at the scale of the input signal, m = 0, i.e. X m * ,i. (See previous section.) Magnitude thresholding is normally carried out to remove noise from a signal, to partition signals into two or more (and not necessarily noisy) components, or simply to smooth the data. It involves the reduction or complete removal of selected wavelet coefficients in order to separate out the behaviour of interest from within the signal. There are many methods for selecting and modifying the coefficients. The two most popular are hard and soft thresholding. Unlike scale-dependent smoothing, which removes all small scale coefficients below the scale index m* regardless of amplitude, hard and soft thresholding remove, or reduce, the smallest amplitude coefficients regardless of scale. This is shown schematically in figure 3.13(b). To hard threshold the coefficients, a threshold, A, is set which is related to some mean value of the wavelet coefficients at each scale, e.g. standard deviation, mean absolute deviation, etc. Those coefficients above the threshold are deemed to correspond to the coherent part of the signal, and those below the threshold are deemed to correspond to the random or noisy part of the signal. Hard thresholding is of the form: W hard = I

{o

W.I

IWil < A IWil;:::A

(3.70)

Hard thresholding makes a decision simply to keep or remove the coefficients. (Hard thresholding was performed in the example of figure 3.12.) Soft thresholding

Everything discrete

keep

99

remove

/

(a)

Scale cutoff at scale index m* i.e. sequential index i = 2M-m*

retain

A _. remove remove

-A _.

retain

(b)

o

o

(c)

-A 0 A

.

-A 0 A

Figure 3.13. Scale-dependent smoothing and coefficient thresholding. (a) Scale-dependent smoothing.

(b) Amplitude thresholding. (c) The relationship between the original coefficients and hard (left) and soft (right) thresholded coefficients.

100

The discrete wavelet transform

recognizes that the coefficients contain both signal and noise, and attempts to isolate the signal by removing the noisy part from all coefficients. Soft thresholding is of the form (3.71) where all coefficients below the threshold, A, are set to zero and all the coefficients whose magnitude is greater than A are shrunk towards zero by an amount A. Figure 3.13(c) shows a schematic diagram of the relationship between the original and thresholded coefficients for both hard and soft thresholding. The retained coefficients shown in the figure are kept as they are when hard thresholding is employed and their magnitude is reduced by A when soft thresholding is employed. Figure 3.14 shows examples of both hard and soft thresholding of a test signal composed of two sinusoids plus Gaussian white noise (figures 3.14(a) and (b)). The wavelet coefficients obtained from a Daubechies D 10 decomposition are shown in figure 3.14(c). These have been hard thresholded for thresholds set at A = 2, 4, 6 and 8 respectively. The reconstructions corresponding to each of the thresholds are shown in figures 3.14(d)-(g). We can see that for low thresholds some of the high frequency noise is retained, whereas for high thresholds the signal is excessively smoothed. From visual inspection, an optimum threshold would seem to lie somewhere in the region of A = 4. One commonly used measure of the optimum reconstruction is the mean square error between the reconstructed signal and the original signal and, in fact, it is found to be minimum near to this value of the threshold. The corresponding soft thresholded reconstructions are shown in figures 3.14(h) and (i) for A = 2 and 4 respectively. Often we do not know the exact form of either the underlying signal or the corrupting noise. The choice of threshold is therefore non-trivial and we can apply a number of signal and/or noise based criteria in producing a value pertinent to the problem under consideration. The threshold can, for example, be a constant value applied to the coefficients across all scales, some of the scales, or its value can vary according to scale. One of the most popular and simplest thresholds in use is the universal threshold defined as (3.72) where (21n N) 1/2 is the expected maximum value of a white noise sequence oflength N and unit standard deviation, and a is the standard deviation of the noise in the signal under consideration. Thus if the underlying coherent part of the signal is zero (i.e. the signal contains only noise), using AU as the threshold gives a high probability of setting all the coefficients to zero. For large samples the universal threshold will remove, with high probability, all the noise in the reconstruction, but part of the underlying function might also be lost, hence the universal threshold tends to oversmooth in practice. (This method is often known as the VISUSHRINK method as the resulting smooth signal estimate is visually more appealing.) In addition, in practice it is usual to leave the coefficients at the largest scales untouched even if they do not pass the universal threshold. The example of the sinusoids plus

101

Everything discrete

(a)

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Figure 3.14. Hard and soft thresholding. (a) Original time series composed of two sinusoids, both with unit amplitude, one twice the periodicity of the other. (b) Time series in (a) with added Gaussian noise (zero mean and unit standard deviation). (c) Wavelet coefficients in sequential format derived using the Daubechies DlO shown. (d) Hard thresholded coefficients, ,\ = 2 (top), and corresponding reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.)

noise shown in figure 3.14 contains Gaussian noise with (J' = 1 and N = 1024, the universal threshold for this data is then Au = 3.723. Within a wide range of practical data sizes (2 6 _2 19 ), only in about one tenth of the realizations will any pure noise variable exceed the threshold. In this sense universal thresholding method gives a 'noise-free' reconstruction.

102

The discrete wavelet transform 10

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10 .---,---...---,------.------.-----.----.-------,----,-------" Olrr--------------------~

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...: 1000

Figure 3.14 (continued). (e) Hard thresholded coefficients, ,\ = 4 (top), and corresponding reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.) (f) Hard thresholded coefficients, ,\ = 6 (top), and corresponding reconstructed time senes (bottom). (The original time series of figure (a) is shown dashed.) (g) Hard thresholded coefficients, ,\ = 8 (top), and corresponding reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.)

Everything discrete

103

10.----,..--.---....---,...--....----.----.-----.-----,-----"

-10

L...-_.L...-_.L..-_...l...-_...1..-_....l..-_--'-_---'-_---L.._----L._----l....J

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(h)

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(i)

Figure 3.14 (continued). (h) Soft thresholded coefficients, .A = 2 (top), and corresponding reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.) (i) Soft thresholded coefficients, .A = 4 (top), and corresponding reconstructed time series (bottom). (The original time series of figure (a) is shown dashed.)

Universal soft thresholding usually produces a signal reconstruction with less energy than that for hard thresholding using the same threshold value, as the retained coefficients are all shrunk towards zero. This can be seen by comparing figure 3.14(e) with 3.14(i) for). = 4. Hence, it is often the case in practice that the universal threshold used for hard thresholding is divided by about 2 when employed as a soft threshold. Another problem encountered when implementing the universal threshold in practice is that we do not know the value of (J" for our signal. In this case, a robust estimate (} is used, typically set to the median of absolute deviation (MAD) of the wavelet coefficients at the smallest scale divided by 0.6745 to calibrate with the standard deviation of a Gaussian distribution. Thus the universal threshold becomes

\

_ (21nN)I/2 M AD _ (21 )1/2 nN (J" 0.6745

AU -

A

(3.73)

104

The discrete wavelet transform

Many other thresholding methods have been proposed including the mlmmax method, the SURE method, the HYBRID method, cross-validation methods, the Lorentz method and various Bayesian approaches. Some of these produce a global threshold of constant value across scales and others provide a scale-dependent threshold. We do not consider these in detail here. More information is provided at the end of the chapter.

3.5

Daubechies wavelets

As we saw with the Haar wavelet transform earlier in section 3.3.5, the coefficients are ordered in two distinct sequences: one acts as a smoothing filter for the data, the other extracts the signal detail at each scale. The Haar wavelet is extremely simple in that it has only two scaling coefficients and both are equal to unity. In this section we will look at a family of discrete wavelets of which the Haar wavelet is the simplest member-the Daubechies wavelets. The scaling functions associated with these wavelets satisfy the conditions given in section 3.2.4, as all orthogonal wavelets do. In addition to satisfying these criteria, Daubechies required that her wavelets had compact support (i.e. a finite number, N lo of scaling coefficients) and were smooth to some degree. The smoothness of the wavelet is associated with a moment condition which can be expressed in terms of the scaling coefficients as Nk-l

2..= (-1 if ck k k=O

J11

=

°

(3.74)

for integers m = 0,1,2, ... , N k /2 - 1. These wavelets have N k /2 vanishing moments, which means that they can suppress parts of the signal which are polynomial up to degree Nd2 - 1. Or, to put it another way, Daubechies wavelets are very good at representing polynomial behaviour within the signal. Some examples of Daubechies wavelets, their scaling functions and associated energy spectra are given in figure 3.15. The support lengths of the Daubechies wavelets are N k - 1, i.e. the D2 (Haar) wavelet, as we already know, has a support length of 1, the D4 wavelet has support length of 3, the D6 a support length of 5 and so on. We can see from figure 3.15 that the scaling function lets through the lower frequencies and hence acts as a lowpass filter, and the associated wavelet lets through the higher frequencies and hence acts as a highpass filter. In addition, we can see that the spectra are oscillatory in nature, with bumps decreasing in amplitude from the lower to higher frequencies. The magnitudes of the secondary bumps in the spectra reduce as the number of scaling coefficients, and hence the number of vanishing moments of the wavelet, increase. In this chapter we have already looked in detail at the Daubechies wavelet with only two scaling coefficients (the D2 or Haar wavelet). Let us now look at the Daubechies wavelet which has four scaling coefficients, the D4. The 'D' represents this particular family of Daubechies wavelet and the '4' represents the number of nonzero scaling coefficients, N k . (Note that Daubechies wavelets are also often defined by the number of zero moments they have, which equals Nk/2, in which case the sequence runs DI, D2, D3, D4, ..., etc.)

Daubechies wavelets WAVELET

SCALING FUNCTION

o

o

_2'--_ _---"

0.5

2 L -_ _---'

o

2 r-,------,-----,

2

0.5

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o

(a) 02 (Haar).

0 Llllo..................... 012345618910

0.5

-2 o~--!----:------'! -2 0~--!----:-----'! 2 3 2 3 2

0.5

o

0 Lioo...................... 012345678910

(b) D4 2 r-T--r--.-~

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r----,----,

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(,) D20 Figure 3.15. A selection of Daubechies wavelets and scaling functions with their energy spectra. All

wavelets and scaling functions are shown only over their respective support-outside their support they have zero value. Note also that only the positive part of the spectrum is given; an identical mirror image is present at negative rrequencies.

106

The discrete wavelet transform

From equation (3.20), we know that the scaling equation for a four-coefficient wavelet is

¢(t)

=

co¢(2t) + Cl ¢(2t - 1) + c2¢(2t - 2) + c3¢(2t - 3)

(3.75)

and, from equation (3.23), that the corresponding wavelet function is

1jJ(t)

=

c3¢(2t) - c2¢(2t - 1)

+ c,¢(2t -

2) - co¢(2t - 3)

(3.76)

To find the values of the scaling coefficients for the D4 wavelet we use equations (3.21), (3.22) and (3.74). From equation (3.21) we get (3.77) from equation (3.22) we get .2 (0

and from equation (3.74) with m

.2 - 2 +Cl,2 +(2.2 +(3=

(3.78)

0 we get

Co - Cl

+ C2 -

C3

=

and again using equation (3.74), this time setting m -1 Cl

+ 2C2 -

3C3

=

(3.79)

0 =

1, we get (3.80)

0

F our scaling coefficients which satisfy the above four equations are Co

1 +J3 =--4-

Co

=

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CI

=

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C2

J3

=--4-

C3

1- J3 =--4-

and so do

14

J3

J3

34

One set leads to ¢(t) and the other to ¢( -t). We will adopt the first set, which are Co = 0.6830127, CI = 1.1830127, C2 = 0.3169873 and C3 = -0.1830127 respectively. The scaling coefficients for the Daubechies wavelet system for larger numbers of coefficients are found by numerical computation. The coefficients for Daubechies wavelets up to D20 are given in table 3.1. We can compute the scaling function from the D4 coefficients using equation (3.75). To do this we must rewrite it as (3.81) where the subscript} is the iteration number. Then choosing an arbitrary initial shape for the scaling function ¢o(t) we find ¢I (t), then using ¢I (t) we find ¢2(t), and so on, iterating until ¢j(t) = ¢j_1 (t) (or at least until ¢j(t) is close enough to ¢j_1 (t) for our purposes). Once we have an approximation for ¢(t) we could define the wavelet directly using equation (3.76). However, to perform the D4 wavelet transform of a discrete signal in practice we are not required to compute the wavelet or scaling functions directly; rather we simply employ the scaling coefficients within the multiresolution algorithm (equations (3.36) and (3.37)) in the same manner as we did for

Dauhechies wavelets

107

Table 3.1. Daubechies wavelet coefficients D2 to D20

-0.34265671 -0.04560113 0.10970265 -0.00882680 -0.01779187 4.71742793e - 3

D2 1 1 D4 0.6830127 1.1830127 0.3169873 -0.1830127

D12 0.15774243 0.69950381 1.06226376 0.44583132 -0.31998660 -0.18351806 0.13788809 0.03892321 -0.04466375 7.83251152e-4 6.75606236e - 3 -1.52353381e - 3

D6 0.47046721 1.14111692 0.650365 -0.19093442 -0.12083221 0.0498175 D8 0.32580343 1.01094572 0.89220014 -0.03957503 -0.26450717 0.0436163 0.0465036 -0.01498699

D14 0.11009943 0.56079128 1.03114849 0.66437248 -0.20351382 -0.31683501 0.10084647 0.11400345 -0.05378245 -0.02343994

DIO 0.22641898 0.85394354 1.02432694 0.19576696

0.01774979 6.07514995e - 4 -2.54790472e - 3 5.00226853e - 4 D16 0.07695562 0.44246725 0.95548615 0.82781653 -0.02238574 -0.40165863 6.68194092e 0.18207636 -0.02456390 -0.06235021 0.01977216 0.01236884 -6.88771926e -5.5400454ge 9.55229711e -1.66137261e -

4

3 4 4 4

D18 0.05385035 0.34483430 0.85534906 0.92954571 0.18836955 -0.41475176 -0.13695355 0.21006834

0.04345268 -0.09564726 3.54892813e - 4 0.03162417 -6.67962023e - 3 -6.05496058e - 3 2.61296728e - 3 3.25814671e - 4 -3.5632975ge - 4 -5.5645514e - 5 D20 0.03771716 0.26612218 0.74557507 0.97362811 0.39763774 -0.35333620 -0.27710988 0.18012745 0.13160299 -0.10096657 -0.04165925 0.04696981 5.10043697e -0.01517900 1.97332536e 2.8176865ge -9.69947840e -1.64709006e 1.32354367e -1.87584156e -

3 3 3 4 4 4 5

the Haar wavelet. In this case, the approximation coefficients are computed using

1 Sm+l,n =

!2 VL

N/(-I

2..=

Ck S m,2n+k

k=O

1

= =

V2 [cOSm,2n + CI S m,2n+ 1 + C2 S m,211+2 + C3 S m,211+3] 0.483Sm,211

+ 0.837Sm,2n+l + 0.224Sm,2n+2 -

0.129Sm,211+3

(3.82a)

We therefore take the product of a four-digit sequence of the signal by the scaling coefficient vector (1/ V2)Ck (the lowpass filter) to generate the approximation component at the first scale. The mechanics of this are shown schematically in figure 3.16. The four-digit sequence, (l/V2)clo slides across the signal at scale index m in jumps of two generating, at each jump, an approximation component at

108

The discrete wavelet transform

So,o

SO,l

1

SO,2

SO,3

SO,4

SO,5

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• • • • I

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-----

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Figure 3.16. Filtering ofthe signal: decomposition. The original signal at scale index 111

0 (i.e. SO,/1) is filtered to produce the approximation coefficients SI,I1" This is done by sliding the lowpass filter along the signal one step at a time. Subsampling removes every second value. The diagram shows only the retained values, i.e. effectively the filter coefficient vector jumps two steps at each iteration. Next, the sequence SI,/1 is filtered in the same way to get S2,/1 and so on. The detail coefficients are found by using the same method but employing highpass wavelet filter coefficients. =

Dauhechies wavelets

109

the next scale (m + 1). To generate the corresponding coefficients for the wavelet we use the reconfigured scaling coefficient seq uence (1/V2)h k (the highpass filter), where hk = (-llcNk -I-1o as follows:

~11+ I,n

1 =

M

Nk-I

L

y2 k=O

b kSm,2n+k

= -0.129Sm,2n - 0.224Sm,2n+ I

+ 0.837 Sm,2n+2 -

0.483Sm,2n+3

(3.82b)

As an example, we will now consider the D4 wavelet decomposition of the signal Xi = (1,0,0,0,0,0,0,0) which we input into the multiresolution algorithm as So,w It has N = 8 components, so we iterate the transform decomposition algorithm three times. After the first iteration the transform vector becomes (0.483, 0.000, 0.000,0.224, -0.129,0.000,0.000,0.837) containing four approximation components followed by four detailed coefficients, i.e. (SI,O, SI,1, SI,2, SI,3, TI,o, TI,1' T I ,2, T I ,3). Remember that we employ wraparound to deal with the edges of the signal. When either of the filter pairs is placed at 2n = 6, the 2n + 2 and 2n + 3 locations are not contained within the signal vector (i = 0, 1, ... , N - 1). Thus, the last two filter coefficients of 0.224 and -0.129 (approximation) or 0.837 and -0.483 (detail) are placed back at the start of the signal vector, the first of each coinciding with the unit value at the beginning of the signal, hence producing respectively the values 0.224 or 0.837. These values are placed at location SI3,- and T I3, respectively. The next iteration is performed on the remaining four approximation components SLO, SI , j, S12' , SI3 , producing two approximation components followed by two detailed wavelet components. These are added to the four detailed coefficients obtained at the previous iteration to give the vector (0.204, 0.296, -0.171, 0.354, -0.129, 0.000, 0.000, 0.837). The third and final iteration produces (0.354, -0.065, -0.171, 0.354, -0.129, 0.000, 0.000, 0.837). This vector has the form (S30, T 30 I, T IO, ' -, , , T 20 ,., T 2,. T I I, T 12 , T I3 ), where the signal mean can be found from the first coefficient, i.e. S3:0/23/2 = 0:354/2 3/ 2 = 0.125. We know that the Daubechies D4 wavelet has two vanishing moments, thus it can suppress both constant signals and linear signals. This means that for an infinitely long constant or linear signal all the coefficients are zero. However, end effects occur if the signal is of finite length. We will use the linear ramp signal (0, 1,2, ... ,31), shown in figure 3.l7(a), as an example. The first iteration produces the transform vector w(l) =

(SI,O, SI,I, SI,2' ... ,SI,IS' TI,o, TI,I, T I ,2' ... , TI,IS)

= (0.897,3.725,6.553,9.382,12.21,15.039, 17.867,20.696,23.524,26.352,29.181,32.009, 34.838,37.666,40.495,40.291,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0, -11.314)

110

The discrete wavelet transform 40,------r-------,

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24

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u

8

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10 100 0

I~.

0 uu u U u -100

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(e)

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o -100 0

10

nn

u

U U

8

o hn -lin _no

u

16

24

32

-10 0 0

8

u

16

24

32

(h)

(g)

Figure 3.17. The multiresolution decomposition of a ramp signal using Daubechies wavelets. (a) Original signal. (b) Decomposition I (04 wavelet). (c) Decomposition 2 (04 wavelet). (d) Decomposition 3 (D4 wavelet). (e) Decomposition 4 (D4 wavelet). (f) Decomposition 5, full decomposition (D4 wavelet). (g) Full decomposition using 06 wavelets. (h) Full decomposition using 012 wavelets. (Note change in scale of the vertical axes.)

where all the detail coefficients at the lowest scale, T I 10 are zero except for the end coefficient T[ 15' This nonzero coefficient is caused by the signal wraparound where the wavelet filter vector (l/V2)b k placed at the end of the signal encounters the sequence 30, 31, 0, 1. We can see that the computation for the corresponding detail component is T I15 ,

=

-0.129 x 30 - 0.224 x 31

+ 0.837

x 0 - 0.483 x 1 = -11.314

A histogram of the coefficients is shown in figure 3.17(b). Notice also the edge effect in the approximation coefficients where they all increase linearly in value except the last

Dauhechies wavelets

111

one due to the wraparound with the scaling filter. The second iteration produces the transform vector W(2) =

(3.804,11.804,19.804,27.804,35.804,43.80,52.196, 52.981, 0,0,0,0,0,0, 1.464, -15.321, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,11.314)

°

where we see that the next set of detail coefficients (T2, to T 2,7) are zero except those at the end (figure 3.17(c)). There are two nonzero coefficients now due to the edge effect in the smooth coefficients at the previous scale. If we repeat the decomposition to the largest scale we get the fully decomposed signal W(S) =

(87.681, -31.460,6.405, -29.530,0,0,3.624, -21.149, 0, 0, 0, 0, 0, 0, 1.464, - 15.321 , 0, 0, 0, 0, 0,0,0,0,0,0,0,0,0,0,0,11.314)

As we get 'edge' coefficients at all scales, we get no zero coefficients at scales m = 4 and m = 5 where there are respectively only two (T4 ,o, T4 ,1) and one (Ts,o) coefficient. The full D2 (Haar) decomposition of the ramp signal is W(S) =

(87.7, -45.3, -16.0, -16.0, -5.7, -5.7, -5.7, -5.7, -2.0, -2.0, -2.0, -2.0, -2.0, -2.0, -2.0, -2.0, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7, -0.7)

Notice that there are no zero coefficients, as we would expect, since the Haar wavelet has only one vanishing moment and therefore does not suppress linear signals. On the other hand, all Daubechies wavelets with more than four scaling coefficients have more than two vanishing moments and all can therefore suppress linear signals. For example, the D6 wavelet can suppress mean, linear and quadratic signals. The D6 decomposition of the ramp signal, which employs the scaling coefficients (CO,Cl,C2,C3,C4,CS) = (0.470,1.141,0.650,-0.191,-0.121,0.0498), is W(S) =

(87.681, -2.259, -29.201,25.344,0.149,9.909, -22.224,6.2484,0,0,0,0,0,4.22, -17.636,4.766, 0,0,0,0,0,0,0,0,0,0,0,0,0,0,-15.175,3.861)

Notice that this time the two end coefficients are nonzero. This is because the filter is longer. It now has six components and hence two end locations of the filter on the discrete signal overlap and wraparound to the beginning of the signal. A D8 decomposition would generate three end coefficients, a D 1 would generate four and so on. The full decompositions of the ramp signal using both the D6 and D 12 wavelet are shown in figures 3.17(f) and (g) for comparison. The more scaling coefficients the wavelet has, the higher the number of its vanishing moments and hence the higher the degree of polynomial it can suppress. However, the more scaling coefficients that a wavelet has, the larger its support length and hence the less compact it becomes.

°

112

The discrete wavelet transform

This makes it less localized in the time domain and hence less able to isolate singularities in the signal (including edge effects). This is the trade-off which must be considered when selecting the best wavelet for the data analysis task. 3.5.1

Filtering

Let us revisit the filtering process described in the last section and shown in figure 3.16. In signal processing, the approximation coefficients at resolution m, Sm,ll' are convolved with the lowpass filter. This done by moving the filter along the signal one step at a time. The approximation coefficients are then subsampled (or downsampled) where every second value is chosen to give the approximation coefficient vector at scale m + 1. The approximation coefficients at resolution m, Sm,/l' are also convolved with the highpass filter and subsampled in the same way to give the detail signal coefficients at scale m + 1. The detail components T,11 + 1,II are kept and the approximation components Sm+ 1,/1 are again passed through the lowpass and highpass filters to give components T m+ 2,/1 and Sm + 2,11' This process is repeated over all scales to give the full decomposition of the signal. Each step in the decomposition filtering process is shown schematically in figure 3.18(a). The sequences (1/ V2)Ck and (1/ V2)b k are contained respectively within the lowpass and highpass filter vectors used within the wavelet decomposition algorithm. The filter coefficients are sometimes referred to as 'taps'. For signal reconstruction (equation (3.42)), the filtering process is simply reversed, whereby the components at the larger scales are fed back through the filters.

Sm,n

T m+2,n

(a)

Sm+2,n

--~ 91-.-----.. . s m+l,n _rlfl gl-.-----.... sm,n ~ ~

(b)

Figure 3.18. Schematic of the signal filtering. (a) Schematic diagram of the filtering of the approximation coefficients to produce the approximation and detail coefficients at successive scales. The subsample symbol means take every second value of the filtered signal. HP and LP are, respectively, the highpass and lowpass filters. In practice the filter is moved along one location at a time on the signal, hence filtering plus subsampling corresponds to skipping every second value as shown in figure 3.16. (b) Schematic diagram of the filtering of the approximation and detail coefficients to produce the approximation coefficients at successively smaller scales. The upsample symbol means add a zero between every second value of the input vector. The coefficients in the HP' and LP' filters are in reverse order to their counterparts used in the decomposition shown in (a). Figures (a) and (b) represent a subband coding scheme.

®

®

Dauhechies wavelets

113

Figure 3.19. Filtering of the signal: reconstruction. The smooth and detail coefficients at scale index are passed back through the filters to reconstruct the smooth coefficients at the higher resolution Sill _ 1,11' This is done by sliding the lowpass and highpass filters along their respective coefficient sequence with zeros inserted between the coefficients, (Refer back to equation (3,42) where, remember, 'n' is the coefficient location index at the lower scale and 'k' is the index at the higher scale.)

111

This is shown in figure 3.18(b). The approximation and detail components at scale index m + 1 are first upsampled (zeros are inserted between their values) and then passed through the lowpass and highpass filters respectively. This time, however, the filter coefficients are reversed in order and are shifted back along the signal. An example of the reconstruction filtering is shown in figure 3.19 where the component Sm,S is found from the sequences Sm+ l,n and ~n+ 1,11" Note that the leading (righthand) filter coefficient defines the location of the Sm,n coefficient. Hence the computation of the Sm,O' Sm,l and Sm,2 coefficients involve components of the filter vector lying off the left-hand end of the upsampled signal at scale m + 1. These are simply wrapped back around to the right-hand end of the signal. We can produce a discrete approximation of the scaling function at successive scales if we set all the values of the transform vector to zero except the first one and pass this vector repeatedly back up through the lowpass filter. This is illustrated in the sequence of plots contained in figure 3.20. Figure 3.20(a) shows the initial transform vector, 64 components long, with only the first component set to unity, the rest set to zero, i.e. (1,0,0,0, ... ,0). After the first iteration, four components can be seen. These are equal to (col V2, cd V2, c21 V2, c31 V2, 0, 0, 0, ... 0) (notice that this is the decomposition filter). The next iteration (figure 3.20(c)) produces 10 nonzero values in the transform vector, the next, 22 nonzero values, and so on. The transform vector components take on the appearance of the scaling function quite quickly. What we are actually doing is reconstructing SO.n based on the initial vector (1,0,0,0, ... ,0) equal to the following transform vectors: (SI,O,Sl,l,SI,2"'" Tl,o, Tl,l' T I,2"") for figure 3.20(b); (S2,O' S2,1, S2,2"'" SI,O' Sl,l, SI,2,'" ,TI,o, TI,I, T I,2,"') for figure 3.20(c); (S3,O, S3,l, S3,2'" ., S2,O, S2,1, S2,2" .. , SI,O, Sl,l' St,2" .. , Tl,o, Tl,l' T l ,2"") for figure 3.l8(d), and so on. Notice that, as we employ wraparound to deal with the edge effects, the final signal shown in figure 3.20(g) is a constant value equal to the signal mean. Figure 3.20(h) shows a high-resolution

114

The discrete wavelet transform

(a)

I

I

I

I

I

I

I

I

I

I

I

I

I

o

(b)

(c)

(d)

c;li1I'I'lTllllllllll~ (e)

(f)

(g)

I

I

I

I

I

(h)

Figure 3.20. The reconstruction of a Daubechies D4 scaling function. (a) The transform vector (1, 0, 0,0,0,0, ...). (b) One iteration, i.e. S1.0 = 1. (c) Two iterations, i.e. S2,O = 1. (d) Three iterations, i.e. S3,O = 1. (e) Four iterations, i.e. S4.0 = 1. (f) Five iterations, i.e. S5.0 = 1. (g) Six iterations of the transform vector with 64 components, i.e. S6.0 = 1. This results in a constant signal equal to the mean due to wraparound. (h) After nine iterations of a transform vector (1, 0, 0, 0, 0, ...) with 4096 components.

Dauhechies wavelets

115

representation of a Daubechies scaling function generated by iterating nine times with an initial transform vector containing 4096 components. You can verify that the constant signal of figure 3.20(g) stems from the wraparound employed when reconstructing from the full decomposition wavelet vector with SM,O set to unity. In this case, for the first iteration, adding a zero to get (SM,O' O) and running the reconstruction filter over both components leads to two approximation coefficients (SM -1,0, SM -I,d equal to ((co + c2)1 V2, (CI + c3)1 V2) = (0.707,0.707). Adding zeros to this vector and iterating again produces four approximation coefficients (0.5,0.5,0.5,0.5) and so on until we obtain the full reconstruction after the sixth iteration which contains constant values of So 11 equal to (11 V2/ = 0.125. We can see this is true for the general case as, from equation (3.74) with m = 0 together with equation (3.21), it can be shown that

L keven

Ck

=

L

Ck

=

1

(3.83)

kodd

Hence, the combined effect of wraparound plus the zeros inserted in the reconstruction process causes SM,O to be multiplied by both (1/V2) 2..:keven Ck and (1/V2) 2..:kodd Ck in turn to find SM _ 1.0 and SM _ 1.1, which both have the same value simply equal to SM,o/V2. This transform vector now has two constant values (the rest zero). The next iteration produces four constant nonzero values and so on. It is easy to prove to yourself that using Daubechies wavelets with different numbers of nonzero scaling coefficients, N b will always result in constant valued transform vector components. If we repeat the reconstruction filtering, this time setting the first detail coefficient to unity, we can generate an approximation to a wavelet. Notice that, in doing so, only the first iteration of the reconstruction algorithm requires the reordered scaling coefficient sequence for the wavelet, h lo and subsequent reconstruction filtering then uses the scaling coefficient sequence Cle Figure 3.21 shows the wavelet reconstruction process. From these examples, where a single component was set to unity, we can see that the transform vector components, when passed back through the filters during the reconstruction process, increasingly spread their influence (information) over the transform vector in terms of either discrete scaling functions (if the original component was an approximation coefficient) or discrete wavelet function (if the original component was a detail coefficient). Figures 3.20 and 3.21 illustrate the discrete approximations of the wavelet and scaling functions at various scales implicit within the multiresolution algorithm. The discrete scaling and wavelet functions at scale 1 are, in fact, the filter coefficient vectors (11 V2)h k and (11 V2)Ck respectively.

3.5.2

Symmlets and Coiflets

Looking back at figure 3.15 we can see that Daubechies wavelets are quite asymmetric. To improve symmetry while retaining simplicity, Daubechies proposed Symmlets as a modification to her original wavelets (also spelled symlets). Symmetric wavelet filters are desirable in some applications, e.g. image coding, as it is argued that our visual system is more tolerant of symmetric errors. In addition, it makes it easier to deal with image boundaries. Daubechies came up with Symmlets by 'juggling with their phase' during their construction. Figure 3.22 contains two examples of

116

The discrete wavelet transform

-

~b I

I

I

I

I

I

I

I

I

(a) -

rl

~ I

T> I

I

I

I

I

I

I

I

(b)

(c)

(d)

Figure 3.21. The reconstruction of a Daubechies D4 wavelet. (a) One iteration with only Tl. o initially set equal to 1. (b) Two iterations with only T 2 .0 initially set equal to 1. (c) Three iterations with only T 3.0 initially set equal to 1. (d) Four iterations with only T 4 •0 initially set equal to 1.

Symmlets together with their scaling functions. They have N k /2 - 1 vanishing moments, support length N k - 1 and filter length N k . However, true symmetry (or antisymmetry) cannot be achieved for orthonormal wavelet bases with compact support with one exception: the Haar wavelet which is antisymmetric. Coiflets are

(a)

(b)

Figure 3.22. Symmlets and their associated scaling functions. (a) S6. (b) S IO. (The scaling functions are shown dotted.)

Translation invariance

(a)

(b)

117

(c)

Figure 3.23. Coiflets and their associated scaling functions. (a) C6. (b) C12. (c) C18. (Scaling functions are shown dotted in figures.)

another wavelet family found by Daubechies. They are also nearly symmetrical and have vanishing moments for both the scaling function and wavelet: the wavelet has N k /3 moments equal to zero and the scaling function has N;)3 - 1 moments equal to zero. They have support length N k - 1 and filter length Nk.. Three examples of Coiflet wavelets are shown in figure 3.23. The number of coefficients, N lo used to define Coiflets increase in multiples of six.

3.6

Translation invariance

Translation invariance (or sh(lt invariance) is an important property associated with some wavelet transforms but not others. It simply means that if you shift along the

signal all your transform coefficients simply move along by the same amount. However, for the dyadic grid structure of the discrete wavelet transform this is clearly not the case: only if you shift along by the grid spacing at that scale do the coefficients become translation invariant at that scale and below. Even for the discretization of the continuous wavelet transform, the transform values are translation invariant only if shifted by any integer multiple of the discrete time steps. This is illustrated in figure 3.24 where a simple sinusoidal signal is decomposed using the discrete orthonormal Haar wavelet and a continuous Mexican hat wavelet discretized at each time step. The original signal shown on the left-hand side of figure 3.24(a) is composed of 10 cycles of a sinusoid made up of 1024 discrete data points. The middle plot in figure 3.24(a) contains the Haar transform coefficients of the signal plotted in their dyadic grid formation. The right-hand plot contains the continuous wavelet transform using the Mexican hat wavelet. The coarse structure of the dyadic grid is evident when compared with the smooth, high resolution Mexican hat transform plot. We will assume the signal continues indefinitely and shift the signal window along to see what happens to the transform plots. The variation in the coefficients of the Haar transform with displacement of the time window along the signal can be seen by looking down the central plots in the figure. Shifting the window forward (or back) along the signal by a scale of 2111 produces the same coefficients in the transform plot at and below that scale shifted in time by 2111 • This can be observed in figures 3.24(b)-(d) where the size of the shift is indicated in the transform plot

118

The discrete wavelet transform

m~ I

Haar wavelet coefficient lot

-r-"::'::===:::"..,

scale index

Mexican Hat wavelet transform lot

m=5-

(a)

r

1 0. a,.,..,...~~......

signal mean component

..1'------...,.

(b)

(e)

L (d)

(e) Figure 3.24. Translation im·ariance. (a) Original signal, 1024 data points. (b) Shirt by 64 data points. (c) Shirt by 128 data points. (d) Shirt by 256 data points. (e) Shift by 1/8 cycle.

of figure 3.24(a). The coefficient level at and below which the coefficients remain the same as the original is indicated by the arrows at the left-hand side of each transform plot. Shifting the signal by an arbitray scale (not a power of two) leads to a completely different set of coefficients, as can be seen in figure 3.24(e), where the signal is shifted by an eighth ofa sinusoidal cycle (12.8 data points). The translation invariance of the

Biorthogonal wavelets

119

continuous transform plots is obvious by looking down the right-hand figures where, for any arbitrary shift in the signal, the wavelet transform values are simply translated with the shift. We can make a translation invariant version of the discrete wavelet transform. Known by a variety of names including the redundant, stationary, translation invariant, maximal overlap or non-decimated wavelet transform, we simply skip the subsampling part of the filtering process described in section 3.5.1. This results in the same number of wavelet coefficients generated at each step, equal to the number of signal components N. The decomposition is the same as that shown in figure 3.16 except that every value is retained as the filter moves one step at a time along the signal. In addition, the filters have to be stretched through the addition of zeros between coefficients, hence this algorithm is called the 'lJ trow,,' algorithm from the French 'with holes'. An average basis inverse can be performed which gives the average of all possible discrete wavelet transform reconstructions over all possible choices of time origin in the signal. This is sometimes useful for statistical applications including denoising. In addition to being translation invariant, the redundant discrete wavelet transform is extendable to signals of arbitrary length. We do not consider the redundant discrete wavelet transform herein, rather the reader is referred elsewhere at the end of this chapter.

3.7

Biorthogonal wavelets

For certain applications real, symmetric wavelets are required. One way to obtain symmetric wavelets is to construct two sets of biorthogonal wavelets: 'l/Jm ,11 and its dual, 'l/Jm,/l" One set is used to decompose the signal and the other to reconstruct it. For example, we can decompose the signal using '1jJ/11,11 as follows: (3.84) and perform the inverse transform using 'l/Jm ,11' 00

00

L L

x(t) =

T,n,I1;j;/11,I1(t)

(3.85)

m=-oo 11=-00

Alternatively, we can decompose the signal using (3.86) and reconstruct using 00

x(t) =

L /11 = -CXJ

(3.87) 11 = -CXJ

120

The discrete wavelet transform

Biorthogonal wavelets satisfy the biorthogonality condition: oo

f

-00

~

'l/Jm,1I (t)'l/Jm',II' (t) dt

=

{I

0

if m = m' and n = n' , otherwIse

(3.88)

Using biorthogonal wavelets allows us to have perfectly symmetric and antisymmetric wavelets. Further, they allow certain desirable properties to be incorporated separately within - the decomposition wavelet and the reconstruction wavelet. For example, 'l/Jm ,11 and 'ljJm ,11 can have - different numbers of vanishing moments. If'ljJm,1I. has more vanishing moments than 'l/Jm,1' then decomposition using 'l/Jm ,11 suppresses higher order poly, nomials and aids data compression. Reconstruction with the wavelets 'l/Jm,1I with fewer vanishing moments leads to a smoother reconstruction. This can sometimes be a useful property, for example in image processing. Figure 3.25 shows three examples of compactly supported biorthogonal spline wavelets and their duals commonly used in practice, together with their associated scaling equations. '"'"

(a)

(b)

Figure 3.25. Biorthogonal spline wavelets. (a) Biorthogonal (1,5) spline wavelets: ¢(t), 'Ij;(t) (left) and ¢(t), {fJ(t) (right). (b) Biorthogonal (2,4) spline wavelets: ¢(t), 'Ij;(t) (left) and ¢(t), {fJ(t) (right). (c) Biorthogonal (3,7) spline wavelets: ¢(t), 'Ij;(t) (left) and ¢(t), {fJ(t) (right). (Scaling functions shown dotted.)

Two-dimensional wavelet transforms

3.8

121

Two-dimensional wavelet transforms

In many applications the data set is in the form of a two-dimensional array, for

example the heights of a machined surface or natural topography, or perhaps the intensities of an array of pixels making up an image. We may want to perform a wavelet decomposition of such arrays either to compress the data or to carry out a wavelet-based parametric characterization of the data. To perform a discrete wavelet decomposition of a two-dimensional array we must use two-dimensional discrete wavelet transforms. We can simply generate these from tensor products of their one-dimensional orthonormal counterparts. The most common (and simplest) arrangement is to use the same scaling in the horizontal and vertical directions. These give square transforms. (Other forms are possible, e.g. rectangular transforms where the horizontal and vertical scaling vary independently and also twodimensional transforms which are not simply tensor products. These are outside the scope of this text.) The two-dimensional Haar scaling and wavelet functions are: two-dimensional scaling function (3.89a) two-dimensional horizontal wavelet (3.89b) two-dimensional vertical wavelet (3.89c) two-dimensional diagonal wavelet

'l//(t 1, t2)

=

'IjJ(td'IjJ(t2)

(3.89d)

Remember that in both the last chapter and this we have been using t as our independent variable, either temporal or (less common) spatial. In this section tl and t2 represent spatial coordinates. The multiresolution decomposition of the two-dimensional coefficient matrices can be expressed as (3.90a) (3.90b) (3.90c) (3.90d) where k I, k 2 are the scaling coefficient indices and nl, n2 are the location indices at scale m + 1. (Compare with the one-dimensional case of equations (3.36) and (3.37).) We can simply use discrete versions of the two-dimensional wavelets at scale index 1 to perform

122

The discrete wavelet tramform

Ca)

(b)

Co)

Figure 3.26. The two-dimensional discrete Haar wa\'elet at scale index 1. (a) Scaling function. (b) Horizontal wavelet. (c) Vertical wavelet. (d) Diagonal wavelet.

the multiresolution analysis of the array. The discrete two-dimensional Haar scaling and wavelet functions in matrix form at scale index m = I are scaling function

vertical wavelet

J

H: :] horizontal wavelet

J

~[I

-I]

2 1 -I

diagonal wavelet

J 1[

1

2 -I

-: ]

As these are at scale index I, they are simply tensor products of the scaling and wavelet filter coefficients Ck/ J2 and bd J2. These discrete functions are shown schematically in figure 3.26. Note the! factor before each matrix is simply the square of the 1/J2 factor preceding the corresponding one-dimensional functions. For discrete scaling and wavelet functions at larger scales, this normalization factor becomes 1/211I • These four 2 x 2 matrices are required for the Haar wavelet decomposition of the two-dimensional array. The general idea of the two-dimensional wavelet decomposition is shown schematically in figure 3.27. The original input array is represented by Xo defined at scale index 111 = O. As with the one-dimensional case, its components are input as the approximation coefficients at scale index 0, i.e. the matrix SQ. After the first wavelet decomposition, a decomposition array is formed at scale index I which is split into four distinct submatrices: the vertical detailed

Two-dimensional wavelet transforms

123

ORIGINAL ARRAY

2M _2 M components

--------------------.

,, ,,

scale index 1

8\

; ,,

,-------------------_., ...------------------r--------------------: , ,, , ,, ,

,---------------------

scale index 2

, , , ,, ,

r---------. r---------.

.'S ,

., v ::T

" 2::

.'

2,,

:_--------~ :_--------~ r---------. r---------.

:'T 2:::: T 2:: h

d

:._-------~ :._-------~ r------------------

, ,, , ,,

._--------------------

etc. Figure 3.27. Schemalic diagram oflhe malrix manipulalion required to perform wal'elel decomposition on a I\\'o-dimensional grid.

components TY, the horizontal detailed components T~I, the diagonal detailed components ~' and the remaining approximation components 8 1, The decomposition array is the same size as the original array, As with the discrete wavelet decomposition of one-dimensional signals, the detail coefficients are subsequently left untouched and the next iteration further decomposes only the approximation components in the

124

The discrete wavelet transform 0.8

0.6

0.'

0.2 00

2

1

1 • 2

3z9aOCi0&<9

°

Figure 3.28. A two-dimensional signal containing a single spike.

submatrix St. This results in detail coefficients contained within submatrices T 2, T~ and at scale index 2 and approximation coefficients within submatrix 8 2 , This procedure may be repeated M times for a 2 M x 2 M array to get a number of coefficient T"/II, T"III an d T"III 0 fsize ' 2M-/II x 2M-/II , were h . su b matnces 111 = 1 , 2 1 " "M . Let us look at a very simple example, a matrix with a single nonzero component

Tl

Xo =

o

0

o

0

o

1

o

0

o

0

o

0

o

0

o

0

(3.91)

This array could represent a single pixel turned on in a display or perhaps a spike protruding from a flat surface. We can visualize it in figure 3.28. The two-dimensional Haar multi resolution decomposition of this array is carried out by scanning over it with each of the four discrete wavelet matrices in turn. This is shown schematically in figure 3.29. On the first iteration (from scale 111 = 0 to 1), the scaling function 2 x 2 matrix is scanned over the input array. We require four of these matrices to cover this 4 x 4 array. The components of the original array are multiplied in turn by the scaling function array to give the resulting matrix product St which is placed in the top right-hand quadrant of the first iteration matrix. Similarly, the vertical wavelet 2 x 2 wavelet matrix is scanned over the array and each value is placed in the top right-hand quadrant of the first iteration array. This is illustrated in the T]' matrix marked on the right-hand of the figure. The procedure is repeated for the horizontal and diagonal wavelet components, placing the products in the bottom left-hand and bottom right-hand quadrants respectively. The normalization factor of 1/2 has been left out of the matrix manipulation until the end to aid clarity. Hence, after the first iteration, the resulting wavelet transform matrix is

W(l)

=

0.5 0, -0.5 0 n! 0 .. ........... n.5 0 0.5 n 0 0

"

0 0

,

(3.92)

0 0

The next iteration uses only the approximation components in the top left-hand quadrant (shown within the dotted box). The four component values of this array

Two-dimensional wavelet transforms

125

original input array components shown shaded

1 2

-0

D 0.5

0 -0.5

0

0

-0.5

0

0

0

0 0 0

0.5

wavelet transform array after the first iteration

Wi

0

0 0

Figure 3.29. Schematic diagram of the matrix manipulation required to perform the Haar wal'elet decomposition of the spike signal. Components of the discrete scaling and wavelet matrices are shown in bold, components of the original discrete signal are shown shaded.

are then interrogated by each of the scaling and wavelet matrices produces the second iteration matrix

W(l)

=

0.25 0.25 -0.5

o

-0.5

o

o

o

o

0.5

o

o

o

o

0.25 0.25

In

turn. This

(3.93)

This is the second and last iteration we can perform in the decomposition of the 4 x 4 (i.e. M = 2) array. This matrix is made up ofTf, T~I, ~/, T 2, T~, T~ and, in the top lefthand corner, the approximation component 52 which is related to the mean of the original array. Notice that, at each iteration, the resulting matrix has the same energy as the original matrix where the energy is the sum of the squared components

126

The discrete wavelet transform 10 5

o

O 2

4

6

(d)

Figure 3.30. An 8 x 8 'step' array together with its associated Haar decompositions. (a) Original step. (b) Iteration I. (c) Iteration 2. (d) Iteration 3. Note that black dots have been added to lhe non-zero coefficients in plots (c) and (d) to aid clarity,

of the matrix, i.e. 2M _I

2,11 _ I 2,1(_1

E=

wt

L L

=

(XO,i,j)'

j=O

i=O

L

(3,94)

;=0

l

where XO,i,J and ) are, respectively, the elements of the input and wavelet decomposition matrices located on row i and column j. E is equal to I for the array considered above, We can easily verify that E is equal to I for the intermediate decomposition matrix as well as the full decomposition matrix. You can confirm this conservation of energy property in all the examples which follow. Let us look at another simple example, the step shown in figure 3.30(a) and given in matrix form as

2 2 2 2 Xo =

2 2 2 2 2

2 2 2 2 2 2

2 2 2 2 2 2

2 2 2

2 2 2 2

2 2

2 2 2 2

2 2 2 2

2 2

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

(3,95)

This 8 x 8 array can be decomposed through three iterations using the Haar wavelet as follows: first decomposition

W(l)

4

4

4

4

0

0

0

0

4

4

4

4

0

0

0

0

3

3

3

3 0

0

0

0

2

2 2

2 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

I

I

I

I

0

0

0

0

0

0

0

0

0

0

0

0

(3,96a)

Two-dimensional wavelet transforms

second decomposition

127

W(2)

third (full) decomposition

8

8

0

0

0

0

0

0

5

5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1 1 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

1

1

1 0

0

0

0

0

0

0

0

0

0

0

W(3)

:T~----:, __ .. ..

. __ .,

_-

".,,,.,,,.,,,

,

11

'

:Tr"'~'"

"

0

(3.96b)

.. .. .. "

"

"

13

0

0

0

0

0

0

0

~):

0

()

()

()

(}

0

0

0 0 0 0 0 0 0 0 1-- --- 1 a a 0 0 0 0 ~ 0 0 0 a 0 0 0 0 "'"'''''''''''''' """.

..... .. , ... ,

~O

0

0

0

0

0

0

0

' -- -:-

1

1

I

I

0

0

0

0

0.. ... 0.........0 ....... 0 0

0

0

a

"

;~ T f1l

..............

,

"

,

_

(3.96c)

Note that, after the three iterations, only the approximation component and the components in regions corresponding to the horizontal wavelet decomposition (labelled TIl, T~ and T~) contain nonzero elements. This is because the original array only contains a single horizontal feature-the horizontal discontinuity or step. Another thing to notice is that the 64 nonzero elements of the original array have been transformed into only eight nonzero elements, i.e. a compression of the data has taken place. The original arrays, together with the three levels of decomposition, are shown as histograms in figure 3.30. We can now use the detail coefficients to reconstruct the array at different scales. This is shown in figure 3.31 and is performed using the detail coefficients at each scale of interest. In the same way as we did for the one-dimensional signals earlier in section 3.4, discrete detail arrays D;~l' D~;l and D~l can be reconstructed at the scale of the input array using each of the detail coefficients, T~l' T~;l and T~l' A combined detail can be found at scale index m, given by V h Dd111+ D111 D 111= D 111+

(3.97)

As with the one-dimensional case, the array can be represented as a sum of the discrete details plus an array mean. M

Xo =X M

+L D

I11

(3.98)

111=1

where X M is the smooth version of the input array at the largest scale index, M, expressed at the scale of the input array m = O. For example, to reconstruct the

128

The discrete wavelet transform

13r·olro····ol 6"'6"'6"'01 !j"!r:o=ilo 010 0 0 01 ro::·~:o:iro::::o1. 0 0 0 01

scale I vertical detail coefficients

II

1110 0110 0 0 01 o 0 0 0110 0 0 01 o 0 0 0110 0 0 O! 1 1 1 1110 0 0 01 o 0 0..........•............................ 0110 0 0 01 •................

:

:: : : : ............................................................

\

scale I diagonal detail coefficients

scale I horizontal detail coefficients

(a)

000 0 0 0 0 0 000 0 0 0 0 0 / 000 0 0 0 0 0 o 0 0.9,0 0 0 0 0.510.5 0.5 0.5 0.5 ..=.9j-0.5-0.5-0.5-0.5 o 0 0 0 0 0 o 0 0 0 0 0

detail coefficients submatrix

Th1

/l detail coefficient

e

detail matrix

D~

1[1[1 1]] 2 -1 -1 1

\

Haar wavelet at scale index m= I (b)

Figure 3.31. Schematic diagram of the matrix manipulation required to derive the detail matrices at scale index m = 1. (a) The three detail coefficient submatrices at scale index I for the step. (b) Computing the horizontal discrete detail from the detail coefficients at scale index I.

detail array components at the smallest scale (m = 1), each corresponding component in each of the three coefficient submatrices, TIt, Tf and Tr, are used in conjunction with the corresponding Haar wavelets (horizontal, diagonal and vertical) to produce the array details at this level: D7, Df and DY. The m = 1 submatrices are shown in

Two-dimensional wavelet transforms

129

figure 3.31(a). These three 4 x 4 coefficient submatrices are each expanded into an 8 x 8 array corresponding to the horizontal, diagonal and vertical details of the original array using the corresponding discrete wavelets. This is shown schematically in figure 3.31 (b) for the horizontal detail at scale index 1, i.e. D /I. The expansion of the transform coefficients through the discrete Haar wavelet into the detailed components is shown explicitly for two of the T1 components in the figure. We can see that the detail matrices D~l and DY will have all elements equal to zero as both Tf and TY contain only zero elements. Hence the combined detail component matrix is

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

o o o o

0.5 -0.5

0.5 -0.5

0.5 -0.5

0.5 -0.5

0.5 -0.5

0.5 -0.5

0.5 -0.5

0.5 -0.5

o o

o o

o o

o o

o o

o o

o o

o o (3.99)

Similarly, we can get the detail matrices at scale index 2, Dt D~ and D 2, using the scale 2 coefficients, respectively T~, T1 and T 2 and the discrete wavelet at scale index 2. This is shown for D~ in figure 3.32. Note that the normalization factor at scale 2 is 1/2 2 • The scale 3 coefficient submatrices, and T), are simply [3], [0] and [0] respectively. It is easily seen that this gives the detail at the largest wavelet

Tt T1

00000000 00000000 00000000 00000000 j···6::25"···iEis···O:·;XS····6:is·j 0.25 0.25 0.25 0.25 : 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 -0.25 -0.25 -0.2 - 0.25 - 0.25 - 0.25 - 0.25 - 0.25 -0.25 -0.25 -0.2 - 0.25 - 0.25 - 0.25 - 0.25

i-

1 1 1 1 1 1 1 1 -1 -1 -1 -1 ..~./

detail coefficient

-1-1-1-1

/

horizantal Haar wavelet at scale index 2

Figure 3.32. Schematic diagram of the matrix manipulation required to compute the horizontal detail at scale index m = 2.

130

The discrete wavelet tramlol'm

scale equal to

0]

=

0.375 0.375 0.375 0.375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0.375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0.375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0.375 -0.375 -0.375 -0.375 -0.375

0.375 0.375 0.375 0.375 -0.375 -0.375 -0.375 -0.375

0375 0.375 0.375 0375 -0.375 -0.375 -0.375 -0.375 (3.100)

where each component is either 3/2 3 or - 3/2 3 . In fact, this matrix is simply a single large horizontal discrete Haar wavelet scaled by 3/2 3 . The approximation coefficient (=13) leads to a matrix of values all equal to the mean of the original array (=13/2 3 ), i.e.

X,=

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

1.625 1.625 1.625 1.625 1.625 1.625 1.625 1.625

(3.101)

Figure 3.33 shows the reconstructions at each scale together with the mean component. From the figure we can see that if we add them all together, D 1 + D 2 + D 3 + X 3, we get back the original input array Xo . 2

(.)

(bJ

(d)

(oj

(Q

Figure 3.33. The delails oflhe step array. (a) Original step array Xo. (b) D 1 . (e) D 2 . (d) OJ, (e) XJ. (f) Sum of (b), (c), (d) and (e) giving original signal (a).

TlVo-dimel1siona/ll!Clvelel transforms

(b)

(a)

131

(e)

Figure 3.34. An image and its first Iwo decomposition matrices. (a) Original array So- (b) Transform array V{I) containing submatrices $1, 1'], 1'/; and (c) Transform array V(2) containing submatrices S 2, T "J, 'r"), .~I II, 'r"2, 'r"zan d ~J 12'

T'l.

100 50

'.

~'I::

~.

•

•

••

:'''~ 50

100

100

100

100 50

o0....-5-0.........100~

50

100

50

100

50

100

(a:

50

50

100

100

50

50

100

50

100

50

100

(e

100

100

50

50

50

100

(l)

Figure 3.35. Haar decomposition of a surface data set. (a) Original data sel 128 x 128 array of surface heights taken from a river bed. (b) Scale index m = I discrete detail 0]. (c) Scale index 11I= 2 discrete detail 2 _ (d) Scale index IJI = 3 discrete detail 3 _ (e) Scale index /1/ = 4 discrete detail 4 _ (f) Scale index 11/ = 5 discrete detail 05' (g) Scale index 11/ = 6 discrete detail Dt>. (h) Scale index 11/ = 7 discrete detail D7 . (i) Sum or first three discrete details 0 1 + D2 + OJ (= (b) + (c) + (d)). (j) Scale index 11I=3 discrete approximation XJ = Xo - (OJ + D2 + DJ ) (=(a) - (b) - (c) - (d)). (Greyscale used in all images: maximum = white, minimum = black.)

°

°

°

132

The discrete wavelet tramlol'm

Figures 3.34 and 3.35 show examples of much larger data sets, both using the Haar wavelet transform. Figure 3.34 shows an image ('Lena') together with its first two decomposition matrices, where the approximation and detail coefficient submatrices can be clearly seen. Figure 3.35 shows an example of the Haar decomposition of a more irregular array. The 128 x 128 array shown in figure 3.35(a) contains the heights of a measured rough surface. The details of this array, at scale indices 111 = 1-7, are shown in figures 3.35(b)-(h). Figure 3.35(i) contains the summation of the first three details and figure 3.35(j) shows the resulting approximation at scale index m = 3 when these details are subtracted from the original array. The blocky nature of the Haar decomposition is noticeable from the plots. The two-dimensional Haar wavelet is very simple in its form and, as with their one-dimensional counterparts, there are more complex forms of Daubechies wavelets and, of course, many other wavelet families to choose from. Some examples of these wavelets are given in figure 3.36. Using these wavelets will result in overlap of the wavelet at the array edge and therefore would require the use of wraparound or other methods to deal with the data edges.

r,

-~~,

.- •..-~. i1-J

(0)

"

"I '

: .....~ ~1If..)., .,', _

Figure 3.36. Examples of two-dimensional orlhonormal wa\'elels: Daubechies, Symmlets and CoiOets.

(a) 02 Haar wavelet. (b) D4. (c) 08. (d) 012. (e) D12, another perspective. (f) 020. (g) 5ymmlet 58. (h) Coiftet C6. (Wavelets shown viewed at various angles and elevations.)

Adaptive transforms: wavelet packets

3.9

133

Adaptive transforms: wavelet packets

As we saw in chapter 2 (figure 2.29) the resolution of the wavelet transform is not uniform in the time-frequency plane. The Heisenberg boxes expand in frequency and contract in time as fluctuations at smaller and smaller scales are explored. The short term Fourier transform (STFT), on the other hand, covers the time-frequency plane with tiles of constant aspect ratio (figure 2.30). We also looked briefly at matching pursuits which offer another way of extracting time-frequency information. In this section we will consider another method which can adapt to the signal and hence allows for more flexibility in the partitioning of the time-frequency plane: the wavelet packet transform. Wavelet packet (WP) transforms are a generalization of the discrete wavelet transform. Wavelet packets involve particular linear combinations of wavelets and the wavelet packet decomposition of a signal is performed in a manner similar to the multiresolution algorithm given earlier for the discrete wavelet transform. The difference is that, in the WP signal decomposition, both the approximation and detailed coefficients are further decomposed at each level. This leads to the decomposition tree structure depicted at the top of figure 3.37. Compare this with the schematic of the wavelet decomposition given in figure 3.6. At each stage in the

scale index / m=O , - - - - - - - - - - - - - - - - - - - - - - - - - - , m=l m=2

original time series

'--------:;7'""'<::,---------L.----7"""":,-------------'

'-----_~-----L..-___,...,,...._-!---~-----L..-____,..;;,___------'

m=3 '-----~--L-....".....--J-----,.;------'-_____"

"-----~--'---_rc_---'-------,.;------'-_____,,.------J

_--'0)

SSSS n, TSSSS n, etc

~.II.

wavelet packet tiling of time-frequency plane at each scale

rr~m.1I t..;time

Figure 3.37. Schematic diagram of wavelet packet decomposition.

wavelet packet vectors

134

The discrete wavelet transform

wavelet algorithm, the detailed coefficients are simply transferred down, unchanged, to the next level. However, in the wavelet packet algorithm, all the coefficients at each stage are further decomposed. In this way, we end up with an array of wavelet packet coefficients with M levels each with N coefficients. A total of N coefficients from this M x N array can then be selected to represent the signal. The standard wavelet transform decomposition coefficients are contained within the WP array, shown by the bold boxes in figure 3.37. A new nomenclature is employed in the figure to indicate the operations that have been performed on each set of coefficients. S produces the approximation components of the previous set of coefficients by lowpass filtering, and T the detail components through highpass filtering. We simply add the letter S or T to the left-hand end of the coefficient name to indicate the most recent filtering procedure. SSTSn , for example, corresponds to the original signal lowpass filtered, then highpass filtered then passed twice through the lowpass filter. Notice also that the subscript contains only the location index n. The scaling index m is omitted as it is obviously equal to the number of letters Sand T in the coefficient name. As with the original wavelet transform, the number of coefficients at each scale depends upon the scale, with one coefficient in each coefficient group at the largest scale M and N /2 coefficients at the smallest scale m = 1. Hence, the coefficient index spans 1 n = O, 1, ... , 2M -111 - . At each stage in the decomposition, the wavelet packet algorithm partitions the time-frequency plane into rectangles of constant aspect ratio. These become wider (in time) and narrower (in frequency) as the decomposition proceeds. This is shown schematically at the bottom of figure 3.37 for each scale. A variety of tilings of the time-frequency plane is possible using the wavelet packet coefficients. For example, we could keep all the coefficients at a level and discard all the others. This would tile the plane in boxes of constant shape, just like one of those shown at the bottom of figure 3.37. Other tilings are possible, some examples of these being shown in figure 3.38. The standard wavelet transform is just one of all the possible tiling patterns. Figure 3.39 contains the wavelet packet coefficient selections corresponding to the tiling patterns of figure 3.38. wavelet transfonn tiling

high frequency

small scales -.-._

II

frequency

II

I -----'----+--'----------'---

low frequency

large scales·-··-· __

·======:::::J: time

(a)

(b)

(c)

Figure 3.38. Schematic diagrams of allowable wavelet packet tiling of the time-frequency plane. The right-hand tiling is that used in the wavelet transform algorithm and corresponds to the components contained in the bold boxes shown in the previous figure.

Adaptive transforms: lI!(/ve!et packets

135

,/ original time series

scale index

---

m=O I

I

'-----------=-~o::----------'

m=! m=2

---------IC=::::::~=:::::I:=~;;:=:::J /~ /~ IC=:;~=r:'-::;~::::J=::::;:==:JI=:::=:;::::J1 ./".

./".

./".

./".

I

I

_I ~I

I

""j\j\""j\j\""j\ ]

I'

I

I

I

I

I

I

I

I

m=5 .L...LLLLLLLLLL1LLJILLJILLJLLJLLJ-Ll--'--'--' 1--,--, I..LLJ II (aJ ,/ original time series

scale index 01=0

I

01=1

I

01=2

I

01=3

I

--------

~

/

/"-. , /"-. I I j

I

j\

r

/"-. I

I

I

~

/"-. j\

m=4

01=5

I

/

'\

j\

------------

j\

I

I

j

I

j\

I

I

I

I

I

I

I I

(b)

,/ original time series

scale index m=O

I

m=l

I

01=2

I

01=3

I

----

~

/

/"-.

/"-.

L

I

'\

m=4

I

01=5

I I I

--------

I

j\

-- j \ I

I

I

I

~

/

/"-.

/"-. _I.

[

j

j\

I I

I

j

I

J

I

I

I

C

I

I

I

I

j\

I

I

(c)

Figure 3.39. The choice of wa\'elet packet components leading 10 the lilings shown in figure 3.38.

136

The discrete wavelet tramlol'm

The optimal or
5(1')

=-

L

(3.102)

p,log(p,)

where, for this case, Pi are the normalized energies (i.e. squared magnitudes) of the wavelet packet coefficients under consideration. Low entropies occur when the larger coefficient energies are concentrated at only a few discrete locations. The minimum possible entropy of zero occurs when Pi = I for only one value of i, the other probabilities being zero. In this case all the information needed to represent the signal is condensed within a single coefficient. The maximum entropy occurs when there is an equal distribution of coefficient energies. In this case Pi = liN and the signal information is evenly spread throughout all the coefficients. We can see that Pi acts as a discrete probability distribution of the energies. (More information on the Shannon entropy measure together with an illustrative figure is given in chapter 4, section 4.2.4.) The set of N wavelet packet coefficients which contain the least entropy are selected to represent the signal. That is, we want the signal information to be concentrated within as few coefficients as possible. To find these coefficients the WP array, such as the one we saw in figure 3.37, is inspected from the bottom upwards. At each scale, each pair of partitioned coefficient sets (the 'children') are compared with those from which they were derived (their 'parent'). If the combined children's coefficients have a smaller entropy than those of their parent then they are The entropy of this block, together with the entropy assigned to its pair is less than its parent above hence it is selected.

These two children blocks have more entropy than the one directly above them. hence they arc not selected.

The entropy ofthis block, plus the entropy assigned to its pair have less entropy than their parent above and hence is selected. The combined entropy of these children is assigned to their parent in order that entropy comparisons can be made further up the tree.

The combined entropy ofthese

children is less than their parent These two children blocks have more entropy than their parent directly above

them, hence they arc not selected. Similarly, the next pair and the end pair at this level are not selected for the same reason.

above and hence they arc selected. However, the value of their combined entropy is assigned. to their parent in order that entropy comparisons can be made further up the tree.

Figure 3.40. Wavelet packet coefficient selection.

Adaptive transforms: IIlavelet packets

137

2,--------,------,----,--------,-----.-,------,-,

o 2 :'0----+.,0,-------±2"0----o3"'0----."'0,-------±,"0----;60~--'

IL--------

x, I

2

r

~/~-'"

1/\,

'-, fn--r'l • --rW , 3

6

II II

II II II II

II

A

n- ~

.Il-

,Jl

n

n

n

II II II II

II II II II

.Sl

'w

II II

II II II II

II II

IL

II

WP

WT

Original Signal

~

-

m=l

m=4

-

m=2

m=3

m=5

m=6

(b)

Figure 3.41. Wa,'elel packet decomposition of a simple signal. (a) Signal (lOp) with wavelet packet decomposition (below). The coefficient values are plotted as heights. The scale indices. 11/ = I to 6. are given down the left-hand side of the plol. Trace WP contains the best selection or wavelet packets and trace WT contains the wavelet transrorm decomposition ror comparison. (b) The time-rrequency tiling associated with each wavelet packet decomposition in (a). Larger coefficient values are plotted darker.

138

The discrete wavelet transform

retained. If not, the parent's coefficients are retained. When the children are selected their entropy value is assigned to their parent in order that subsequent entropy comparisons can be made further up the tree. This is shown schematically in figure 3.40. Once the whole WP array has been inspected in this way, we get an optimal tiling of the time-frequency plane (with respect to the localization of coefficient energies). This tiling provides the best basis for the signal decomposition.

(0)

I

I

I

I

(d)

(,)

Figure 3.41 (continued). (c) The time-rrequency tiling associated with the best wavelet packet decomposition (len) and wavelet decomposition (right). (d) The 16 largest coefficients rrom (c): wavelet packet decomposition (len) and wavelet decomposition (right). (e) The reconstruction or the signal using only the 16 largest coefficients given in (d): wavelet packet (len) and wavelet (right).

Adaptive transforms: IIlavelet packets

139

Figure 3.41 illustrates the wavelet packet method on a simple discrete signal composed of a sampled sinusoid plus a spike. The signal is 64 data points in length. A Haar wavelet is used to decompose the signal. Figure 3.41 (a) shows the wavelet packet coefficients below the original signal for each stage in the WP algorithm. The coefficients are displayed as histograms. The bottom two traces contain the coefficients corresponding to the best wavelet packet basis and the 'traditional' discrete wavelet basis respectively. The WP tiling of the coefficient energies in the time-frequency plane for each scale is given in figure 3.4J(b). The larger coefficient

1

~ /~

2\J J r 3

I

IL

x,

,

...JLr-

~I

4

s 6

II II I [ II II II II II II II II II

I I I I I . I II II II II II II I I I I I I II II II II II II

I I I I I I I I

WP WT

Ca)

-

(b) Figure 3.42. Wa\'elel packet decomposition of a simple signal using a Oaubechies 020 wavelel. (a) The wavelet packet decomposition of the signal shown at the top of figure 3.31 (a). (b) The time-frequency tiling associated with the best wavelet packet decomposition (len) and wavelet decomposition (right).

140

The discrete wavelet transform

energies are shaded darker in the plot. In figure 3.4I(c) the optimal WP tiling is compared with the wavelet transform tiling of the time-frequency plane. The plots in figure 3.41(d) outline the 16 largest coefficients in both time-frequency planes of figure 3.4 I(c). These are used to reconstruct the signals shown in figure 3.41(e). The 16 largest wavelet packet coefficients contain 98.6% of the signal energy. whereas the 16 wavelet transform coefficients contain 96.5% of the signal energy. The wavelet packet reconstruction using the selected coefficients is visibly smoother than the reconstruction using the traditional wavelet transform coefficients. Figure 3.42 contains the same signal as figure 3.41. However, this time the WP decomposition is performed using a Daubechies D20 wavelet (refer back to figure 3.15). Using this wavelet results in a different tiling of the time-frequency plane for the WP method (compare the left-hand plots of figures 3.42(b) and 3.41 (c». Again, the 16 largest coefficients are used in the signal reconstruction. The oscillatory parts of both reconstructions shown in figure 3.42(d) are visibly smoother than their Haar counterparts in the previous figure (figure 3.41(e». We expect this as the D20 is more smoothly oscillatory than the Haar. Note, however, comparing figures 3.42(c) and 3.4I(d), we see that the signal spike leads to a single high frequency tile for both Haar decompositions but respectively five and four high frequency tiles for the 020 wavelet. The more compact support of the Haar wavelet has allowed for a better localization of the signal spike, but it does makes it less able than the D20 to represent the smooth oscillations in the signal. The energies of the reconstructed signals for the 020 decompositions using only the largest 16 coefficients are 99.8% (WP) and 99.7% (WT), indicating the data compression possibilities of the techniques.

I I

(e)

(d)

Figure 3.42 (continued). The 16 largest coefficients rrom figure 3.31 (c): wavelet packet decomposition (left) and wavelet decomposition (right). (d) The reconstruction or the signal using only the 16 largest coefficients given in figure 3.31 (d): wavelet packet (left) and wavelet (right).

Endnotes

3.10

Endnotes

3.10.1

Chapter keywords and phrases

141

(You may find it helpful to jot down your understanding of each of them.) discrete wavelet transform wavelet/detail coefficients wavelet frames tight frame orthonormal basis dyadic grid scaling function approximation coefficients inverse discrete wavelet transform discrete approximation continuous approximation signal detail multiresolution scaling equation scaling coefficients compact support fast wavelet transform decomposition algorithm reconstruction algorithm

3.10.2

wraparound scale index sequential indexing level indexing hard thresholding soft thresholding scale thresholding Daubechies wavelets subsampled upsampled Symmlets eo ifie ts translation invariance redundant/stationary/translation invariant/non-decimated/maximal overlap/discrete wavelet transform biorthogonal wavelets wavelet packet transforms Shannon entropy measure

Further resources

Papers describing the discrete wavelet transform at an introductory level include those by Kim and Aggarwal (2000), Depczynski et al (1997), Asamoah (1999) and Graps (1995). The paper by Williams and Armatunga (1994) contains a good explanation of the derivation of the Daubechies D4 scaling coefficients and multiresolution analysis. They present a clear account of the wavelet filtering of a signal using matrices. Other useful introductory papers are those by Sarkar et al (1998), Meneveau (1991a), Jawerth and Sweldens (1994), Newland (1994a-c), Strang (1989, 1993) and the original paper on multiresolution analysis is by Mallat (1989). In his book, Newland (1993a) gives a more detailed account of the conditions that must be satisfied for discrete orthonormal wavelets. Diou et al (1999) and Toubin et al (1999) provide some useful information on projecting the details and approximations determined at one scale at another scale. Chui (1997) provides a little more mathematical detail in a readable account of discrete wavelet transforms and their role in signal analysis. Strang and Nguyens' (1996) text concentrates on the connection between wavelets and filter banks used in digital signal processing. Daubechies' (1992) book provides a good grounding in all aspects of wavelet transforms, containing among other things useful text on wavelet frames and wavelet symmetry. It also contains more information on the construction of Symmlets, Coiflets and biorthogonal wavelets. This chapter has concentrated on two compact,

142

The discrete wavelet transform

discrete orthonormal wavelets: the Haar wavelet and the Daubechies D4 wavelet. There are many other wavelets we have not considered, most notably the Shannon, Meyer, and Battle-Lemarie wavelets. More information on these can be found in, for example, Chui (1997), Mallat (1998), Daubechies (1992) and Blatter (1998). More examples of the multiresolution analysis of simple discrete signals can be found in the book by Walker (1999). Books providing a more mathematical account of discrete wavelet transforms are those by Hernandez and Weiss (1996), Benedetto and Frazier (1994), Chui (1992a,b), Walter (1994) and Percival and Walden (2000). There is also a lot of useful information on the web. The appendix contains a list of useful websites from which to begin a search. The book by Starck et al. (1998) contains a lot of good introductory material on two-dimensional discrete wavelet transforms, covering many of their practical applications including remote sensing, image compression, multiscale vision models, object detection and multiscale clustering. A nice illustrative paper on the application of two-dimensional wavelet transforms is that by Jiang et al (1999), who investigate the three-dimensional surface of orthopaedic joint prostheses. We considered only square two-dimensional transforms in this chapter, where the horizontal and vertical scalings were kept the same. If they are allowed to vary independently, we get rectangular transforms. Also possible are two-dimensional transforms which are not simply tensor products but wavelets constructed 'intrinsically' for higher dimensions (Jawerth and Sweldens, 1994). There are many applications of wavelet packets cited in the rest of this book. Quinquis (1998) provides a nice introduction to wavelet packets, while the paper by Hess-Nielsen and Wickerhauser (1996) and Wickerhauser's (1994) book provide a more in-depth account. We mentioned briefly the redundant wavelet transform, a variant of the discrete wavelet transform which produces N coefficients at each level and is translation invariant. This has been found useful in statistical applications: see for example Coifman and Donoho (1995), Lang et al (1996) and Nason and Silverman (1995). A concise summary of the various wavelet thresholding methods developed over recent years is to be found in the paper by Abramovich et al (2000) together with a comprehensive list of references on the subject. In addition, the book by Ogden (1997) provides a more detailed overview together with numerous examples. The universal threshold method is detailed in the paper by Donoho and Johnstone (1994) together with another global method, the minimax thresholding method. Donoho and Johnstone (1995) also developed a scheme which uses the wavelet coefficients at each scale to choose a scale-dependent threshold. This method is known as the SURE or SureShrink method after 'Stein's Unbiased Risk Estimate' on which it is based. They suggest a HYBRID method to be used in practice for decompositions where, at some scales, the wavelet representation is sparse, i.e. a large number of the coefficients are zero (or very near zero). The HYBRID method uses the SURE method, but defaults to the universal threshold for scales where the data representation is sparse due to the noise overwhelming the little signal contributed from the nonzero coefficients. A number of other methods have been proposed: including those based on cross validation, e.g. Nason (1996), Jansen and Bultheel (1999); Bayesian approaches, e.g. Chipman et al (1997), Vidakovich (1998), Abramovich et al (1998); and the Lorentz curve (Katul and Vidakovich,

Endnotes

143

1996, 1998). Translation invariant denoising is considered by Coifman and Donoho (1995) who compute the estimated signal using all possible discrete shifts of the signal. See also the papers by Donoho and Johnstone (1995, 1998), Donoho (1995), Johnstone and Silverman (1997), Hall and Patil (1996), Efromovich (1999), Downie and Silverman (1998), Moulin (1994), Krim and Schick (1999) and Krim et al (1999). We will come across the use of thresholding extensively in the subsequent application chapters of this book. In particular, thresholding is revisited in chapter 4, section 4.2.3, where another thresholding method, the Lorentz threshold, is explained. Often we want to compress the signal into as few detail coefficients as possible without losing too much information for data compression applications such as speech and audio transmission. We can do this by finding a wavelet which decomposes the signal into a few large amplitude detail coefficients, which we retain, and many coefficients close to zero, which we discard. How many coefficients we set to zero affects the subsequent quality of the reconstruction. In practice for larger data sets and more suitable wavelets we can get good compression ratios without the loss of significant detail, where the term 'significant' is determined by the user. Here we have not considered compression schemes, including quantization and encoding, but rather the reader is referred in the first instance to the simplified explanation of Walker (1999) and the more detailed treatments in Chui (1997) and Mallat (1998). Note that all examples in this chapter have used wraparound to deal with the signal edges, i.e. the part of the wavelet spilling over the end of the signal is placed back at the start. This is the simplest and one of the most common treatments of edge effects for a finite length signal, and it results in exactly the same number of decomposition coefficients as original signal components. However, it is not always the best for the application. Other methods were shown in chapter 2, figure 2.35. In addition, there are families of boundary wavelets which are intrinsically defined on finite length signals (see for example the discussions in Jawerth and Sweldens (1994) and Chui (1997)). Take particular care when using off-the-shelf wavelet software packages as they may employ other boundary methods as the default setting, for example zero padding, which results in slightly more than N coefficients resulting from the full decomposition. In addition, we have employed one version of the scaling coefficient reordering b k (equation (3.25)). Again this is very much software dependent and you will find alternative ordering employed by various software packages.

Chapter 4

Fluids

4.1

Introduction

A fluid is a non-rigid interconnected mass which may in general exhibit either laminar or turbulent flow. Laminar flows are characteristic of slow-moving or highly viscous flows where the fluid particles move in an ordered fashion, sliding over themselves in sheets (or laminae, hence 'laminar'). Turbulent flows, on the other hand, are characteristic of fast-moving or low-viscosity flows, where small disturbances in the flow quickly blow up causing the fluid particles to move in an unpredictable fashion, mixing themselves up from one point in the flow to the next. Almost all real flows of interest to scientists and engineers are turbulent: the flow of water in rivers and pipelines; the flow within hydraulic machinery, e.g. turbines and pumps; atmospheric wind flows and ocean currents; and the flow or air around buildings, moving vehicles and aircraft. Turbulence manifests itself as a multiscale cascading phenomenon, where fluctuations (eddies) over a large range of scales are superimposed on a mean flow, e.g. the buffeting experienced on a windy day. Over the past decade, the wavelet transform has emerged as a particularly powerful tool for the elucidation of fluid signals (e.g. velocities, pressures and temperatures), both temporal and spatial, covering a variety of pertinent problems from the Kolmogorov scaling of high-Reynolds-number homogeneous turbulence to the nature of vortex shedding downstream of bluff bodies. Many researchers have made use of the wavelet transform's ability to probe simultaneously both the spectral and temporal (or spatial) structure of turbulent fluid flows. This chapter begins with a basic outline of the wavelet-based statistical methods used extensively in the analysis of fluid flows. The chapter is then split into two main parts: the first details the wavelet analysis of jets, wakes and coherent structures in engineering flows and the second considers geophysical flows. The choice of the most appropriate wavelet to use in the analysis of fluid data depends very much on the nature of the data itself. As we shall see in the forthcoming sections, in general, discrete dyadic grid orthonormal wavelets are used for statistical measures of turbulence, where ensemble averaging is usually a necessity due to the translation invariance of this type of transform. Often the Haar wavelet is employed due to its relationship with velocity structure functions used in traditional turbulence Copyright © 2002 lOP Publishing Ltd.

analysis. In addition, it is compact in the time domain, its short support eliminating edge effects. Due to their high resolution in the wavelet domain, continuous wavelets arc normally employed for feature detection in nows with recognizable coherent structures. The Mexican hat wavelet is often used when compactness in the time domain is important or when modulus maxima-based analysis is undertaken. Complex varieties of wavelet (very often the Morlet wavelet) are used when phase information is important, for example in vortex shedding flows. As we saw in chapter 2, the standard Morlet wavelet, with its wavepaeket structure, has better compactness in the frequency domain than the single-humped Mexican hat.

4.2

Statistical measures

Traditionally, turbulent statistical measures are often calculated in Fourier space. However, important temporal information is lost owing to the non-local nature of the Fourier modes. As a result wavelets have been utilized to quantify the temporal and spectral distribution of the energy in new statistical terms, such as wavelet variance, skewness, flatness, etc. These statistical measures are generally computed for discrete orthonormal wavelet expansions, which some authors believe are preferable because orthogonality both reduces the number of wavelet coefficients and suppresses undesired relationships between them. Wavelet-based statistics enable both scale- and location-dependent behaviour to be quantified. In this section we will look brieny at some basic wavelet-based statistics developed for the analysis of turbulent now signals. We will consider mainly the manipulation of discrete transform coefficients Tm ," generated from full decompositions using real-valued, discrete orthonormal wavelet transforms of the type we covered in chapter 3. In addition, we will assume that the mean has been removed from the signal and that it contains N (=2 M ) data points. 4.2.1

Moments, energy and power spectra

The pth order statistical moment of the wavelet coefficients Till ,II at scale index m is defined as 2M "'_I

I: (T""n )" T" ) _ n~O ( 11/,11 /1/ - --'----:=2TM",:::-,,--

(4.1 )

where only the coefficients at scale 111 arc used in the summation. The brackets (-. ')/1/ denote the average taken over the number of coefficients at scale 111, hence the 2M -/1/ term in the denominator. For example, the commonly used coefficient variance at scale index m is 2M lit_I

I:

11=0

Copyright © 2002 lOP Publishing Ltd.

(T"",,)' (4.2)

The wavelet coefficient variance is simply the average energy wrapped up per coefficient at each scale 1J1. Do not confuse equation (4.2) with Pm, the mean energy per IInil lime in the signal at scale m (given below in equation (4.7», also known as the scale-dependent power. A general dimensionless moment function can be defined as (4.3) where the pth order moment is normalized by dividing it by the rescaled variance. For example, the scale-dependent coefficient skewness factor is defined as the normalized third moment:

FJ

11/

(T,;,."),,, 3/2 ((T,;""),,,)

=

(4.4)

and, similarly, the scale-dependent coefficient flatness factor is defined as

r:

(T,;"II)m ? 2 ((T"""),,,)

=

m

(4.5)

The flatness factor gives a measure of the peakedness (or flatness) of the probability distribution of the coefficients at each level. It is well known that for Gaussian distributions the flatness factor is equal to 3. Values higher than 3 occur for distributions with more pronounced tails. The flatness factor increases as the flow signal becomes morc intermittent (e.g. Meneveau, 1991a; Mouri el ai, 1999). We now consider the wavelet-based scale-dependent energy defined as 2.11 !II_I

L

(T",.,,)'

(4.6)

/',.1

11=0

Notice that this equation is slightly different in form from that given in chapter 3, equation (3.52) (where an integer time step was assumed), as the sampling time, /',.1, has now been added. The scale-dependent energy per unit time, or scaledependent power, is Pili = EIII/T where T is the total time period of the signal. Hence, as T = 2M !::J.t, it can be written as 2M 11I_1

(TII/,n) 2

I:

Pm

11=0

(4.7)

= ---"=,-::-.",,---

2"

Therefore, as long as a signal has zero mean, both the total energy and total power of the signal can be found by summing £11I and Pm respectively ovcr all scale indices m. We can construct a wavelet power spectrum for direct comparison with the Fourier spectrum as follows: I 2/11.6.t 2'\/

PwU;,,)

= -:;.

In 2

III-I

L

11=0

Copyright © 2002 lOP Publishing Ltd.

2

(T,,,,,,)

12'I/f::J.t /',.1 = -:;.

In 2 E",

(4.8)

The term (2 m ~t) / In 2 stems from the dyadic spacing of the grid. The temporal scale of the wavelet at scale index m is equal to 2m ~t. Often the Haar wavelet is used in turbulence studies and this temporal scale is taken as its representative period, hence the associated frequency is 1m = 1/(2m ~t). (However, note that we can easily modify this expression to take into account a characteristic frequency of the mother wavelet, such as the spectral peak, h, or bandpass centre frequency, h. For example, by employing h, the scale-dependent frequency becomes 1m = IcI(2 m ~t) and hence Ie would appear in the denominator of equation (4.8).) A Taylor expansion of1m = 1/(2m ~t) gives the discrete incremental change in frequency associated with the discrete change in scale index m:

~r = r' ~ Ym

Jm

r" (~m)2 r'" (~m)3 r"" (~m)4 m+Jm 2! +Jm 3! +Jm 4!

+

...

(4.9a)

Truncating at the first term with 1m = 1/ (2 m ~t) and remembering also that the scale index is an integer (i.e. ~m = 1) we get ~Im = -ln2/(2m ~t) = -In(2) xlm, where the negative sign is ignored in practice. It is common in the literature for the Taylor expansion to be truncated at the first term in this way-even though it has a simple limit, i.e.

~r

Ym

=

1 [-1 (2) 2m ~t n

+

In(2)2 _ In(2)3 2 6

+

In(2)4 24

+

... J

1m 2

(4.9b)

Again, ignoring the negative sign we see from the previous expression that the incremental discrete change in the frequency is equal to half the frequency itself. In fact, this is obvious without employing a Taylor expansion, i.e. ~Im =lm+I -1m

=

1 2m + I ~t

1

(4.9c)

that is, the incremental discrete change in the frequency is equal to half the frequency itself. This is for the forward difference Im+ I - 1m. For the backward difference, 1m - 1m-I, it is equal to 1m (try for yourself). So now we have three options for ~Im (Im/2, In(2)lm and 1m) which are all equally valid. In the rest of this text we will use ~Im = In(2)lm as it is the most commonly used form in practice (e.g. Katul et aI, 1994; Katul and Parlange, 1995; Kulkarni et aI, 1999). The addition of the (2 m ~t)/ln(2) term in equation (4.8) is therefore required for the discrete summation over the frequency range to equal the total power in the signal. The total length of signal 7 is present in the denominator of equation (4.8) as the area under the power spectrum represents the average energy per unit time. As 7 = 2M ~t we can rewrite the power spectrum as 2M

Pw(fm)

=

- m

_I

In(2~~-m ~

(Tm,n)2

(4.10a)

or simply in terms of the wavelet coefficient variance as

Pw(lm) Copyright © 2002 lOP Publishing Ltd.

~t

=

2

In(2) (Tm,n)m

(4.10b)

or in terms of the scale-dependent signal power as I

(4.]Oc)

Pw(J",) = In(2)[,,, P'"

Compare the wavelet power spectrum for the dyadic grid orthonormal transform given by equations (4.8) and (4.IOa)-(4.IOc) with the definition of the power spectrum for the continuous wavelet given in chapter 2, section 2.9, as

1

Pw(J) = r}; C

fT

•0

T(J, b)

2

(4.11 )

db

g

where Cg is the admissibility constant for the particular wavelet used, .t;. is a characteristic frequency of the mother wavelet defined at scale a = I, the frequency f is equal toj;/a and the derivative df = -da/a2 Figure 4.1 contains a vortex shedding signal taken downstream of a cylinder in an open channel flow together with associated wavelet and Fourier-based power spectra. Both continuous Mexican hat and discrete Daubechies D4 wavelets have Ca) 0.17,---,----.-----,---,---r--.-----,----,-----,

-0.17 ':----:':--~-----,J':----:'::--r_'::------f.,___--::'::------:' o 10 W ~ ~ ~ ffi W W t (sec)

(b)

IO,-------r-,------. .,-..- - , - - - - - - .-.."

l .E

!

, '

:

O.It-==o
O.otI-------"'~.:..pH____\+_'__l\c O.OOII-----+'-+.,.-;+------"'f-J'++

"

~ 1.1O-41------,---++-.;-,+--------c~-+'----'

0.

t.t0-5t----4-,-,---,f-----S+---,--~-"-'-:-_,_.,__1

t.t 0-<> L.-

O.Ot Fourier -0 continuous

--'---'--'---

---'-'--'--_-'-

0.1

-'

10 frequency (Hz)

........ discrete

Figure 4.1. Power spectra of a vortex shedding velocity signal. (a) Velocity signal taken downstream of

a cylinder in open channel flow. (b) Fourier and wavelet (Daubechies D4 and Mexican hat) power spectra of the signal in (a). After Addison el al2001 ASCE JOIIJ'lla! of Engineerillg Mechanics 127 58-70. © ASCE 2001, reproduced with permission of the publisher. Copyright © 2002 lOP Publishing Ltd.

been used to construct the wavelet spectra. Note the distinct peak in the Fourier spectrum at the vortex shedding frequency of 0.133 Hz. The continuous Mexican hat spectrum also peaks at this value, although smearing of the spectrum around the maximum is evident. The Mexican hat spectrum was produced using a fine resolution, translation-invariant discretization of the temporal location parameter,

b, and a fractional power-of-two scale for the wavelet scale, a. Close inspection of the plot corresponding to the discrete orthonormal Oaubechies wavelet transform, however, reveals the coarser resolution due to the dyadic grid structure, i.e. integer

power-of-two translations and dilations. Other examples of wavelet power spectra occur later in this chaptcr: see for example figures 4.2,4.14 and 4.33. Qiu el al (1995a) provide some words of caution in the application and interpretation of wavelet-based power spectra resulting from dyadic orthonormal wavelet transforms. They found that orthonormal wavelet analysis can lead to false spectral slopes and suggested that wavelets of similar shape to the coherent structures are used and that non-orthonormal wavelet analysis is used when high resolution is important. They illustrated the problem by employing the Haar, Oaubechies 012 and LcmarieMeyer-Battle (LM B) wavelets to construct energy spectra from atmospheric turbulence measurements. One commonly used statistical measure of the energy distribution across scales is the normalized variance of the wavelet energy. This is called the ftuctuation intensity (FI) and is defined as

FT = [(r,;""),,, - «T,~""),,,)2J'/2 '"

(T2m,/1 )m

(4.12)

which measures the standard deviation of the variance in coefficient energies at scale index m. It is also sometimes referred to as the coefficient of variation (CY) (e.g. Katul el ai, 1994; Kulkarni el ai, 1999). It follows from equation (4.12) that skewness and ftatness measures may also be found for the scale-dependent energies (e.g. Yee el ai, 1996). The ftuctuation intensity provides a measure of the variability of the signal energy at scale index m. Another way to present this variability is by constructing the dual spectrum, which combines the power spectrum, Pw(f",), and the ftuctuation intensity converted to suitable units:

Dw (1,,, )

=

2'" 6.1 E", [

In(2)T

I + F1",

I=

( . )[ I Pw 1", I + FT",

(4.13)

The dual spectrum is a plot of both PwU;,,) and DwU;,,) which provides information concerning both the contribution to the energy at various scales and its associated spatial variability (i.e. its variance-see Meneveau, 199Ia,b). An example of a dual spectrum is given in figure 4.2. It is derived from the streamwise velocity signal taken downstream of a cylinder in a wind tunnel experiment using LM 8 wavelets. The slight increase in the relative variance of the local energies at the larger wavenumbers is due to the increasingly intermittent nature of the kinetic cnergy distribution here. This is confirmed by the ftatness factor plot for the same signal given in figure 4.3. Note that the wavenumber is used in the plots, where the Copyright © 2002 lOP Publishing Ltd.

10'

,

, 4

4

10'

4

10' ~

i;J

""0~

10"

4

,

0-

" ,,

10°

"

10-2

,4

•

10-4 2-10-3

10-1

10-2

10°

/oj wavenumber

Figure 4.2. Dual spectrum of the turbulent flow behind a cylinder (stream wise one-dimensional spec-

trum). Solid line-usual Fourier spectrum. Circles-wavelet spectrum. Triangles-wavelet mean energy at every scale to which one standard deviation (computed from the spatial fluctuations) has been added. Arter Meneveau 1991b J Fluid Mec" 232 469-520. With kind permission from Cambridge University Press.

a o

a

a a

8

•a 10-2

a o

a a

• • to-I /oj wavenumber

8

10°

Figure 4.3. Flatness factors of the w3l'elet coefficient computed from laboratory data. The circles are for the boundary layer flow and the squares are for the turbulent wake (cylinder). After Meneveau 199Jb J Fluid Mech 232 469-520. With kind permission from Cambridge University Press.

Copyright © 2002 lOP Publishing Ltd.

wavenumber is the spatial frequency, k, of the flow structures which has a similar form to the temporal angular frequency w. That is, the reciprocal of the length scale, y, multiplied by 21r:

k

_ 21r m -

y

(4.14)

The denominator y represents the physical distance between structures. We can estimate y from velocity signals for flows which have relatively low turbulence intensities compared with their mean advective velocities using Taylor's frozen flow hypothesis. This states that, if the turbulent field is changing slowly enough with respect to the mean velocity, then measuring data at a point as the turbulent field advects past is equivalent to taking a linear section through the field. Thus characteristic time periods, p, in the velocity signal can be converted to spatial separations y = up, where u is the mean advective velocity. If, as before, we set the temporal scale 2m !:1t = p (= Ilfm) then y = u2m !:1t = 2m !:1y, where !:1y is the spatial increment set by the mean velocity and the sampling time. In addition, the scale index frequency is related to the wavenumber through the mean velocity and the 21r factor as k m = (21rlup) = (21rlu)fm' The power spectrum in terms of wavenumber k m is then (4.15a) As !:1y

= u !:1t, this can be rewritten as (4.15b)

where Pm = Emir is the scale-dependent energy per unit time, or scale-dependent power, given above in equation (4.7). It can also be written (most commonly) in terms of the coefficient variation as (4.16) Remember that (T~,n)m is the sum of the coefficient energies at scale m normalized by 2M - m, i.e. the energy per coefficient, hence the disappearance of the 2 - m factor in the denominator going from equation (4.15b) to (4.16), as Pm is the sum of the coefficient energies at scale m normalized (in a global sense) by 2 M . The wavelet co-spectrum or cross-spectrum between two variables u and v is defined as

(4.17) v then equation where (Tm,n)u denotes the wavelet coefficients for variable u. (If u (4.17) reduces to the standard power spectral density function defined by equation (4.15a).) Katul et al (l998b) have used wavelet co-spectra to analyse the relationship Copyright © 2002 lOP Publishing Ltd.

between turbulent velocity and skin temperature perturbations above a grass-covered forest clearing. 4.2.2

Intermittency and correlation

The intermittency at each scale can be viewed directly using the intermittency index proposed by Farge (1992). This allows the investigator to visualize the uneven distribution of energy through time at a given wavelet scale. The intermittency index, Im./I' is defined as

I

= (Tm"i

(4.18)

(T') m,/I 11/

1/1./1

Im,1I is the ratio of local energy to the mean energy at temporal scale 211I D.f. For example, a constant value of 1m ,/I = I for all m and 11 means that there is no flow intermittency at all, whereas a value of 10 at a specific set of indices 111 and 11 means that, at that location in the signal, there is ten times more energy contained within the coefficient at that location than for the temporal mean at that scale. Figure 4.4 contains a plot of the intermittency indices for the vortex shedding signal shown in figure 4.1 (a). High magnitude values of 1m ,,, can be observed intermittently at the lower scales. The correlation between the scales can be measured using the pth moment scale correlation R~;I defined as (111-1)_1

P

"'"'" L..J

(4.19)

B 11I'[11 Sf'Ill-I,ll

11=0

where B~", , is the pth order moment function (defined below) and [~J - requires that the integer part only be used. In order to pair all the coefficients at the smaller scale (index 117 - I) with those at the larger scale (index 117), the sum is taken over the number of coefficients at the smaller scale, e.g. from 11 = 0 to 2M -(m-l) - I.

,

400

,

, ,.

• 300 I

~ 200 100 I-

,

j

,. ,

I.

J 10

"20

• I

L

I

L

JL

•

II

L

.J.'

30

.-

j

,I,

J

40

50

j

J....I.. 60

70

80

time (t)

Figure 4.4. Intermittency indices according to w3\'elet scale for the flow downstream of a cylinder in an open channel. The traces from bottom to top correspond to scale indices 111 = I, 2, 3, 4 and 5. The index plols have been displaced by 100 units 10 aid viewing. Arter Addison et a/200 I A SeE JOlll'lla/ of Engineering Mechanics 127 58-70. © ASCE 2001, reproduced with permission orthe publisher. Copyright © 2002 lOP Publishing Ltd.

(For more information see Yamada and Ohkitani (1991) and Yee ef al (1996).) B;;,." is the pth order moment function defined as (4.20)

Note that this has a similar form to the intermittency index when

p~2,

except that

B~;1.11 has a normalized sum at each scale, i.e.

LB~;"II "

(4.21 )

= I

whereas the sum of the intermittency indices at scale

f11

is equal to the number of

coefficients at that scale, i.e. M "'"" L.t I m,1I -~ 2 -

/1/

(4.22)

" Figure 4.5(c) shows the scale correlation that exists within the turbulent velocity signal shown in figure 4.5(a) takcn from within the atmospheric boundary laycr by Yamada and Ohkitani (1991). At large scales there is no obvious correlation. However, at small scales there is a general increase in all the pth moment scale correlations. This is in contrast to the pth moment correlations shown in figure

4.5(d) for the phase-randomized signal of figure 4.5(b). Phase randomization of a signal is performed by randomizing the phases of the Fourier components of the signal then taking the inverse Fourier transform. It destroys the correlations between levels in the signal and provides a useful benchmark signal for detecting such correlations. Moriyama ef al (1998) also examined scale correlation in a study of the density fluctuations in granular flows through pipes at various flow rates. They found that low density flows exhibited a Gaussian distribution for the wavelet coefficients at all scales. However, for higher density flows the distribution became noticeably non-Gaussian. Figure 4.6 shows examples of the two probability density functions (PDFs). The authors found no correlation between scales for the low density flows (figures 4.6(a) and 4.7(a)) and concluded that the time series signal from these flows are equivalent to a random signal. Extended tails were, however, found for

the probability distributions corresponding to the high density flows (figure 4.6(b)) which contain significant correlation across scales (figure 4.7(b)). 4.2.3

Wavelet thresholding

Thresholding techniques are used extensively in the analysis of fluid flows to partition the signal into a coherent and 'more random' turbulent part, sometimes referred to as the strong and weak signal components respectively. We saw a variety of thresholding methods in chapter 3 (refer back to chapter 3, section 3.4.2) and many of these have been employed in the analysis of turbulent fluid signals. Examples of wavelet thresholding used in the analysis of both spatial and temporal fluid signals can be found in Hagelberg and Gamage (1994), Turner and Leclerc (1994), Higuchi ef al (1994), Farge ef al (1996) (see figure 4.31), Katul and Vidakovich (1996,1998), Briggs and Levine Copyright © 2002 lOP Publishing Ltd.

3 0104

k

2

(b)

(al

8 6

ts

6

S

t,z 4

4

,z

£ 2

£ 2

0

1 10

large scales

(el

-2 0

\

5

j

10

small scales

(d)

Figure 4.5. The pth moment correlations between scales m and III - I. (a) Original signal. (b) Phase randomized signal. (c) plh moment correlations of original signal (a). p = 1 (squares); p = 2 (circles); p = 3 (triangles); p = 4 (diamonds). (d) plh moment correlations of phase randomized signal (b). Note that authors use levels j instead of scale 11/ where j = M - III and the nomenclature C(p)(j) instead of R~:" i.e. large levels j correspond to small scales. After Yamada and Ohkitani (1991). Reproduced with the kind permission of the Physical Society of Japan and the authors.

(1997), Hagelberg el al (1998), Katul el al (\ 998a), Szilagyi el al (1999) and Kailas and Narasimha (\ 999). Figure 4.8 shows an example of the partitioning of the vortex shedding signal of figure 4.1. The partitioning is performed using both hard thresholding and scale-dependent thresholding. The original coefficients from a Daubechies D4 decomposition are shown in sequential format in figure 4.8(a). The signal was 8192 data points in length. Figure 4.8(b) (top plot) shows the small scale coefficients set to zero where the threshold scale index was set to /11' = 6. Subsequent wavelet reconstruction using the remaining coefficients gives the strong signal. The weak signal is reconstructed from the coefficients below /11'. The bottom plot of figure 4.8(b) contains the original signal plotted together with the weak and strong parts of the signal. Figure 4.8(c) plots both the hard thresholded coefficients and corresponding signal partitions obtained using Donoho and Johnstone's universal threshold, with & derived from the median absolute deviation of the wavelet Copyright © 2002 lOP Publishing Ltd.

-

.• = 1514 ~

• = 14 =13 ~ 12 =11 = 10

10-1

=13 = 12 =11 =10

10-"

~9

g

.~

~8

~9

10-2

~8

'"->:

~

"e '"

-Gaussian

Gaussian ~ IS

"e '"

10-3

10-<

10-5

.

--••

:f...,

"

~

L--'-'-_~~_~~-'---~

..

~,

Hr' ~--7--~:----:---;' -10 -5 0 5 10

-{;-4-20246 nonnalized

normalized

(b)

Figure 4.6. PDFs of wa\'elet coefficients for sparse uniform flows. (a) Sparse uniform flow. (b) Dense uniform flow. The abscissa slands for Ihe wavelet coefficient which is normalized 10 have unit variance, the ordinate for the probability of finding the coefficient (j = level index). After Moriyama el al(1998). Reproduced with the kind permission of the Physical Society of Japan and the authors.

coefficienls at the smallest scale. Comparing figures 4.8(b) and (c), we can see thaI scale thresholding smoothes the signal in the strong part and leaves remnants from the vortex shedding process in the weak part. On the other hand, hard thresholding retains much of the high frequency components of large amplitude in the strong 10-2

10-2 .q -I

~q=l

"q = 2

'q~2 "q~3

"q~3

10-1

~--

.q=4 . q =5

10- 1

· . • ·· •• i • , • I • • • I

10°

! ,

I

~~ 10°

10-1 L-_ _~_ _~_ _--'

o

5

j

large scales

(aJ

10

\

'q=4 "q=5

·· , , , , • , , , , , ,• ., ,• .. • . • " •• I

10-1 '---~--~-----' 10 15 o 5

15

j

small scales (b)

Figure 4.7. Correlation between adjacent scales of the sparse and dense uniform flows. (a) Sparse uniform flow. (b) Dense uniform flow. Note lhal level indexing is used: j = level index; q = moment. CJq) is the qth scale moment correlation. C)q) is equivalent to Rf,,, given in the text for plh moment scale indexing. After Moriyama el al (1998). Reproduced with the kind permission of the Physical Society of Japan and the authors.

Copyright © 2002 lOP Publishing Ltd.

1

.

0

,

11'1-·'

(a) -1

0

1000

2000

3000

4000

5000

6000

7000

8000

1000

2000

3000

4000

5000

6000

7000

8000

1

o IT -1

0

0.3 .------,-----,------.----.------r----.----,-------, 0.2 0.1

o -0.1

-o.3L.-------'------'-------'--------'------'-------'-------'------.L.-----' 10 20 40 o 50 60 70 80 30 (b)

l,..-----,-----.------r---,----,....------r------.---,--,

-1 '--_----'-_ _----'--_ _--'---_ _. L . -_ _' - - _ - - - - ' - _ _----'-- _ _--'--'

o

1000

2000

3000

4000

5000

6000

7000

8000

0.3 .---------,----,-------,----.-------.------.----,---------, 0.2 0.1

o -0.1

(c)

-0.3 L.-_-----'_ _----'-_ _----'--_ _----'--_ _--'---_ _--'----_ _...l...-_---' o 10 20 30 40 50 60 70 80

Figure 4.8. Scale and hard thresholding of a vortex shedding signal. (a) Original coefficients. (b) Scale thresholded coefficients (top) and associated partitioning of signal (bottom). Top trace = original signal; middle trace = strong signal component; bottom trace = weak signal component. (c) Hard thresholded coefficients (top) and associated partitioning of signal (bottom). After Addison et al 2001 ASCE Journal ofEngineering Mechanics 127 58-70. © ASCE 2001, reproduced with permission of the publisher. Copyright © 2002 lOP Publishing Ltd.

energy loss

relationship for evenly distributed energy

\

relationship for turbulent signal

"-..

Lo .-- ----------(= 0.20)

L:::::::::==========::::::==:'l__J

0 0

proportions of coefficients rejected

Po (= 0.75)

Figure 4.9. A schematic of the Lorentz cune used in wavelet thresholding. The optimal proportion Po is determined from the tangent parallel to the diagonal.

part of the signal and removes the vortex shedding fluctuations from the weak part, leaving a more evenly distributed noisy weak signal. Lorentz thresholding has been suggested as another method for setting a global threshold for the analysis of coherent structures within turbulent fluid signals (Katul and Vidakovich, 1996; Katul el ai, 1998a). In contrast to other methods, the Lorentz threshold does not assume a probabilistic structure for the wavelet coefficients. It uses the fact that the energy in the wavelet domain is not evenly distributed over the coefficients. If we plot the proportion of energy loss against the removal of each of the smallest energy coefficients in turn, we obtain a Lorentz curve. A schematic of a Lorentz curve is shown in figure 4.9. As the energy is not evenly distributed throughout the coefficients for turbulent signals this curve is convex. If the energy were distributed evenly we would get the diagonal line shown in the figure. The tangent to the Lorentz curve with the same slope as the diagonal locates the point on the curve where the energy lost by removing a single coefficient is equal to the average energy in the coefficients. At this point the gain (in parsimony) by thresholding an additional wavelet coefficient is smaller than the loss in energy. This tangent corresponds to Po and L o. The example in figure 4.9 shows that thresholding at this point removes 75% (Po = 0.75) of the coefficients but removes only 20% (L o = 0.2) of the energy. An example of Lorentz thresholding is given in figure 4.10. Figures 4.10(a) and 4.10(b) both contain a section of a turbulent time series shown together with its strong and weak components found through hard thresholding using the Lorentz threshold. The data were acquired by a laser Doppler anemometer within a turbulent channel flow downstream of a bluff block obstacle. (For more information see Addison el al (2001).) A Haar and Symmlet(l2) wavelet were used respectively in figures 4.10(a) and 4.10(b). For the Haar decomposition only 244 coefficients out of 4096 making up the original signal were used in the strong reconstruction. This represents only 6% of the coefficients obtained from the decomposition. However, 95.1 % of the signal energy is contained within this signal. The weak signal is composed of the remaining coefficients and represents only 4.9% of the signal Copyright © 2002 lOP Publishing Ltd.

· 0.6

,....------,------.-----,.------r-----r------.------r-----r-------,

sao

1000

1500

2000

2500

3000

3500

4000

4500

0.6 ,....---,..---.-----,.---,.---,....---,....---.-----.---------,

sao

1000

1500

2000

I-

-

I-

-

I-

..,

2500

3000

3500

4000

4500

.J (c)

1 0.5

~

-

0 -0.5 (d)

0

sao

1000

1500

2000

2500

3000

3500

4000

4500

(e)

Figure 4.10. Lorentz thresholding of a turbulent velocity signal. (a) Original signal (top) with strong signal (middle) and weak signal (bottom). Lorentz thresholding used with a Haar wavelet. The strong signal contains 95.1 % of the energy of the original signal, using only 6% (244 out of 4096) of the coefficients. (The mean has been removed from the original signal.) (b) As (a) but using a

Copyright © 2002 lOP Publishing Ltd.

energy. The smoother Symmlet wavelet uses even fewer coefficients in the reconstruction and contains slightly morc energy in the strong signal reconstruction. The thresholded Symmlet coefficients are shown in figure 4.10(d), where it can be seen that most of the large amplitude coefficients are to be found at lower scales. Figure 4.10(e) contains an enlargement of the first quarter of the signal showing the acquired data points together with the wavelet estimates of the strong signal. The blocky nature of the Haar reconstruction is obvious from the plot. 4.2.4

Wavelet selection using entropy measures

Ifwe have no preset requirements for the wavelet used in the analysis ofa signal, such as vanishing moments or smoothness, then an entropy measure can be employed for the selection of the most suitable wavelet. The Shannon entropy is defined for a discrete probability distribution Pi, i = 1,2, ... , N,

S(p)

= -

L

Pi 10g(Pi)

(4.23)

where Li Pi = I. (Refer back to chapter 3, section 3.9, where we looked briefly at the role of entropy measures in selecting the 'best' set of wavelet packet coefficients.) The maximum entropy possible from a distribution occurs when the data set has an equal probability distribution Pi at every i, i.e. when the information is evenly spread across the signal. Any other distribution results in an S(p) less than the maximum. The more clustered the distribution the lower the entropy. The minimum entropy occurs when all the information is contained in a single location, i.e. at only one value of i, where

Pi = I (see figure 4.11). This entropy measure is extended to the wavelet coefficient energies where we usually want to contain as much information from the signal in as few wavelet coefficients as possible. Hence we look for the wavelet which gives us the maximum entropy for the squared coefficients. To utilize the Shannon entropy measure in the selection of the optimal wavelet, the normalized wavelet coefficient energies t'~111 , are used where

Tmil

2

-?

,

-

Til/II,

?

LIII LII Ti,I,II

(4.24)

The denominator is the total energy in the signal defined in terms of the wavelet coefficients, hence Lm,II f,~,,11 = 1. The normalized wavelet coefficient energy t,~,,11 is the relative proportion of the total energy contained within the coefficient TII/,w

Symmlet( 12) wavelet. The strong signal contains 96.3% of the energy of the original signal, using only 4.7% (193 Ollt of 4096) of the coefficients. (c) The Lorentz curve corresponding to the Symmlet decomposition. The whole curve is shown on the left and the last eighth of the curve on the right. (d) The strong and weak coefficients for the Symmlet decomposition. (e) Close-up of the strong signal reconstructions lIsing the Haar (top) and Symmlet (bolt om) wavelets. The reconstructed signal is shown as a bold line and the acquired data points are shown as light grey circles. The velocity signal was acquired within a uniform channel (water) flow using a laser Doppler anemometer. Lorentz thresholding was carried out following the method of Katul and Vidakovic (1996).

Copyright © 2002 lOP Publishing Ltd.

Box 1

Box 2

Box 3

Box 4

0.25

0.25

0.25

0.25

- ~ Pi 10g(Pi) = 0.60206 i=l

0.80

0.10

0.10

0.00

- ~ Pi 10g(Pi) = 0.27753 i=l

1.00

0.00

0.00

0.00

- ~ Pi 10g(Pi) = 0 i=l

N

(maximum entropy)

N

(an arbitrary intermediate entropy)

N

where

(minimum entropy)

Pi probability associated with box i N = number of boxes, here equal to 4. base 10 logarithums used. also define Pi 10g(Pi) = 0 whenPi = 0

Figure 4.11. Schematic illustration of Shannon entropy measure. Three signals each containing four data points, (0.25, 0.25, 0.25, 0.25), (0.80, 0.10, 0.10, 0.00), (1.00, 0.00, 0.00, 0.00), are shown with each data point contained within a box. The signals have been normalized so that the sum of their components equals 1, analogous to a discrete probability distribution.

We then define the Shannon entropy measure in terms of the normalized wavelet coefficient energies as (4.25) m

n

Numerous authors have used the Shannon entropy measure to select the best wavelet in the analysis of turbulent signals; see for example Briggs and Levine (1997) and Katul and Vidakovich (1996).

4.3 4.3.1

Engineering flows Jets, wakes, turbulence and coherent structures

Coherent structures are large-scale organized motions that exist in turbulent fluid flows and which influence a number of fluid-related processes including mixing, noise, vibration, heat transfer and drag. There has been much research carried out on the twin problems of separating them from background turbulence and characterizing their properties. Recently wavelet analysis has joined the toolbox of methods used in their investigation. Bonnet et al (1998) have described the use ofwave1et transforms, together with a large number of other eddy structure identification methods (conditional sampling, pattern recognition, proper orthogonal decomposition, stochastic estimation, topological concept-based methods, full field methods) in an investigation of a shear layer generated at the interface of two fluid streams of different velocities. They mentioned the utility of the wavelet transform in providing information concerning the location of vortical structures in the flow, i.e. how their Copyright © 2002 lOP Publishing Ltd.

!

I

0.050 ::> 0.025

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Figure 4.12. Velocity time series taken at various centreline locations downstream of an orifice plate in a pipe. This material has been reproduced from Addison P S 1999 Proceedings 01 the Institutioll of Mechanical Engineers, Part C. Jouma! oj Mechanical Engineering Science 213 217-229, figure 4, by permission of the council of the Institution of Mechanical Engineers.

spatial energy is distributed as a function of physical location and length scale. The structure of the near field of a three-dimensional wall jet has been investigated by Sullivan and Pollard (1996). They compared wavelet analysis to three other methods (proper orthogonal decomposition, Gram-Charlier estimation, linear stochastic estimation) for the identification of coherent flow structures from multipoint measurements made in the three-dimensional flowfield. They used the peaks in the wavelet power spectral density plots from a number of simultaneously measured signals to form cross-sectional maps of the spectral peaks across the flowfield normal to the jet axis. By doing this, they found that they could locate secondary flows within the flowfield corresponding to localized vortex motion. Figure 4.12 shows three velocity signals taken within a low Reynolds number pulsed flow downstream of a pipe orifice plate (Addison, 1999). The axisymmetric, periodic vortices shed from the orifice plate produce the regular oscillatory velocity time series shown in figure 4.12(a). This highly organized motion breaks down to a more complex flow regime as it advects downstream (figures 4.12(b) and (c)). The Mexican hat wavelet transform plots relating to the velocity signals are shown as both contour plots and surfaces in figure 4.13. The regular oscillatory nature of the initial vortex shedding is clearly seen in the smoothly undulating transform plot of figure 4.I3(a). In addition, high frequency background turbulent activity can also be seen at the smaller wavelet scales towards the bottom of the plot. Slightly further downstream , the vortices begin to interact with each other in a process of merging and disintegration. The subsequent development of larger scale structures within the flowfield can be seen in the transform plots of figures 4.13(b) and (c). The Fourier and wavelet power spectra associated with the transform plots of figure 4.13 are given in figure 4.14. The appearance of the wavelet spectrum as a smoothed version of the Fourier spectrum is evident in the plots. This is most obvious in figure 4.14(a), where there is a marked smearing of the spectral peak in the wavelet spectrum. According to Bonnet ef al (1998) the smooth wavelet-based spectrum is an advantage when analysing single realizations of the flow. The structure of the eddies in a free jet flow has been investigated using continuous wavelet transforms by Li and Nozaki (1995). Using Mexican hat wavelets they analysed velocity time series taken from a number of locations within the jet at both the jet centreline and the jet edge (mixing layer). In subsequent research, they used a wavelet cross-correlation function based on the Mexican hat (Li and Copyright © 2002 lOP Publishing Ltd.

".,

;;p

~ 0.16

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Figure 4.13. Wavelet transform plots of the velocity time series taken downstream of an orifice plate. (a) 10mm downstream (0 ~ b ~ 2.56s, 0.01 ~ a ~ 0.64s). (b) 75mm downstream (0 ~ b ~ 2.56s, 0.01 ~ a ~ O.64s). (c) 150mm downstream (0 ~ b ~ 10.24s, 0.02 ~ (I ~ 2.56s. Note change in the scale ranges of the axes from plots (a) and (b». This material has been reproduced from Addison P S 1999 Proceedings of Ihe InSlill/lion ofMechclllical Ellgineers, Part C, JOUI"I/(lI of Mechanical Engineering Science 213 217-229, figure 5, by permission of the council of the Institution of Mechanical Engineers.

Nozaki, 1997) and the Morlet wavelet (Li, 1998a) to determine the relationship between two simultaneously measured signals at different locations within the jet flowfield. The wavelet cross-correlation function between two signals x and y is defined for the continuous wavelet transform as

C"y(a, T')

=!T.OfT TAa, b)' Ty(a, b + T') db

(4.26)

where T,(a, b) and Ty(a, b) are respectively the wavelet transforms of signals x and y; the asterisk denotes the complex conjugate; T is the time period of the signal (taken long enough for the signal statistics to settle down) and T' is the delay between the two signals. C,.,.(a, /) is then the cross-correlation between the wavelet coefficients of each signal over a time delay T at scale a. If the correlation between points in the same signal is sought (i.e. the wavelet auto-correlation function) then the same signal is used as x and y in the above equation. Figure 4.15 shows a (Mexican hat) wavelet cross-correlation function together with the traditional cross-correlation Copyright © 2002 lOP Publishing Ltd.

0.1 1 10 frequency (Hz)

100

(bJ

(oj 0.01 r--~,----~-~------, 0.001 ~ ... 1·\0....

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Figure 4.14. Wa,'elet and Fourier spectra of the "elocity time series taken downstream of an orifice plate. (a) 10 mm downstream. (b) 75 mm downstream. (e) 150 mm downstream. The ordinate is the turbulent energy in arbitrary units and the abscissa is the frequency in Hz. Circles indicate the wavelet power spectrum curve. This material has been reproduced from Addison P S 1999 Proceedings of fhe Institutioll of Mechanical Engineers, Parr C, Jot/mal of Mechanical Engineering Science 213 217-229, figure 7, by permission of the council of the Institution of Mechanical Engineers. WRxy (a;t:)

68 62 56 50 44 "

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o

38 32 26 20 14 8

-j).2 -j).4 -j).6 -j).8 -I

E"o·i~

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o

10 20 30 40 50 60 70 80

Figure 4.15. Wavelet cross-correlation of two test signals. Test signals:

X(I) = kl[ sin (2;0) + sin (;)] e-

k

y(I) ~ kl[ cos c;o) +cos (~,)] e-

+01 [sin

'

k '

C'(I ~12))

+ 01 [cos

+sin

C,(I1;8)) +sin (n(I: 3))]

(1ri~)) + cos (nl~))

+cos

("n]

After Li and Nozaki (1997). Reproduced with kind permission of the Japan Society of Mechanical Engineers.

Copyright © 2002 lOP Publishing LId.

y

d

"'

~ t::l

WRxy (a;'t)

255 230 205 180

1 0.8 0.6 0.4

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0.2

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(b)

-11.2 -11.4 -11.6 -11.8

t o

-1

_.~-~--'-

50

100 150 't(ms)

200

250

Figure 4.16. Wal'elet cross-correlation jet flow signals. (a) Schematic of the experimental configuration. (b) Wavelet cross-correlation plot corresponding to locations xlf! = 13 and 2/j"xjd = 3. Arter Li and Nozaki (1997). Reproduced with kind permission of the Japan Society of Mechanical Engineers.

function for the test signal defined in the figure. Figure 4.16(a) shows a schematic of the experimental jet flow investigated by Li and Nozaki, and figure 4.l6(b) shows the crosscorrelation functions (wavelet and traditional) for locations xl d = 13 and 2/',.xl d = 3. It can be seen from the wavelet cross-correlation plot that the large-scale periodic motion at scale a = 140 ms contains three small-scale motions with a = 30 ms. Hudgins el at (1993) have provided an example of cubic spline-based cross-correlation functions for II and IV velocity fluctuations in atmospheric turbulence, and Benaissa et at (1999) have employed a wavelet cross-spectral density function in an analysis of the dissipation of a passive scalar in a heated boundary layer. The vortex shedding downstream of an elliptical plate has been studied by Abe el at (1999). They used the Morlet wavelet (wo = 6) to analyse velocity signals downstream of the plate on both the major and minor axes. Figure 4.17 shows a typical power spectrum obtained from the velocity signals on each of these planes, together with a diagrammatic sketch of the plate (minor plane denoted z, major Copyright © 2002 lOP Publishing Ltd.

z

~ a)

(a)

(b)

o

100

200 I(Hz]

300

y

400

Figure 4.17. Power spectrum of the horizontal velocity fluctuation u. Ca) Signal taken at location (x/D,y/D,zID) = (4.0,0.0,2.0) in the minor plane. (b) Signal taken at location (xl D,yl D, zl D) = (4.0,2.0,0.0) in the major plane. D is the minor diameter of the elliptic plate. Disc aspect ratio AR = 3.0. After Abe el al (1999). Reproduced with kind permission of the Japan Society of Mechanical Engineers.

plane denoted y). From the spectrum we see that the frequency of vortex shedding from the major plane is approximately 80 Hz and the minor plane 45 Hz. Figure 4.18 shows the Morlet wavelet transform of the velocity signal on the minor plane. Both the real part of the transform and the modulus are shown. The dominant vortex shedding scale of a/ t::./ ~ 24 is evident in the modulus plot. (t::./ = 0.5 ms is the digitization time interval of the velocity signal.) This corresponds to the vortex shedding frequency for the minor plane (as seen in figure 4.17). However, a Fourier spectrum of the wavelet transform values at the level a/ t::./ = 24 (figure 4.19) reveals a broad spectrum which peaks in the range 15-20 Hz, i.e. approximately one-fifth of the vortex shedding frequency. A similar low frequency unsteady fluctuation was also found for the major plane at approximately one-fifth of the vortex shedding

~

50

100 10

o

512 blat

1024

Figure 4.18. Morlet wal'elef transform of tI at (4.0, 0.0, 2.0) in the minor plane for AR = 3.0. Top: real part. Middle: modulus, where only values greater than 30% of maximum plotted. Bottom: velocity fluctuation II. Arter Abe el (II (1999). Reproduced with kind permission of the Japan Society of Mechanical Engineers.

Copyright © 2002 lOP Publishing Ltd.

o

5

50

10

100

I[Hz] Figure 4.19. Power spectrum of the fluctuation components of the modulus. (a) In the minor plane at aj!J.f = 24. (b) In the major plane at a/6.1 = 41. After Abe el af (1999). Reproduced with kind permission of the Japan Society of Mechanical Engineers.

frequency of 45 Hz. Further analysis by the authors revealed that, although both low frequency fluctuations were at approximately one-fifth of the vortex shedding frequency, the unsteadiness is in phase on both sides of the wake but out of phase in the different planes. Further details of the work are given by Kiya and Abe (1999). Boniforti el al (1997) have employed the Morlet wavelet in a study of transitional shear flows downstream of a cylinder in an unsteady flow. They used filtering of the wavelet energy map to select structured events (i.e. those 'marked by distinct individuality'). This was done by setting those energy map values far from local maxima to zero. The conditional frequency spectra they subsequently constructed gave useful information which allowed the sclection of eigenmodes linked to further degrees of freedom in a dynamical model of the wake. Higuchi el al (1994) made use of wavelet maps, energy maps and structure maps based on Mexican hat wavelets to investigate the two-dimensional wake occurring behind a pair of flat plates in water flow. As the spacing of the plates became closer, the asymmetry of the wake increased. This increase in asymmetry in the wake is shown in figure 4.20, which contains a flow visualization and mean velocity map of the wake flow at spacing ratios of slh = 2 and sl h = I, where s is the spacing between the plates and h is the plate width. Figure 4.21 contains the time series taken behind the plates spaced at slh = 2, together with its corresponding scalogram and Fourier spectrum. Figure 4.22 shows an energy map for plates spaced at slh = I, together with its corresponding structure map. The structure map is constructed from the energy maps using a hard thresholding algorithm where the local threshold, A, is based on the local maxima in wavelet energies within the scalograms. (Full details of the algorithm are given in Higuchi et al (1994).) The authors state that the following features of this complex asymmetric flow can be observed more clearly on such structure maps: the absence of local maxima at durations between the primary vortices and small scale activity, the lack of apparent correlation between them, the occasional interruptions in the sequence of vortices, fluctuations in frequency and duration and the small asymmetry of the primary shedding frequencies. The authors also used the structure maps in the construction of conditional power spectra.

Copyright © 2002 lOP Publishing Ltd.

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Mouri el al (1999) employed orthonormal wavelet transform statistics to investigate an experimental velocity signal taken from within isotropic turbulence generated downstream of a biplanar grid in a wind tunnel. The signal decomposition was performed using four separate wavelets (Haar, Meyer, Harmonie, Oaubeehies 020) and the authors computed a number of measures including flatness factors and seale correlations. Figure 4.23 shows the results of the flatness factor calculation obtained for each of the wavelets used. Good agreement between the plots for each wavelet exists. However, the Haar does exhibit a slight discrepancy which Mouri and his eo-workers put down to the poor localization of the Haar wavelet in the wavenumber regime. The flatness factor plots all show an increasing trend at small seales due to the intermittency of the flow, whereby the energy associated with small seales is dominated by a small number of wavelet coefficients. This is shown in figure 4.24, where the proportion of wavelet coefficients found above both one and two standard deviations of the coefficient values at each level are plotted. This proportion decreases with decreasing seale as expected, since more energy is concentrated in fewer coefficients at the smaller seales. They also compared their findings with both those of a numerically generated isotropic turbulence field and an analytical solution for the field associated with a vortex tube. Using, among other techniques, the flatness factor and intermittency index, Camussi and Guj Copyright © 2002 lOP Publishing Ltd.

0.02

20.02

40.02

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lis (el Figure 4.21. (a) Velocity trace, (b) wavelet scalogram and (c) Fourier spectrum in the wake oflWO flat plates spaced at sill = 2, x/II = 3, yllI = -3.5. The timescale for the signal matches thai of the map. After Higuchi et at (1994). Reproduced with kind permission of the authors. See also colour section.

(1997) found universal properties in fully developed turbulent flows at low Reynolds numbers. Using Battle-Lemarie wavelets, they developed a conditional averaging technique based on the wavelet transform which allowed them to identify the time signatures of coherent structures. They also investigated a jet flow and found both coherent structures, i.e. vortex rings, and intermittent structures similar to those

found in homogenous grid turbulence. Later, Guj and Camussi (1999) showed that, when used to investigate the temporal dynamics of coherent structures and the evolution of energy at each scale, wavelet statistical measures exhibit almost universal properties. However, when used to investigate the characteristic size and Copyrighl © 2002 lOP Publishing Ltd.

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(e)

time (s)

Figure 4.22. (a) Energy map and (b) and (e) structure maps on the narrow side of a wake behind a double plate at sill = I, x/II = 3, yJh = -2. The detailed map (e) corresponds to the fourth line of (a) and (b). After Higuchi el at (1994). Reproduced with kind permission of the authors. See also colour section.

shape of turbulent structures no universality is observed and, in fact, a strong dependence on the flow Reynolds number and/or flow conditions is observed. See also Camussi and Guj (1999). There are many other papers concerning the application of wavelet transform analysis to jets, wakes and coherent structures. Sutherland and Linden (1998) have used Daubechies D20 wavelets in a study of both wave radiation and the coupling of coherent structures in a wake shed from a tall thin vertical barrier in a stratified fluid (salt water). Ohmura et at (1995) employed Daubechies DIO wavelets for both Copyright © 2002 lOP Publishing Ltd.

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Figure 4.23. Flatness factor versus wavelet scale. The bases used are (a) Harr's, (b) Daubechies', (c) Meyer's, (d) harmonic with cp = 0 and (e) harmonic with cp = ni2. The wavelet scale is normalized by the Kolmogorov length scale 77. Filled circles are for the longitudinal velocity component. Open circles are for the transverse component. Solid lines denote the flatness factor of the velocity increments. L = integral length scale. A = Taylor microscale. After Mouri et al 1999 Journal of Fluid Mechanics 389 229-254. With kind permission from Cambridge University Press.

noise reduction and the extraction of frequency components from an experimental signal of Taylor-Couette flow en route to a region of chaotic turbulence. Van Milligan et af (1995) have detailed a number of wavelet-based correlation measures including a delayed cross-coherence and cross-bi-spectrum, applying them to a model of drift wave turbulence relevant to plasma physics. Within a paper concerning the analysis of the near field flow structure in a turbulent jet, Li (1998b) defined a number of wavelet-based flow parameters including a wavelet Reynolds stress function, wavelet triple velocity correlation function, and wavelet skewness and flatness factors. Iuso et af (1996) have also used the Mexican hat in a wavelet formulation of Reynolds stresses. They found that this wavelet is particularly suitable for finding Copyright © 2002 lOP Publishing Ltd.

§

······················e··w··g···(j··o.. 'i'.. e e e e il2. >2(il2).

0.3

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+

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il2. k>2(il2). J.

J

0.12':-'-'-....4.L...1..1.~6.L...1..1.~8 ........~1O~~12~.u14 log2 (MYJ)

Figure 4.24. Number fractions of wavelet transforms with U},k > (u 2 )j and U},k > 2(u 2 )j. Filled circles are longitudinal components. Open circles are for transverse components. Dotted lines indicate values expected from Gaussian distribution. Note that the authors use the nomenclature il],k to mean the squared wavelet coefficient at location k and level j. In addition, (il 2)) is the variance (average energy) of the coefficients at level j. After Mouri et al 1999 Journal of Fluid Mechanics 389 229-254. With kind permission from Cambridge University Press.

the regular structures in images associated with vortex pairs. Yilmaz and Kodal (2000) have investigated turbulent coaxial jet flows using Morlet wavelet transforms, as have Walker et al (1997), who used the Morlet wavelet to investigate multiple acoustic modes and shear layer instabilities which characterize a supersonic jet. Jordan and Miksad (1998) have examined intermittent events in a wake and Jordan et al (2000) have used a method based on wavelet ridges to demodulate transitional wake instability modes where there is no well defined carrier frequency. Gordeyev and Thomas (1999) have found interesting phase shifting behaviour of the subharmonic instability within a forced laminar jet shear layer using a Morlet wavelet decomposition of velocity signals. They provided a Hamiltonian formulation of the problem, and found good agreement with the experimentally observed phenomena and this model. Hangan et al (1999) have developed a wavelet pattern recognition method and used it to investigate the relationship between small (incoherent) and large (coherent) turbulent scales. Specifically, they employed the method in the study of velocity data fields taken in the near region of two wake generators-a solid circular cylinder and a porous mesh strip. Poggie and Smits (1997) have analysed the wall pressure fluctuations in a Mach 3 flow over a blunt fin using the continuous wavelet transform. They partitioned the signal into two parts: one associated with the characteristic timescale of the shock crossing events and the other associated with the relatively smaller scale turbulent fluctuations. 4.3.2

Fluid-structure interaction

The interaction of fluids with structures is a common phenomenon which manifests itself, within an engineering context, as a number of problems including wind loading on buildings, flow-induced vibration of bridge decks and both the loading and scouring at bridge piers. Gurley et al (1997) have discussed the use of the wavelet transform in a more general paper concerning analysis and simulation tools in wind engineering. They illustrated the potential uses of wavelet-based methods for Copyright © 2002 lOP Publishing Ltd.

transient and evolutionary phenomena, using the specific example of a wind velocity signal measured just after the hurricane eye has passed the measuring instrument. In

another related paper, Gurley and Kareem (1999) have discussed the applications of both the discrete and continuous wavelet transform to earthquake, wind and ocean engineering, including fluids engineering problems such as the transient response of buildings to wind storms, the analysis of bridge responses to vortex shedding and the correlation between pressure measured at a building rooftop and upstream. Both papers give details of two wavelet methods for the simulation of non-stationary wind velocity signals. An example of a measured and corresponding synthetic signal is shown in figure 4.25. Hajj and Tieleman (1996) have suggested using the wavelet transform to characterize the intermittent nature of wind events to model pressure variations on low rise

structures. The authors detailed the advantages of using a wavelet-based approach over a conventional Fourier approach to the problem. They illustrated their ideas briefly using the Daubechies D4 wavelet to decompose a sample wind velocity time series. In later work using the Morlet wavelet transform, Hajj el al (1998) found correlations between energetic events in the atmospheric wind and low pressure

25 20 t5 to

50

to

20

30

40

50

60

to

20

30

40

50

60

(aJ

25 20 t5 to

50 (b)

Figure 4.25. Measured and synthesized wind velocity signals. (a) Measured and (b) synthesized using W
Copyright © 2002 lOP Publishing Ltd.

Figure 4.26. Dimensions of the experimental building and the pressure tap local ions. After Hajj e/ at (1998). Reproduced with kind permission of Academic Press Ltd.

peaks that occur at pressure taps placed over a large area ora low rise building (figure

4.26). Figure 4.27(a) shows a velocity time trace taken by a cup-vane anemometer at a site near the building. Figure 4.27(b) shows the pressure trace taken at location 2 during the experiment. Figures 4.28(a) and (b) contain the energy density plots corresponding to the velocity and pressure signals of figure 4.27. Figure 4.28(c) shows the cross-scalograms for the velocity and pressure data. The cross-scalogram is defined by Hajj el at as T,(a, b)' Ty(a, b), where T,(a, b) and Ty(a, b) are the transform of two signals X(I) and Y(I) respectively and the asterisk represents the complex

~

12

8

(al

-I <.J~

-2 F-=--4~1-

-3~~~§ Tap tr,2

-4 (b)

Figure 4.27. Velocity and pressure time traces. (3) ti velocily time trace. (b) Pressure I race al lap location 2. After Hajj e/ a/ (1998). Reproduced with kind permission of Academic Press Ltd.

Copyright © 2002 lOP Publishing Ltd.

rr:::......--......,..,."..---.. 5 fJlM'1I'"""""::"'=' 4 3 II

2

I

°O':--""20~O:--'40~O:--'60~O:--='· I(S)

(aJ

(b)

(e)

Figure 4.28. Wavelet energy density and cross-scalogram plots. (a) Energy density plot of velocity signal. (b) Energy density plot of pressure trace. (c) Cross-scalogram plol. Arter Hajj et al (1998). Reproduced with kind permission of Academic Press Ltd.

conjugate. This type of scalogram gives peaks where the fluctuations from the two time series coincide in scale and time. The cross-scalogram shows that there is a

feature occurring at In(a) = 4 (corresponding to a frcquency of 0.05 Hz) at bctween 450 and 550 seconds. The cross-scalograms show the relationship between the timelocalized fluctuations of thc velocity and low pressure peaks. This was found to be the case for all four pressure tappings and both the It and v velocities considered in the study. Hajj (1999) also analysed data from the same source using Daubechies D4 wavelets to show that the energy of each of the scales of turbulence within the wind signals varied significantly with time. He provides quantitative measures of this intermittency and shows that some of the scales contribute significantly to the

total cnergy over short periods of time. Marshall el at (1999) have used both the first and second derivative of Gaussian (DOG) wavelets to study gust-induced bending moments in trees. After initially attempting to use the bending moment time series

to trigger the conditional sampling of the vertical and horizontal velocities, they inverted the procedure and used vertical velocity events to trigger the conditional

sampling of the bending moment and horizontal velocities. The trigger for the conditional sampling was the detection of a 'Honami gust'-an upward--
Daubechies tight frame wavelet-in the elucidation of vibration signals from a cylinder in a cross flow. Jubran el at (1998a,b) extended this work to cover the chaotic nature of vortex shedding from a cylinder in a cross flow.

4.3.3

Two-dimensional flow fields

A number of studies have been conducted concerning the wavelet analysis of twodimensional flow fields-either imaged data or numerically simulated flow fields. In this section we review a few of these studies. Copyright © 2002 lOP Publishing Ltd.

~-=~ ~

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The digital images of a mixing layer have been analysed by Kailas and Narasimha (1999) using two-dimensional Mexican hat wavelets. Figures 4.29(a) and 4.30(a) contain the original images of the laminar and turbulent mixing layers respectively. Thresholded versions of the original images are plotted with the originals to reveal

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the scale-dependent structure of the images. The threshold was set so as to capture details uniq ue to that scale, while omitting unnecessary clutter. The roll-up of each vortex in the laminar mixing layer is obvious in the d = 0.02 plot of figure 4.29. The homogeneous nature of the small-scale structure present within the turbulent mixing layer is clearly seen at the smallest wavelet scales (d = 0.0 I, 0.02 and 0.04) in figure 4.30. The large-scale structural organization of the turbulent mixing layer becomes more obvious at larger scales. Kailas and Narishima followed up this work in a study of the coherent structures evident in the vorticity fields of numerically simulated jets, both with and without heating (Siddhartha ef ai, 2000). Using a

--6--4-20246 vonicity_< (wavelcl)

--6--4-20246 vonicity_«Fourier linear)

Figure 4.31. Comparison of nonlinear wavelet filtering (left) with linear Fourier filtering (right) of vorticity for the same compression rate. Top: total vorticity. Middle: coherent parI. Bottom: incoherent part. (Note that the original figure is in colour; see colour section.) Arter Farge ef at (1999). Reproduced with kind permission of the authors.

Copyright © 2002 lOP Publishing Ltd.

two-dimensional wavelet which imitated the vorticity field topology, Vorobieff and Rockwell (1996) investigated two-dimensional, PIV (particle image velocimetry) vorticity fields generated beneath a delta wing. In particular they focused their attention on the distinct topological structures associated with the onset of vortex breakdown and found the wavelet transform to be an effective pattern recognition tool for this kind of study. Ishizawa and Hattori (1998) separated out the turbulent and coherent subregions of two-dimensional magnetohydrodynamic fields using Meyer wavelets where each of the separated regions could be defined by its own distinct spectral slope. Rightley ef a{ (1999) have filtered images of shock-accelerated gas curtains using two-dimensional Mexican hat wavelets. They found that their

wavelet analysis produced clear evidence for the growth of scales both smaller and larger than initial disturbances imposed on the flow. Everson ef a{ (1990) have also used two-dimensional Mexican hats to study two-dimensional spatial dye

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Copyrighl © 2002 lOP Publishing Ltd.

concentrations in turbulent jets from laser-induced-fluorescence images to gain an insight into the spatial-scale-dependent structure of the flow field. Do-Khac er al (1994) have used both continuous and discrete orthonormal wavelet analysis in a study of two numerically generated, two-dimensional turbulent fields: a growing planar mixing layer and homogenous turbulence (see also Basdevant er ai, 1993). Using a third-order B-spline wavelet, Tsujimoto el al (1999) analysed a direct numerical simulation of channel flow in an investigation concerning the grouping of quasi-streamwise vortices. They found evidence for clusters of the quasistreamwise vortices appearing intermittently within the flowfield. Farge er al (1999) used an iterative method based on hard thresholding to extract coherent vortices from two-dimensional turbulent vorticity fields. The method is designed in such a way that it results in the discarded wavelet coefficients of the vorticity field exhibiting a Gaussian distribution. This was tested using moment functions (including skewness and flatness) of the reconstructed signal using the weak coefficients. When this reconstructed signal is of a Gaussian form, it is assumed to contain the noisy part of the signal and the remaining strong signal is assumed to contain the coherent part. Figure 4.31 shows a realization of the two-dimensional vorticity field, together with its decomposition into coherent and incoherent parts, using both quintic spline wavelets and conventional Fourier techniques. We can see from figure 4.32, which contains the PDFs of the coherent and incoherent parts, that the incoherent part of the vorticity field defined by the wavelet filtering more closely matches the Gaussian PDF than does the Fourier filtered field.

4.4

Geophysical flows

The meteorological community has been particularly active in the application of wavelet-based methods to the analysis of fluid flows. Many of the papers already cited earlier in section 4.2 come from this source. Meyers er al (1993) have provided an introduction to the use of wavelets (specifically the Morlet wavelet) in Oceanography and Meteorology. They illustrated their discussion with examples of the wavelet decomposition of simple signals into modulus and phase before applying wavelets to the analysis of Yanai waves. They commented both on the 'nontrivial task' of interpreting the phase of complicated signals and on the edge elfects of the data on the transform plots. The rest of this section covers the application of wavelet transforms to a variety of geophysical flow phenomena. 4.4.1

Atmospheric processes

A number of workers have investigated the power spectra of time series measurements acquired in the atmospheric boundary layer. Some researchers have investigated the power spectra of the measurements and the deviation from the expected -5/3 Kolmogrov scaling in the inertial subrange. Others have concentrated more on the detection and interrogation of coherent structures in the flow. Figure 4.33 shows a plot of the power spectra (Haar, wavelet and Fourier) of three signals (u, IV velocities and temperature) taken in the atmospheric boundary layer by Katul Copyright © 2002 lOP Publishing Ltd.

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•

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2 2 t> 10"~

"-

10-' W-' 10-3 2

10-2 2

(b)

to-I 2

10'

2

10 1 2

102

wavenurober(rn)

Figure 4.33. Fourier and wavelet power spectra. (a) Comparison between Fourier (solid line) and Haar wavelet (closed circle) power spectra for the longitudinal (II) and vertical velocity (IV). The Ii spectrum is shifted by two decades to permit comparison with II' spectrum al small wave numbers. Taylor's hypothesis is used to convert the time domain to wave number domain. The -5/3 power law (dotted line) predicted by K41 is also shown. (b) Same as for (a) but for temperature. After

Katul et al (1994). Reproduced with kind permission of the authors.

ef al (1994). There is good agreement between the Fourier and wavelel spectra. The

-5(3 signature of the inertial subrange is evident in all spectra, especially for that of the 11 velocity where it eXlends over a large portion of Ihe curve, Figure 4.34 shows a plot of the fluctuation intensity (called the coefficient of variation, CV, by the authors) for the three signals. An increase in the fluctuation intensily with increasing wavenumber indicates increasing turbulent energy activity at smaller scales. Also

noticeable in the plot is that the flucluation inlensity for the temperature signal is much larger than those of the velocities, possibly indicating that the temperature is not simply advected by the flowfield, even at small scales. Szilagyi ef al (1996) Copyright © 2002 lOP Publishing Ltd.

5.5 5.0 4.5 4.0 3.5 3.0 .

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-{l.5 '-:-~----=-:--~-----=c-:-----::-:~2 3

to-I

2 3

100

wavenumber

2 3

101

2 3

102

(m- l )

Figure 4.34. Fourier and wavelet power spectra. The coefficient of variation (CY) as a [unction of wave number for longitudinal and vertical velocity as well as temperature. The dotted line is the CY assumed by Fourier analysis. After Katul el a/ (1994). Reproduced with kind permission of the authors.

investigated the effect of turbulent intermittency on the shape of the power spectrum of an atmospheric boundary layer turbulent time series. Using Daubechies 04 wavelets, they showed that the local wavelet spectrum in the inertial subrange was sensitive to intermittency in the flow. They defined the strength of the intermittency in terms of the wavelet variance. For regions of weak intermittency the local wavelet spectrum was found to have a slope flatter than that of Kolmogorov's -5/3 law, whereas regions of strong intermittency exhibited slopes greater than -5/3. The average slope tended to the -5/3 law. Katul and Parlange (1995) have analysed the -I power law that arises in heat flux measurements under unstable atmospheric conditions. Large values of wavelet flatness factor (much greater than 3) indicated strong non-Gaussian statistics of the signals in the -I power law range. This they linked directly to the widening of the gradient probability density function of the flux measurements. Using the Haar wavelet transform, Katul and Chu (1998) computed the power spectra associated with velocity signals acquired both from the atmospheric surface layer and laboratory open channel flow in a study of the emergence of a universal slope of -I in turbulent power spectra at physical scales larger than the inertial subrange (with its own well documented slope of -5/3). See also Hudgins el al (1993) who make use of cubic spline-based cross-spectra and cross-sealog rams in their investigation of velocity fluctuations in atmospheric turbulence. The large-scale intermittent structures involved in the exchange of heat and mass within and above plant canopies have been investigated using wavelet decomposition of a variety of signals by a number of research workers. These structures usually appear in the signals (e.g. temperature, vapour density, velocity) as distinct large amplitude excursions, or jumps from the mean value. Many researchers have used the locations oflocal maxima of the wavelet variance of the continuous wavelet Copyright © 2002 lOP Publishing Ltd.

600,..---------,-----..., straw mulch, z=9.6 em 14:50-15:00 PST,August 23,1994

-----------",-.... I

400

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-

~

200

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I I I I

I

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0.1 a (s)

Figure 4.35. Determjnation of optimum

(I

=

am at the scale of maximum wavelet variance. From

Chen W, Novak M D, Black T A and Lee X 1997 'Coherent eddies and temperature structure functions for three contrasting surfaces. Part 1: ramp model with finite microfront lime', Boundary-Layer

Meteorology 84 99-123. Reproduced with kind permission of Kluwer Academic Publishers and the authors.

transform to identify the timescales of coherent structures. We saw 2, section 2.9, that the wavelet variance is defined as

c?(a)

=~JT T

IT(a,b)I' db

In

chapter

(4.27)

0

where T is the duration of the signal segment under investigation. Obviously the longer r is, the more accurate a value of 0"2(0) is obtained for a stationary signal. The wavelet variance is, in fact, a measure of the average energy associated with scale a (refer back to the end of section 2.9 of chapter 2.) Note that some authors omit T from the denominator, some add Cg , and others add Cg andf, to the denominator in their definition of wavelet variance. If Cg andf, are both added we get back to the wavelet power spectral density of equation (4.11). It is reasonable to use any of these definitions as T, Cg and j~ act only as constant rescaling factors, and only the locations of the peaks in a'(a) are required in order to identify the dominant scales in the signal. Chen et al (1997b) decomposed temperature signals using the Mexican hat wavelet in an attempt to detect ramp structures in turbulent Row signals acquired above a variety of surfaces. The average recurrence times determined from the wavelet analysis agreed well with those found from the analysis of third-order structure functions of the temperature. Figure 4.35 shows the wavelet variance plot derived from one of these temperature signals. The peak in the variance plot was used by the authors to select the dominant scale, which they denoted am' The wavelet coefficients at this scale were then plotted together with the original signal and used to locate coherent structures in the Row. This is shown in figure 4.36 for three surfaces: bare soil, straw mulch and forest. The dotted line in each of the three plots corresponds to the wavelet coefficients at scale am for each signal (solid lines). Copyright © 2002 lOP Publishing Ltd.

............ F(a",b)

- - T(I)

3

bare soil

0

-3 5

., straw mulch

~

u 0

,...

~

0

"-

-5

0

2

1

3

5

6

Douglas-fir forest

0

-5

5

4

.....

0

50

100

150

200

1(')

Figure 4.36. Typical segments of air temperature time series (solid lines) and the wa\'ele' transform values at that level of am (dotted lines). E,'cry zero crossing with a negative slope of the dotted line signifies a ramp event. From Chen W, Novak M D, Black T A and Lee X 1997 'Coherent eddies and temperature structure functions for three contrasting surfaces. Part I: ramp model with finite microrront time', Bowulary-Layer Mefeorology 84 99-123. Reproduced with kind permission of

Kluwer Academic Publishers and the authors.

Each zero crossing of the coefficient trace with negative slope signifies a ramp event. Gao and Li (1993) have also used wavelet variances from a Mexican hat wavelet decomposition of thermal and velocity fields at an atmosphere-forest interface to identify coherent structures. Their data consisted of simultaneously sampled temperatures and vertical velocities at six heights: two within and six above a deciduous forest. They plotted scalograms at each height over the measurement duration and used the scalogram data to construct wavelet variance plots of both temperature and vertical velocity signals in order to locate coherent structures within the flow field. Qiu et al (1995b) employed representative sections of flow signals as the basic shape for pseudo-wavelets in a study of turbulence patterns above three different vegetation layers: orchard canopy, forest canopy and maize canopy. These pseudowavelets were simple linear ramp shapes which matched with the global structure of the ramp structures evident in the signal. Although, these pseudo-wavelets do not satisfy the mathematical constraints required of them to be considered wavelets, the authors used them in the same way as wavelets to find coherent structures from within the signal. Collineau and Bruinet (1993a,b) used four continuous wavelets (Haar, Mexican hat, a wave shape and a ramp shape) to detect coherent motions in a forest canopy. Lu and Fitzjarrald (1994) used the peaks in the scale-dependent wavelet variance of the continuous Haar wavelet to identify coherent structures Copyrighl © 2002 lOP Publishing Ltd.

above a midlatitute deciduous forest. It is also worth noting that wavelets have also been employed to characterize the physical gaps that occur in the forest canopy itself (see chapter 7, section 7.4.2). Howell and Mahrt (1994b) have extracted energycontaining events from a wind velocity signal taken above flat terrain in near neutral

conditions using an adaptive multiresolution data filter. Qi and Neumann (1997) used both the Haar and Oaubechies 020 wavelets in both standard multiresolution and non-decimated formats in an assessment of timescale-dependent errors of the bulk aerodynamic formula for turbulent heat fluxes used in general circulation models (GCMs). They focused their investigation on the errors arising between a GCM simulation and various eddy correlation flux measurements, including those of

wind, sensible heat, latent heat and CO" above a forest canopy. Hayashi (1994) used Meyer wavelets in an investigation of the turbulent transport of momentum

in the atmospheric surface layer. The study found that the large scale fluctuations were mainly responsible for the large downward transport of momentum. Yee ef 01 (1996) used a wide range of wavelet statistical measures to analyse the dynamical characteristics of concentration fluctuations in a dispersing plume. They used their results to construct a conceptual model of the turbulent transport, stirring and mixing regimes in a dispersing plume. Katul and Vidakovich (1996) have used the Shannon entropy measure to pick the optimal Oaubechies wavelet for Lorentz thresholding-based partitioning of the attached and detached eddy motion in atmospheric turbulence signals. See also Katul ef a/ (I998a,b). Szilagyi ef a/ (1999) have provided a method for finding the principal timescale of coherent structures using the scale-dependent wavelet coefficient energies. These are normalized, to find the relative contribution at each scale, and corrected for the presence of signal noise using their own 'global' method. They illustrated their new method using Oaubechies (04,06,08) and Symmlet (S8, S12, S16) wavelets to transform wind velocity measurements taken above bare and vegetated surfaces. Their method, which decomposes the energy contribution at each scale into an organized and random eddy motion, gives very similar results to denoising using Donoho and Johnstone's (1994) thresholding method. Turner and Leclerc (1994) have used the Haar wavelet in the conditional sampling of velocity-time series data to identify coherent eddy structures within the 'more random' flow field of atmospheric turbulence. This was performed by thresholding the time series at arbitrary multiples of the standard deviations of the wavelet coefficients at each scale. These were used to produce a strong signal from those coefficients above the threshold together with a weaker background signal from those coefficients below the threshold. Kulkarni e1 a/ (1999) have used a variety of (Haar) wavelet-based statistical measures to investigate intermittent turbulent transport in the atmospheric surface layer over a monsoon trough region including power spectral density, coefficient of

variation of energy (fluctuation intensity), flatness factor, and an isotropy (ISO) coefficient defined as

,

ISO(k ) =

'" where u, vand

IV

2a;.(k",) ~(k",)

+ ~(k",)

(4.28)

arc respectively the longitudinal, transverse and vertical components

of velocity; km is the wavenumber corresponding to wavelet scale 111; and O"~(km) Copyright © 2002 lOP Publishing Ltd.

1730 ="""""'7--r--,-,-nTT1n

O. 5

1430

o

1135

0

O. 5

0830

0330

0030 '-'-'-'-.L.Ju.J........ I 2 3 4

5

6

7

8

9

10

11

12

scale (m)

Figure 4.37. The isotropy (SO(K",) with scales at different obsen'ational hours. (The bottom axis is actually scale index m.) From Kulkarni J R, Sadani L K and Murthy B S 1999 'Wavelet analysis or intermittent turbulent transport in the atmospheric surface layer over a monsoon trough region', Boundary Layer Me/eorology 90 217-239. Reproduced with kind permission Kluwer

or

Academic Publishers and the authors.

denotes the wavelet variance of the

II

velocity signal at scale index

111.

If all three

components exhibit the same variance, the anisotropy coefficient is equal to unity.

The contour plot in figure 4.37 shows the variation in isotropy with scale index 111 over a 17 hour period. Kulkarni and his co-workers used regions of sharp decrease in the isotropy coefficient to define separation between small-scale eddies and large-scale eddies in the flow field. The small-scale eddies exhibit a high degree of three-dimensionality, whereas the larger-scale structures become increasingly two

dimensionaL The temporal evolution of the flatness factor for the same data is shown in figure 4.38. The plot is constructed using equation (4.5) to compute the flatness factor for each scale at various times throughout the observation period. A contour plot is then generated using this computed data with the scale index as the horizontal axis, the time as the vertical axis and the flatness factors used to form the contours. The increase in the flatness factor with smaller scales, indicating increasing non-Gaussianity, can be seen in the plot together with the variation in

this statistical measure throughout the day. In the study, the sampling frequency of the sonic anemometer used was 8.42 Hz and the mean of the longitudinal velocity was 3.025 m s-'. Thus the scale indices 111 in the flatness plot relate to physical distances (8.42/3.025) x 2'" metres. Using both symmetric and antisymmetric spline wavelets and dyadic wavelet transforms, Hagelberg and Gamage (1994) examined both scale thresholding and hard thresholding in a study of velocity, temperature and buoyancy flux signals taken from within the atmospheric boundary layer. They employed a combination of both thresholds to separate out the coherent and non-coherent part of the signals. They went on to use wavelet variance to detect the dominant scale in each of the Copyright © 2002 lOP Publishing Ltd.

1430

1135

0830

0330

0030

,-","---l-l.-'-",--""_-,""

1

2

3

4

5

6

7

8

9

10

11

12

scale (m)

Figure 4.38. The flatness factor FF( K",) with scales at different observational hours. (The bottom axis is actually scale index m.) From Kulkarni J R, Sad ani L K and Murthy B S 1999 'Wavelet analysis of intermittent turbulent transport in the atmospheric surface layer over a monsoon trough region', Boundary Layer Meteorology 90 217-239. Reproduced with kind permission of Kluwer Academic Publishers and the authors.

signals. The wavelet transform was then interrogated at this scale to form an intermittency index based on the extrema in the wavelet transform occurring at this maximum scale of wavelet variance. Hagelberg et a/ (1998) have used hard

thresholding of two-dimensional Mexican hat wavclet eoefficicnts to analyse the scale distribution of coherent water vapour structures in the marine atmospheric

boundary layer as measured by a shipboard Raman lidar. Using the minimum entropy criterion described previously in section 4.2.4, Briggs and Levine (1997) selected the Daubechies 08 from the Daubechies family 02 to 020 in a study which used wavelet thresholding methods to filter two-dimensional meteorological forecast fields. Mahrt (1991) used both wavelet variance and skewness of a Haar wavelet decomposition of aircraft-measured vertical velocity data in an investigation of the

eddy asymmetry in the sheared heated atmospheric boundary layer. Gamage and Blumen (1993) havc employcd wavelets to inspect temperature records containing atmospheric cold fronts. They stated that the wavelet transform is particularly appropriate for such frontal signals which arc characterized by relatively isolated frontal gradients and by aperiodic and intermittent disturbances. Weng and Lau (1994) have used both the orthogonal Haar and continuous Morlet wavelets scalograms to probe satellite-collected infrared radiance data for the organization of convection processes over the Western Pacific. Grivet-Talocia and Einaudi (1995) have developed a wavelet-based automatic algorithm for the detection, identification and extraction of gravity waves from atmospheric pressure traces. Demoz et a/ (1998) used the Morlet wavelet in a study of the dynamical processes in cirrus clouds. They interrogated the Morlet scalograms to elucidate combined information on the Copyright © 2002 lOP Publishing Ltd.

intensity and phase of the undulations in vertical air velocity-time series which they interpreted as convective cells and, at larger scales, gravity wave trains. Other studies which have applied wavelet-based techniques to large scale geophysical flow data include the investigation of the El Nifio Southern Oscillation (ENSO) by Kestin et al (1998) and the investigation, using the Mexican hat wavelet, of meteorological variables (wind speed, pressure and electrostatic field) by Takeuchi et al (1994). 4.4.2

Ocean processes

A clear and concise study of the grouping characteristics of wind waves using the Morlet wavelet transform has been presented by Lui (2000). He used the transform plot to identify local wavegroups in a time series of surface elevation corresponding to wind-generated waves measured in nearshore areas of the Great Lakes. He found a linear relationship between group energy and duration. He also showed that mean maximum group wave height and significant wave height differ by around 17% and stressed the implications this has for engineering design. Finally, he used his results to illustrate the non-stationarity of the data, linking his findings to an earlier study concerning the characteristics of waves on the Atlantic Ocean (Liu, 1994). Shen et al (1994) studied wind-generated ocean surface waves using Morlet wavelets and produced visual evidence from the wavelet scalograms which showed the wave action conservation law not to hold for long fetches. Using a decomposition of wave data based on complex Morlet wavelets, Savtchenko et al (1998) studied the relationship between the phase of the dominant waves and moments of the first appearance of burst and sweep events in corresponding velocity fluctuation data. Their results allowed them to identify strong contributions to the Reynolds stress during various atmospheric conditions in the coastal marine atmospheric boundary layer. However, they found no evidence for the correlation of wave phase with the bursts and sweeps. This supports their conclusion that it is not only the wavefield that plays a role in the generation of these ordered motions but that large scale eddies in the windfield playa part too. Willemsen (1995) has used wavelet packets to quantify the lag and scaling of maximum wave heights with wind speed over the ocean and, in smaller-scale laboratory experiments, Jinshan et al (1998) have employed Morlet wavelet-based decomposition in a novel method to detect and quantify breaking waves from wave height signals. The analysis of subtidal sea fluctuations has been performed by Percival and Mofjeld (1997) using non-decimated discrete wavelet transforms. In their discussion they outlined the implications for coastal inundation forecasting using such waveletbased analysis techniques. Quinquis et al (1996) have used Daubechies DIO wavelets to analyse the fine structure extracted from oceanographic celerity profiles. Machu et al (1999) analysed both sea surface heights and ocean colour (chlorophyll pigment) data to detect Rossby waves appearing on the ocean surface. R6denas and Garello (1997) have used both continuous and discrete wavelet analysis to detect and characterize oceanic internal waves from SAR (synthetic aperture radar) ocean image profiles and Gourdeau (1998) has used wavelet analysis in a study of internal tidal waves in the ocean. The analysis of sea surface temperature time series in the Copyright © 2002 lOP Publishing Ltd.

Indian Ocean has been considered by Meyers and O'Brien (1994) and the analysis of two-dimensional sea surface temperature fields off the coast of Japan and their effect on the turbulent mixing processes is described by Ostrovskii (1995). Applications of wavelet transforms to other geophysical processes including cloud structure and surface temperatures are given in chapter 7, section 7.4.

4.5

Other applications in fluids and further resources

Two comprehensive introductory papers on the role of wavelet transforms in the

analysis of fluid flows are those by Farge (1992) and Meneveau (199Ia). See also Farge el al (1996) which details the applications of wavelets and wavelet packets to the analysis, modelling and computation of turbulent flows. A number of edited texts contain papers concerning the modelling and analysis of fluid flows. WaFelels in Geophysics edited by Foufoula-Georgiou and Kumar (1994) contains a number of papers concerning the wavelet analysis of geophysical flows. The book concerning wavelets in physics edited by van den Berg (1999) contains a number of papers concerning fluid turbulence. The reader is also directed to the papers cited in other chapters of this book which concern: turbulent flows within combustion engines (chapter 5, section 5.8.3); blood flow (chapter 6, sections 6.4.1 and 6.5); subsurface flows, river runoffs and liquid-bubble column flows (chapter 7, sections 7.4.2, 7.4.4 and 7.5.2 respectively); and turbulent plasmas (chapter 7, section 7.5.3). A number of attempts have been made to probe the multifractal nature of turbulent flows using wavelet transforms. See for example Roux el al (1999) who used the wavelet modulus maxima method to detcct vorticity filaments in turbulent swirling flows. They discriminated between vortex filaments and background pressure fluctuations by examining the profile of the modulus maxima lines: specifically the a-scales corresponding to the beginning of the modulus maxima line and its peak value, and the magnitude of the peak value itself. They effectively filtered the signal using thresholds based on these three criteria. They also considered the multifractal nature of the data using the modulus maxima. There are many other papers concerning the fractal and multifractal nature of turbulence and the use of wavelet transforms in describing them. Some of these are cited in chapter 7, section 7.2, where the link between wavelet analysis and fractal geometry is explored (see especially figure 7.12). In addition, the reader is referred to the paper by Roux el al (1999), which contains over 120 references to work in this field. There are many other examples in the literature of the application of wavelet-based techniques to fluid flow problems. Liandrat (1996) has described wavelet algorithms for the analysis and modelling of turbulence and Trevino and Andreas (1996) have discussed the application (and limitations) of wavelet analysis applied to nonstationary flows. Liu and Chen (1995) have detailed wavelet-based numerical methods for the simulation of one-dimensional and two-dimensional advection-diffusion phenomena, and

Elliot and Majda (1994) have described a wavelet-based method for the stochastic simulation of turbulent diffusion. Gerritsen and Olsson (1996) have developed easy-la-implement mathematical criteria based on antisymmetric quadratic spline Copyright © 2002 lOP Publishing Ltd.

wavelets to detect discontinuities, sharp gradients and spurious oscillations within a numerical model of fluid flows. Anderson and Diao (1995) give details of a two-dimensional wavelet transform method for the analysis of images used in holographic particle velocimetry, and Kishida et al (1999) have used three-dimensional helical wavelets to investigate the local interactions in the nonlinear energy transfer process within three-dimensional homogeneous, isotropic turbulence.

Copyright © 2002 lOP Publishing Ltd.

Chapter 5

Engineering testing, monitoring and characterization 5.1

Introduction

Wavelet analysis has been applied to a variety of pertinent problems in engineering. In this chapter, we review a selection of these, including the assessment of machine processes behaviour; condition monitoring of rotating machinery; the analysis of nonlinear and transient oscillations; the characterization of structural impacting; the interrogation of NDT signals; and the characterization of rough surfaces. As with the fluid problems described in the previous chapter, the choice of the most appropriate wavelet to use in the analysis of engineering problems depends very much on the nature of the data itself. Both discrete and continuous (usually complex) wavelets have been used to monitor rotating machinery such as gears, shafts and bearings. Discrete wavelets are favoured when, for example, a small number of data are required as input to a classifier such as a neural network. Continuous wavelets are favoured when high temporal resolution is required at all scales. Complex continuous wavelets are well suited to the free vibrations of plates and beams. The temporal records of such vibrations quickly exhibit a high degree of complexity due to the superposition of multiple wavegroups (from multiple reflections at the specimen edges) whose group velocity is frequency dependent. Complex continuous wavelets are able to unfold these signals in time and frequency, allowing for the decoupling of vibration modes. Most surface characterization work has used discrete wavelets, whose coefficients are used to determine scale dependent surface characteristics such as a power law (fractal) scaling of a surface. Finally, as we might expect, discrete wavelet transform coefficients are particularly useful for signal compression problems . . . m engmeenng.

5.2

Machining processes: control, chatter, wear and breakage

The objective of any machining process is the efficient production of a part of specific shape with acceptable dimensional accuracy and surface quality. The monitoring of machining processes is therefore an important problem in manufacturing engineering. There is a considerable economic incentive to develop a reliable monitoring Copyright © 2002 lOP Publishing Ltd.

( .5

-1.0

-0\ I

0.5

1.0

\j5

(a)

1.0 0.5 -1.5

-1.0

O.

-0.5

1.0

1.5

-'{j.

(b)

10•

Figure 5.1. Caussian modulated sinusoid: the w3yelet used by Khraisheh et (II in their metal cutting study. (a) Real part. (b) Imaginary parI. After Khraisheh et at (1995). Reproduced with kind permission of Academic Press Ltd.

technique, hence a considerable research effort has centred around this problem over recent years. Success has been limited by the inherent problems associated with the monitoring signals which are typically affected by process defects (e.g. chatter and wear), working conditions, process noise and sampling noise (Wu and Ou, 1996). Recently, a number of researchers have attempted to tackle the problem using wavelet transforms. Khraisheh el al (1995) used a modulated Gaussian (i.e. Morlet type) wavelet in an experimental investigation of chatter vibrations occurring during

metal cutting. The complex wavelet they used in the study is shown in figure 5.1. Figure 5.2 contains one of the time series from the study together with its wavelet transform plot. The transform plot is partitioned into two regions: white for high energy regions and black for low energy regions. The cutting process caused builtup edges (BUEs) to appear on the cutting tool, which broke off intermittently during the cutting process. The occurrence and breakage of these edges is evident in the wavelet transform plot of figure 5.2 as white vertical patches extending to the high frequencies. In addition to the detection and occurrence of BUEs during the cutting process, Khraisheh and his colleagues showed that the wavelet transform was good at detecting the boundary of transient regions in these signals. The frequency information found in the wavelet transform of the signals suggested that Copyright © 2002 lOP Publishing Ltd.

200 150 100 50 0 -50 -100

-150 1.74

1.76

1.78

1.80

1.82

1.84

1.86

1.88

1500

!,.,

1000

"" ""

500

u

C'

'"

1.740

1.765

1.790

1.815 time (s)

1.840

1.865

1.890

(b)

Figure 5.2. Time series based analysis of primary chatter in metal cutting. (a) The time history for tool acceleration in the x direction. (b) The wavelet transform. White represents rich signal energy and

black is poor signal energy. (Feed rate = 0.254 mm/rev, width of CUI = 5.08 mm, spindle speed =200rpm.) Arter Khraisheh el at (1995). Reproduced with kind permission of Academic Press Ltd.

the cutting process was quadratically nonlinear. The authors went on to detect chaotic motion in the cutting signals for high feed rates of material. Biorthogonal spline wavelets were used by Berger el al (1998) to identify chatter and non-chatter cutting states associated with the orthogonal cutting of stilT metal cylinders using a CNC lathe. They investigated the elTect of both cut depth and cutting frequency (independently). Figure 5.3 shows the cutting force signal (top) for a 2.3 mm cut, together with the detail components for scales I to 5 (labelled d I, d2, etc. in the figure) plus the remaining approximation (labelled as). The authors used a variety of wavelet-based signal amplitude parameters including the standard

deviation, mean absolute and median absolute deviations of the signal details. They found that the ratio of the mean absolute deviations of scales 3 and 4 provided the best way to dilTerentiate between pre-chatter and chatter states. The mean absolute (m.a.) deviation of a series,

Xi,

i

= 0, ...

IN-I

m.a. = -

L

N ;=0

1

N - I, is defined as

Ix; - -
(5.1 )

where X is the mean value of x;. (Do not confuse with the median of absolute deviation often used within thresholding algorithms-sec chapter 3, section 3.4.2.) This statistic was then calculated for the detail sigual components at each scale, denoted m.a.(m). Figure 5.4 shows a plot of the ratio of m.a.(3)jm.a.(4) for these Copyright © 2002 lOP Publishing Ltd.

(b)

Figure 5.3. Biorthogonal 6,8 spline wavelet decomposition of a cutting force signal. (Note scale indexing.) The original signal, S, is at the lOp of the figure. Immediately below s is the signal approximation, as, at scale 5. The detailed signals over the first five scales, clS-dl, are given below as. After Berger ef of (1998). Reproduced with kind permission of Academic Press Ltd.

18 16 14 12 0

.~

•

10

8 6

0_____ 0-

4

-.

2 "'--'-----'-----'---'----'--....!....--'---'--'----' 2.30 2.35 2.40 2.45 2.50 2.55 2.60 2.65 2.70 2.75 2.80 depth of cut (mm)

Figure 5.4. The ratio m.a.(3)/m.a.(4) versus cui depth for the signal in the pre\'ious figure. After Berger et (II (1998). Reproduced with kind permission of Academic Press Ltd.

Copyright © 2002 lOP Publishing Ltd.

signals taken for a series of cut depths from 2.3 mm to 2.8 mm. The chatter state occurs at a 2.8 mm depth of cut and it can be seen from the plot that it has an m.a. ratio markedly higher than the four pre-chatter states at 2.3, 2.5, 2.6 and 2.7 mm. Berger and his co-workers also used the variation in the kurtosis of the detail signals at scale 3 to complement the m.a.(3)(m.a.(4) ratio as a chatter indicator. Kamarthi and Pittner (1997) have compared Fourier transform and wavelet transform-based neural network schemes to determine the wear of lathe tools from the force and vibration (acceleration) signals taken at the tool holder. Subsequently, Pittner ef a/ (1998) proposed a wavelet network scheme to tackle this problem where the two major tasks in the method-sensor data representation and flank wear assessment-are combined within a single computational unit. This is done by determining the wavelet parameters and the neural network weights concurrently during the neural network training process. In a study of flank wear estimation in

turning, Bukkapatnam el a/ (2000) used a Oaubechies 04 wavelet-based filtering technique to smooth vibration signals prior to analysis. The fractal characteristics

of the signals were then computed and related to instantaneous flank wear using a recurrent neural network. See also the earlier study by the same research group where wavelet packets were employed to decompose acoustic emission signals used to monitor the machining process (Bukkapatnam el ai, 1999). A combination of sensory data from CCO camera (images) and microphone (sounds) allowed Mannan ef a/ (2000) to monitor the condition of cutting tools. Symlet wavelets were used to decompose the sound signals, and the energy content of the detail coefficients was used to differentiate between sharp and worn tools. Luo el a/ (2000) have used combination wavelets in a study of vibration signals from machining

processes. Tansel el a/ (2000) have provided details of a neural-network-based method for the off-line evaluation of tool condition. They used a number of preprocessing algorithms to reduce the input data to the neural network and found that a wavelet-based method performed best. See also the earlier work by Tansel ef a/ (1993) and the wavelet-neural-network-based eondition monitoring algorithm developed by Zhou el a/ (1995). Wu and Ou (1996) have used wavelet paekets to monitor both chatter in turning processes and tool wear in drilling proeesses. They developed an automatic feature extraction procedure which sorts the wavelet packets according to energy content and selects the first few packets containing the most energy. The signal is reconstructed from the dominant packets and interrogated using two feature assessment criteria defined by the authors: cross-correlation (time domain) and eross-eoherenee (frequency domain). Figure 5.5 shows Wu and Ou's wavelet packet decomposition of machine tool vibration signals for a turning process during both stable eutting and ehatter. Comparing the two figures we ean see that certain packets contain a significantly larger amount of energy during chatter. Tn faet, the energy in wavelet paekets number 5 and 13 at level 5 beeame dominant during chatter. The energy in these packets was described using a peak-to-valley index (i.e. the differenee between the maximum and minimum paeket eoeffieients). The value of this index was then used to determine whether the vibration signal indicates a state of ehatter or stable eutting. The Haar wavelet was used by Lee and Tarng to monitor tool failure in end milling operations (Lee and Tarng, 1999; Tarng and Lee, 1999). They followed this Copyright © 2002 lOP Publishing Ltd.

60

o

250(ms)

250(ms)

-20 (al

(d)

Figure 5.5. Signal and wavelet packet decomposition during stable cutting and chatter. (a) Stable

cUlling signal and (b) associated wavelet packet decomposition. (c) Chatter signal and (d) associated wavelet packet decomposition. After Wu and Du (1996). Reproduced with kind permission of Academic Press Ltd.

up in a later paper (Lee and Tarng, 2000) employing both a Oaubechies 02 (Haar) and 012 wavelet to monitor spindle motor current. They used the approximation coefficients at level 4 as the best indicator of tool failure. Fu el al (1999) have used a matching pursuit method to predict the onset of drill breakage in the drilling process. They found that the matching pursuit method performed satisfactorily in detecting small drill bit behaviour with three different wavelet dictionaries: Gaussian, Gabor and Haar. Xiaoli (1999) has employed both continuous and discrete wavelet analysis of a.c. servo motor currents in a method for the detection of breakage of small diameter drills. (See also Li, 1998c.) Related studies for the detection of drill breakage using wavelet-transformed acoustic emission signals are described by Li el al (1999) and Xiaoli el al (1997). In another study of the prediction of small drill bit breakage, Mori el al (1999) developed a procedure to extract prefailure information from the cutting force signal. They developed three index functions based on Dauhechies 012 wavelet coefficients at various scales: an energy index)

waviness index and irregularity index. These indices could differentiate the thrust behaviour of three tool states: normal, sawtooth and screeching. A discriminant function was then applied that reduced the three indices to a binary value that identified the state of the drill: 'normal' or 'prefailure'. Finally, in their paper concerned with the machining of millimetre scale optics using ion-beams, Shanbhag el al (2000) employed a wavelet-based deconvolution algorithm to generate an appropriate dwell function for ion-beam rastering. They tested their method by machining a one-dimensional sinusoidal depth profile in a prepolished silicon substrate. Copyright © 2002 lOP Publishing Ltd.

5.3

Rotating machinery

The condition monitoring of rotating machinery attempts to detect and diagnose machinery faults from vibration signals picked up usually from the machine casing. In this section we begin with some recent applications of wavelet techniques to gear diagnostics, where the early detection of gear failure is a prime concern, then

we look at wavelet-based detection and diagnosis of signals from other rotating machinery components, such as shafts, bearings and blades. 5.3.1

Gears

Daubechies 04 wavelets were employed by Paya

el ([/

(1997) in the analysis of vibra-

tion signals acquired from an accelerometer attached to a bearing housing on an

experimental drive line model (figure 5.6). They investigated these signals for various configurations of faulty gears and bearings, including gears with material added, material taken away and faulty bearings. The authors interrogated signal segments consisting of 1024 data points. Figure 5.7(a) shows the wavelet coefficients obtained for a good gear and good bearing: the reference case. Notice that the coefficients are indexed sequentially (chapter 3, section 3.3.4). Figure 5.7(b) shows the coefficients from one of the faulty configurations: a faulty bearing and shaved gear. The dominant wavelet coefficients derived from the faulty signal were then fed into a neural nctwork for classification. The authors found that by preprocessing the data using wavelet transforms prior to using an artificial neural network they could successfully distinguish between all the various fault configurations considered. See also the paper by Sung el ([/ (2000), who used discrete wavelets (D20s) to detect the location of tooth defects in a faulty gear system. A brief account of the use of the Morlet wavelet in vibration analysis for mechanical fault diagnosis is provided by Wang (1996). Dalpiaz el ([/ (2000) have assessed a number of techniques for detecting cracks in gears from the vibration

signal picked up from the gearbox casing of an experimental test rig. They employed cepstrulTI analysis, time-synchronous average analysis, cyclostationary analysis and

disc brake (load)

/

shaft couplings

accelerometer

2 automotive gearbox

rolling clement bearing hOllsings

Figure 5.6. Schematic presentation of the model drive system. Arter Paya ef al (1997). Reproduced with kind permission of Academic Press Ltd. Copyright © 2002 lOP Publishing Ltd.

0.8,---:-;------------,

.g

j

~

0.4

J

O~

'"~ -0.4

0.4

M

'"~ -0.4

-0.80:-'-:2::0"0-'--:4:::-00::-'--=6:COO:-'-"800~'-:1-::OO::0,.J (a)

0.8,--.,,---,----------,

wavelet number

-o.80:--'L2=OO'::-'-4':OO::-'---:60"'0:-'--'::800=-L1"OO'='0=-' (b) wavelet number

Figure 5.7. Sequentially indexed wavelet coefficients for two drive line configurations. (a) Good bearing and gear. (b) Faulty bearing and shaved gear. (A lolal of 1024 coefficients were generated. The smallest scale=wavelel numbers 513~1024, next smallest scale = wavelet numbers 129-512, etc. The first coefficient is related to the signal mean.) After Paya el al (1997). Reproduced with kind permission of Academic Press Ltd.

wavelet analysis in the interrogation of the signal. They found that the Morlet-based wavelet transform is well suited to the detection of transient dynamic effects caused by these localized faults. A Morlet wavelet (wo = 1.75rr = 5.5) was used by Staszewski and Tomlinson (1994) to detect a damaged tooth in a spur gear. They introduced a fault detection algorithm which characterized the differences which arose between damaged and undamaged gears in the wavelet transform modulus plots of their respective vibration signals. In this way, they were able to differentiate between damaged and undamaged gears. In addition, visual inspection of the modulus and phase plots of the wavelet-transformed signals enabled the fault location to be determined. Staszewski and Worden (1997) used Morlet-based wavelet preprocessing, among other techniques, in a neural-network-based classification algorithm for faulty gearboxes. The neural network pattern classifier they developed was applied to vibration signals from a pair of meshing spur gears with a tooth fault. Yoshida ef al (2000) have also used the Morlet wavelet in the detection of tooth surface failure, using it to analyse both vibration data from the gearbox and dynamic strain data taken at the tooth fillet of the gear. They suggested that it may be possible to detect the failure position on the tooth surface from the strain gauge data using the wavelet transform, and found the wavelet transform, superior to Fourier methods when diagnosing the failed tooth and the state of the tooth surface. A concise account of the application of three orthogonal wavelets (Daubechies D4, D20 and the harmonic wavelet) to the detection of abnormal signal transients generated by early gear damage is given by Wang and McFadden (1995). They used the wavelets to decompose residual gear vibration signals obtained by removing all harmonics of the tooth meshing frequency from the time domain average. They noted that, although orthogonal transforms allow for fast algorithms and zero redundancy, there is a distinct lack of resolution in the discrete transform energy scalograms. They suggested the use of non-orthogonal (i.e. highly redundant) wavelets to overcome this problem and used the Morlet wavelet as an example. Wang and McFadden (1996) followed this up with a more detailed study of the use of the Morlet wavelet in the analysis of gearbox signals. They detailed a method to ensure that the number of scales just covers the frequency band of interest and that the redundancy of computation, although necessary for fault identification, is minimized. Copyright © 2002 lOP Publishing Ltd.

lJ~~J~~e~~ • ",' , I rt tl II I

wavelet map - amplitude

I

"'0

0.8

wavelet map - phase

0.8

N

I •

0~ 0.4

•

1"

•• "

t"""

32 0.6

-

0~ 0.4

0.2

0.2

O~~~~'"""'":~_....._ ~

o

90 180 270 360 angular position of shaft [deg.]

0

0

90 180 270 360 angular position of shaft [deg.]

Figure 5.8. Vibration of healthy gear. Synchronous time average with its phase modulation, and amplitude and phase plots. After Boulahbal et at (1999). Reproduced with kind permission of Academic Press Ltd.

Lin and McFadden (1997) have used cubic B-spline wavelets to decompose gear vibration data. They analysed a complex time signal generated by removing various periodic elements of the original vibration signal in the Fourier domain and taking the inverse Fourier transform. In this way, although the signal became complex, the reconstructed signal was deemed to carry most of the information describing the change in the vibration signal caused by the crack. McFadden et al (1999) have also decomposed gear vibration signals using a generalized S transform which has many similarities with the wavelet transform.

Boulahbal et al (1999) have used both amplitude and phase information obtained from a Morlet-based wavelet decomposition to detect cracks in geared systems. Figure 5.8 shows a synchronous time-averaged vibration signal for a healthy gear with 16 teeth. Three bands appear in the wavelet transform amplitude plot: one corresponding to the gear meshing frequency (GMF) of 320 Hz and the other two corresponding to its harmonics. These are indicated by dashed lines in the plot. There arc 16 jumps in phase along the GM F band corresponding to the 16 teeth of the gear. Figure 5.9 shows the same plots for a gear which has one of its teeth with a transverse crack cut into it to a depth of 20% of the tooth thickness. The cut was made at an angle of 90" to the timing mark on the gear. The location of the crack can be seen in both the signal and the corresponding wavelet transform plot beneath. The change in phase can be seen in the wavelet phase plot as a bifurcation in the vertical phase bands at approximately 90° and at a frequency just above the GMF. Boulahbal and his colleagues went on to investigate the use of amplitude and phase maps on 'overall residual' signals obtained by filtering out the GMF and its harmonics from the signal. In addition, they employed a polar representation of the amplitude and phase plots (figure 5.10). These arc formed by wrapping the plots in figure 5.8 around a circle and joining up the two ends. According to the authors, this type of map is 'continuous and very intuitive' due to the periodic Copyright © 2002 lOP Publishing Ltd.

,....,

synchronous time average

phase modulation

l-~~t~~~ I

_ _ wa:elet map - amplitu~e

"'0

0.8

I

wavelet map - phase

0.8

N

N

2 0.6

~

-

0-

0.6

0-

~ 0.4

~ 0.4

0.2

0.2

90

180

270

angular position of shaft [deg.]

360

o

0

90

180

270

360

angular position of shaft [deg.]

Figure 5.9. Vibration of gear with cracked tooth. Synchronous time average with its phase modulation, and amplitude and phase plots. After Boulahbal e/ at (1999). Reproduced with kind permission of Academic Press Ltd.

nature of the signal which repeats itself every revolution of the gear. Tn addition, the polar map squeezes the low frequency components of the map and stretches out the high frequency components. This is useful as normal wavelet plots become eluttered at high frequencies as the correspondingly smaller wavelets pick up more and more detail. The arrows on the two maps show the location of the features in the maps corresponding to the cracked tooth. Note that the phase map shows up the feature Ijl6th of a rotation before the amplitude map. This is because the cracked tooth causes a local increase in speed which is immediately reflected in the phase. The local speed increase causes the next tooth entering the meshing region to sufTer a stronger impact. Hence, the stronger impact appears as a feature in the amplitude map Ijl6th of a revolution later than the phase change for a 16-tooth gear.

Figure 5.10. Vibration of gear with cracked tooth. Amplitude and phase wavelet maps or the synchronous time average signal shown in polar representation. After Boulahbal e/ at (1999). Reproduced with kind permission of Academic Press Ltd.

Copyright © 2002 lOP Publishing Ltd.

5.3.2

Shafts, hearings and blades

The problem of faulty bearings in rotating machinery is addressed by Li and Ma (1997) who have investigated two eharaeteristie bearing defects using wavelet analysis. They employed a decaying exponential (real-only) sinusoidal wavelet to differentiate between bearings with damaged outer races and damaged rollers. The variation of the wavelet coefficients at different scales was used to detect localized bearing defects. For example, the vibration signal from a bearing with a roller defect is shown in figure 5.II(a). The wavelet decomposition of the signal at five scales, 0t, 02, ... ,0, is shown in figure 5.11 (b). There is a noticeable periodic structure in the 03 coefficients. The periodicity of this structure is given at 4.76 ms from the autoeorrelation plot of figure 5.II(c). The frequency of this periodie strueture is 210Hz corresponding well with the frequency of 211 Hz expected from a damaged 1000

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Figure 5.11. Wavelet-based vibration analysis of defective bearing. (a) Vibration signal of a bearing with a defective roller. (b) Magnitude of the wavelet transform coefficients for five a scales. (Dilations in the plot from top 10 bottom are 01 > (12 > (l3 > ((4 > (15') (c) The autocorrelation of the wavelet magnitudes at dilation (13 in figure (b). Reprinted from Li and Ma, NDT & E Inremmiol/a/30(3) 143~149, copyright (1997), with permission from Elsevier Science.

Copyright © 2002 lOP Publishing Ltd.

roller. The authors were able to develop an algorithm for detecting the onset of localized defects on roller bearing elemcnts by examining the frequency of structures in the wavelet coefficient outputs within a narrow frequency band associated with the defects under consideration. Pifieyro ef 01 (2000) found that a simplified technique

based on the Haar wavelet transform proved to be applicable in the early detection of the burst generated in modelled signals during a fault development in roller bearings. However, they failed to obtain useful information when using the technique on laboratory-measured acoustic emission signals. Mori ef 01 (1996) applied a discrete Haar wavelet transform to vibration signals to

predict the occurrence of spalling in ball bearings. They proposed a method of spalling prediction based on the trend of the scale dependent wavelet coefficient maxima which, they noted, increases at small scales just before the occurrence of spalling. Tandon and Choudhury (1999) have provided an overview of vibration and acoustic measurement methods for the detection of defects in roller bearings 0.32

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with kind permission

or the ASME.

Copyright © 2002 lOP Publishing Ltd.

which sets wavelet transform analysis in context with other methods used in this area. Yacamini et al (1998) have developed a method to detect the torsional vibrations associated with a.c. motors and generators from their stator currents. In particular,

they suggested the use of the wavelet transform to analyse the transient torsional vibration signals which occur during the direct on-line start up of induction motors. This is a crucial period of operation as large shaft stresses may exist over short periods of time. Shibata et al (2000) have used discrete Daubechies D8 wavelet transforms to aid in the visualization of sound signals as a fault diagnosis method for rotating machinery. In order to detect wrap-up incidents in napping machines used in the textile industry, Dad et al (1999) performed a discrete wavelet transform

decomposition of vibration signals taken from the bearing housing and input the coefficients into a multilayer neural network for classification. Tsai et al (2000) have developed a set of criteria to detect abnormal loads on servomechanisms.

They employed a Haar-based multiresolution analysis to enhance the detection of abnormal torque events using their method. Lin and Qu (2000) have used a modified soft thresholding techniquc based on the continuous Morlet wavclet transform to analyse vibration signals from both defective rolling bearings and gearboxes. Their 'generalized' soft-thresholding algorithm removes only a fraction of the threshold from those coefficients above the threshold value, whereas those below the threshold arc set to zero. The technique effectively lies somewhere between soft and hard thresholding. According to the authors this method is more applicable to impulse component extraction from mechanical dynamical signals. 0.2

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Figure 5.13 Unsteady pressure signatures. (a) One rotor blade twisted and (b) two rotor blades fouled. air and a,b are, respectively, the wavelet amplitudes for the faulty and healthy conditions. (The signatures are derived from the differences in the level 4 wavelet coefficients.) Arter Aretakis and Malhioudakis (1997). Reproduced with kind permission of the ASME.

Copyright © 2002 lOP Publishing Ltd.

Aretakis and Mathioudakis (1997) used D20 wavelets to diagnose faults in gas turbines. They calculated discrete wavelet scalograms of signals taken from two gas turbines: a 'healthy' one and one with a twisted rotor blade. Three different signals were analysed: wall pressure, compressor case vibration and radiated sound. The

discrete wavelet maps corresponding to the pressure signal from the healthy and twisted blade turbines are shown in figures 5. I2(a) and (b) respectively. The difference between the maps was then computed in order to show up the locations of significant signal departures. Figure 5.12(c) shows this 'difference' map and the locations of the significant differences are shown on figure 5.12(b). The authors focused on the differences of level 4 coefficients as a signature for each type of fault. Two examples of these signatures are shown in figure 5.13. Figure 5.I3(a) contains the signature for four different operating points, A to D, for a single twisted blade. The signature for point A corresponds to a rotor blade with smaller twist, i.e. the same fault but less severe. Figure 5.13(b) contains the signatures at four operating points for two fouled rotor blades. The results indicated that the signature patterns are independent of operating conditions and, in addition, each failure type generates a unique failure signature. The authors suggested that these two properties of the wavelet signature makes it suitable as a diagnostic tool for such faults.

5.4

Dynamics

A number of papers concerning the application of wavelet-based analytical techniques to the investigation and modelling of dynamical signals have appeared in recent years. Applications include the evaluation of dynamic properties and system characteristics; the modelling and control of dynamical behaviour; and the partitioning or decoupling of multiple responses within dynamical systems. A few of these are reviewed briefly in this section.

The detection of system nonlinearities through the identification of damping and stiffness parameters for multi-degree-of-frcedom dynamic systems during transient testing has been carried out by Staszewski (1997, 1998a) using a Morlet wavelet. This wavelet is very effective for this application as it has good support in both frequency and time, which allows the decoupling of the system's various modes of vibration with respect to time. Figure 5.14 shows one of the signals analysed in the study resulting 4.0 r ; - - - - - - - - - - - - - - - - - - ,

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Copyright © 2002 lOP Publishing Ltd.

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from the impulse response of a two-degree-of-freedom model system. The frequency response function and wavelet transform plots-amplitude and phase-eorresponding to this signal are shown in figures 5.15 and 5.16 respectively. Figure 5.17 shows the ridges of the modulus plot and figure 5.18 shows the real parts of the wavelet transform skeletons obtained from these ridges. The decoupling of the modes is evident in the plot. The reconstruction of the modes and subsequent damping parameter estimation is found to be better for the skeleton reconstructions than a simple reconstruction based on the wavclet coefficients within a certain frequency range. The ability of the method to separate modes that are closer in frequency is also tackled in the paper. Kyprianou and Staszewski (1999) have employed cross-wavelet analysis using the Morlet wavelet as an alternative approach to classical input-output analysis based on frequency response functions for nonlinear oscillator system identification. Ruzzene et at (1997) also used Morlet-based wavelet transforms to identify natural frequencies

and damping ratios of multi-degree-of-freedom (MDOF) systems. They tested the technique on a four-degree-of-freedom model, then used it to analyse the acceleration response of a bridge excited by road traffic, wind and low intensity ground motion. In a subsequent paper, Piombo et at (2000) have reported on dynamic tests performed on a simply supported bridge in Northern Italy under traffic excitation. They determined modal parameters using their wavelet estimation technique and verified the accuracy 2 ...--------,

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(b) Phase. After Staszewski (1997). Reproduced with kind permission of Academic Press Ltd. Copyright © 2002 lOP Publishing Ltd.

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of their modal estimates through a modal quality index based on a wavelet cross correlation statistic. Recently, Lamarque el al (2000) have introduced a waveletbased logarithmic decrement formula to estimate damping in multi-degree-of-freedom systems from time domain responses. They then employed their method in a study of the in silu dynamic response of a civil engineering building excited both with harmonic and shock testing (Hans el ai, 2000). Their results demonstrated that the method permits the uncoupling of the eigenfrequency modes of the MDOF system and provides reasonable estimates of the associated damping. Newland (1999a) applied his own wavelet, the harmonic wavelet, to the analysis of bending wave propagation within a steel beam. This complex wavelet is defined as a series of non-overlapping boxes in the Fourier domain (figure 5.19). The inverse Fourier transform of these box functions gives the wavelets in the time domain which, with an appropriate choice of spacing, can be forced to be orthogonal to each other at each scale. Newland applied a smoothed version of this wavelet (using a Hanning window) to the analysis of experimentally measured impulse response signals from a suspended mild-steel beam of rectangular cross section. The windowing of the wavelets improves their temporal localization. However, it forces them to become nonorthogonal and hence the analysis becomes highly redundant. The response signal from the beam, taken close to the point of impact, is shown in figure 5.20(a). The corresponding time-frequency map is given in figure 5.20(b). Copyright © 2002 lOP Publishing Ltd.

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Groups of high frequency bending waves travel faster than low frequency ones. This can be seen in the plots where reflections of the high frequency wave groups occur more often. To reduce the smearing of the information in the time-frequency maps, their ridges are determined. These are shown plotted in figure 5.20(c). Determination of the ridge loca1ions is a non-trivial task in practice and Newland gives some details on how to tackle this problem. Newland also covers the task of phase extraction and provides a second example of the application of the harmonic wavelet to the reflection of acoustic waves in a closed duct. Two more engineering applications of the harmonic wavelet-ground vibrations from underground trains and the dynamic behaviour of soil under earthquake excitations-are described by Newland in a subsequent paper (Newland, 1999b). For more information on the early development of the harmonic wavelet and the generalized harmonic wavelet, see Newland (1993a,b) and (1994d) respectively. Onsay and Haddow (1994) applied the Morlet wavclet to experimental vibration signals from both semi-infinite and free beams. They showed how the modulus and phase of the wavelet transform enables good resolution of the signal over a wide spectral range. This resulted in efficient localization of the complex interference patterns of transient wave groups within the dispersive medium of the beams.

Kishimoto et af (1995) also applied the Morlet wavelet transform to the time-frequency analysis of dispersive flexure waves in a simply supported beam. They found that the dispersion relationship of the group velocities could be evaluated accurately using the information extracted from the wavelet transform. Kishimoto (1995) Copyright © 2002 lOP Publishing Ltd.

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Copyright © 2002 lOP Publishing Ltd.

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suggests the use of the method to determine the velocity and attenuation of ultrasonic pulse echo signals used in non-destructive testing. In a subsequent related paper, again using Morlet wavelets, Inoue er al (1996) reported on a time-frequency analysis of the flexural waves set up by central impacts on a simply supported beam. A method was developed by the authors to determine both the group velocity of the structural waves and the impact sites on the beams by utilizing the arrival times extracted from the wavelet analysis. In a similar study, this time for steel plate elements, Gaul and Hurlebaus (1997) have determined impact site locations using the wavelet decomposition of strain sensor data. A wavelet-based study of transient waves propagating in composite laminate plates has been described by Jeong and Jang (2000). An analysis of the non-stationary response of a rigid block resting on a moving plane has been performed by Basu and Gupta (1999) using a wavelet-based stochastic linearization technique. This simple model was used to show the potential application of wavelet-based analytical tools to the response of slipping structures to earthquake excitation. See also the earlier papers by Basu and Gupta (1997,1998). Gurley and Kareem (1999) mention briefly the application of wavelet transforms to ground motion analysis during earthquakes and building responses to wind events (including vortex shedding) in their paper concerning analysis and simulation tools for wind engineering. Robertson er al (1998a,b) have detailed a discrete wavelet transformbased method for extracting the temporal impulse response functions of structures, and Chen and Wu (1995) have developed a spline wavelet expansion-based finite element method for frame structure vibration analysis. Copyright © 2002 lOP Publishing Ltd.

A study by Dalpiaz and Rivola (1997) compared the effectiveness and reliability of different vibration analysis techniques for fault detection and diagnostics in cam mechanisms used in high-performance packing machines. They compared traditional analysis methods-amplitude probability density (APD), power spectral density (PSD) and time synchronous averaging (TSA)-with a wavelet transform method based on the Morlet wavelet. They found the time--frequency analysis of the wavelet-based method well suited to detecting and precisely locating transient dynamic phenomena from the signal. Mastroddi and Bcttoli (1999) have performcd a wavelet analysis on the output signal of a nonlinear system in the neighbourhood of a Hopf bifurcation. They used the Morlet wavelet-based analysis to point out the linear and nonlinear signatures of the system and suggested its use for aeroelastic applications. Karshenas el al (1999) compared the wavelet power spectrum smoothing method with the Welch method in the random vibration control algorithm of an electrodynamic shaker. They found that the wavelet method achieved twice the power spectrum resolution of the Welch method. Sjoberg el al (1995) included wavelet transformbased methods in a comprehensive paper concerning 'black-box' models of nonlinear dynamical systems. A new wavelet-based method for denoising transient dynamical signals by first projecting them into a multidimensional state space has been described by Effern el al (2000) and applied to both model data and event-related potentials (medical EEG signals).

5.5

Chaos

Nonlinear oscillator systems are capable of the most fascinating behaviour known as chaotic motion, or simply chaos, whereby even simple nonlinear systems can, under

certain operating conditions, behave in a seemingly unpredictable manner (Addison, 1997). The realization that real systems can exhibit this type of non-periodic response has prompted much research work in the area over the past two decades. The ability of wavelet-based methods to characterize chaotic oscillations has received attention from a variety of workers in the field. Both Daubechies and Morlet wavelets have been employed by Staszewski and Worden (1999) to analyse time-series data sets containing a variety of features including coherent structures (fluid turbulcnce), fractal structures (devi!'s staircase and Mandelbrot-Weierstrass function), chaos (Duffing, Henon, Lorenz and Rossler systems) and noise (Gaussian white). Their paper provides a wide ranging overview at an introductory level of the application of wavelets to signals from these types of systems. Figure 5.2l(a) shows the time series of a Duffing oscillator in chaotic mode. This oscillator is a sinusoidally driven, damped nonlinear oscillator with the nonlinearity contained in the cubic spring tenn. The Duffing oscillator investigated by Staszewski and Worden has the form

x + 0.05-': + x 3 =

7.5 cos 1

(5.2)

where ,i: and x are, respectively, the first and second derivatives of x (the displacement if we think of it as a physical mass-spring-damping system). For the set of parameter Copyright © 2002 lOP Publishing Ltd.

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values given in equation (5.2) Ihis forced nonlinear oscillator produces a chaotic response. The Poincare map of figure 5.21(b) was generated by plolting the velocity against displacement once every period of forcing of the oscillator. This map is useful in highlighting the fractal structure of the strange altractor associated with the chaotic system. Figure 5.22 contains the phase portrail of a Gaussian noise signal logether wilh that for the Duffing oscillator. The similarity between the Iwo

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Copyright © 2002 lOP Publishing Ltd.

at small scales is evident, highlighting the self-similarity contained within the two systems. Wong and Chen (2001) provide a clear, well illustrated introduction to the Morlet wavelet transform of the nonlinear and chaotic behaviour of multi-degreeof-freedom systems. They introduce the transform explaining its use, using a variety of simple test signals before considering single, then multiple, oscillator systems based on coupled Duffing oscillators. These systems are considered when subjected to both single impulses and continuous forcing. They conclude by examining the chaotic response of the single-degree-of-freedom Duffing oscillator, contrasting its modulus and phase plots with nonchaotic cases. Their paper is well worth consulting for the many clearly presented diagrams used to illustrate the discussion. In their comprehensive paper on adaptive strategies for recognition, noise filtering, control, synchronization and targeting of chaos, Arrecchi and Boccaletti (1997) have shown how to employ the Daubechies D20 wavelet in a noise reduction strategy to separate noisy contributions from deterministic parts of chaotic data sets. As an example (Boccaletti et aI, 1997), they used the Mackey-Glass delay differential equation configured to produce 7.5-dimensional dynamics with both white and coloured noise added separately. They detailed the effect of wavelet threshold level on the ability of their method to determine the underlying dynamics of the system and recommended it for easy implementation in experimental situations as it does not require information on the correlation properties of the additive noise. Grzesiak (2000) has also employed a wavelet-based denoising technique to filter chaotic data. Grzesiak determined the efficiency of the technique by comparing the correlation dimension of noisy and clean data generated for a variety of chaotic dynamical systems and found that the wavelet method was comparable with other methods commonly used to filter chaotic data. Permann and Hamilton (1992) have performed a wavelet analysis of the time series from a Duffing oscillator in both periodic and chaotic mode. They were able to detect small-amplitude harmonic forcing terms, even when the data were highly nonstationary and of short duration. See also the paper by Permann and Hamilton (1994) who investigated the chaotic behaviour of a weakly damped and weakly forced Morse oscillator using Daubechies D8 wavelets. Lamarque and Malasoma (1996) have constructed wavelet-based exponents, similar to Lyapunov exponents, for the identification of chaotic behaviour. Cao et al (1995) have used wavelet networks to make both short- and long-term predictions of the time series from chaotic systems. Systems they investigated include the MackeyGlass equation, Lorenz system, and the U shiki and Ikeda maps. In addition, they modified the Ikeda map by using one of its parameters as a variable. In this way they were able to investigate parameter-varying systems. Allingham et al (1998) have used a hybrid system to model time series from chaotic dynamical systems. Their system combines continuous optimization with a wavelet matching pursuit method. Gamero et al (1997) have analysed the Lorenz equations displaying chaotic motion using their multiresolution-based information measures for dynamical signals. They also considered the Henon map and an EEG signal displaying an epileptic seizure. Heidari et al (1996) have studied the wavelet transform of deterministic self-similar signals, suggesting its use as a method to interrogate the noisy strange attractor of the Henon map. Masoller et al (1998) have interrogated experimentally measured low-frequency intensity fluctuations from a semiconductor Copyright © 2002 lOP Publishing Ltd.

laser operated near threshold. Using the discrete wavelet transform, they compared their results with the wavelet analysis of a theoretical model, showing that the differences between the results were confined to the 'fast and short' components of the signal. Another experimental study by Russo el al (2000) has concentrated on the dynamical regimes optically induced in a nematic liquid-crystal as the intensity of an incident laser beam increases. Their paper contains a series of Morlet wavelet transform plots generated at increasing values of the control parameter, which

illustrates the jump to stochastic behaviour above a known threshold. Their wavelet-based analysis suggests the presence of a transition towards a chaotic state.

5.6

Non-destructive testing

Non-destructive testing (NDT) is concerned with the interrogation of underlying structural integrity using procedures which do not impair in any way the intended performance of the structure during and after examination. Sonic echo testing is a common method employed in the NDT of structural elements. It involves striking the test specimen (e.g. structural element or material specimen) with an instrumented hammer which records both the input pulse (strike) and subsequent response of the specimen. This response is interpreted as an indirect measurement of the integrity of the specimen. A typical velocity trace from such a test on a foundation pile is shown in figure 5.23 (Watson el ai, 1999). A schematic of the sonic pulse transmission through the pile is shown in the figure. For such a heavily damped system there are rarely multiple longitudinal reflections and the frequency dependences of the group velocities are negligible. Therefore, the temporal isolation of the signal features is more important than their frequency decoupling and hence a Mexican hat was used which is more temporally compact than the standard Morlet (5 < Wo S; 6) or harmonic wavelets used in the study of free beam and plate vibrations. The scalogram corresponding to this velocity trace is shown below. The pile is II m long and the velocity of the stress wave through the pile is 3800 m s-'. Thus we would expect to see the reflection of the end of the pile occur 2 x 11/3800 = 0.0058 s after the initial impulse. The pile toe reflection shows up particularly well in the top right-hand quadrant of the scalogram, as it has a distinctively different shape and appears lower down the scalogram from the initial oscillations which occur at a dilation around a = 10- 4 These oscillations, seen to occur in the top left-hand quadrant of the scalogram just after the input initial pulse, are known as ringdown and are in fact the surface oscillations of the pile head due to the hammer impact. Figure 5.24 contains the reconstructed traces of both wavelet and Fourier filtered traces. Simple scale dependent wavelet filtering was employed where the transform components at a scales less than 0.0001 were set to zero and an inverse wavelet transform performed. The wavelet filtered trace is shown in the top left quadrant of figure 5.24. The Fourier low pass filter cut-off frequency was set to 2.25 kHz, as is the case in practice. The Fourier filtered trace is shown in the top right quadrant of figure 5.24. The two lower plots in the figure show a zoomed-in section of the upper traces in region of the pile toe reflection. Comparing the filtered traces of figure 5.24, it can be seen that the wavelet filtering separates the ringdown oscillations Copyright © 2002 lOP Publishing Ltd.

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0.004

0.005

location 'b' (seconds)

Figure 5.23. Wavelet transform decomposition of a sonic echo signal. Schematic of sonic echo testing of a foundation pile (top left). A Fourier filtered velocity trace (top right) and corresponding wavelet transform plot (bottom) of an 11 In pile in stiff/very stiff clay.

0.004 O'OO6~tim:';s~! :::~~: °llo.OO2

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0

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/

Figure 5.24. Wa\'elet and Fourier filtering of the sonic echo pile signal. W
o

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Figure 5.25. Filtering of a field test sonic echo signal. Field test result (len) and when filtered: wavelet (top right) and Fourier (bottom right). After Watson ef 011999 Joumal ojShock alld Vibrarioll 6 267272. With kind permission or lOS Press.

from the pile toe feature much more effectively than Fourier filtering. Figure 5.25 shows a field signal, together with two filtered versions: one wavelet based, the other Fourier based. This time a discrete orthonormal Oaubechies 08 wavelet was used. Again the wavelet-based filtering better separates the ringdown oscillations from the toe feature. However, note that the dyadic nature of this discrete wavelet transform can present problems in this respect due to its translation invariance.

Figure 5.26 illustrates a more sophisticated filtering method for removing both erroneous ringdown artefacts from the pile signal based on the modulus maxima of the scalogram. Figure 5.26(b) contains the finite element (FE) generated velocity trace of an II m pile in stiff clay. A schematic of the pile is given in figure 5.26(a). A simulated defect in the form of a reduction in section (necking) is present approximately one third of the way down the pile. The location of both the input pulse and echo from the pile discontinuity are highlighted in the figure. The transform plot associated with the wavelet decomposition of the trace is shown in figure 5.26(c). Modulus maxima are found from the original wavelet transform using a simple

algorithm which scans across the transform plot scale by scale and identifies local maxima and minima. The modulus maxima obtained for the scalogram of

figure 5.26(c) is shown in figure 5.27(a). We can see from the modulus maxima plot that the large scale input pulse feature in the signal contains a ridge extending from high to low frequencies. The other maxima lines do not extend as far down into the low frequency range. A close approximation to the signal can be reconstructed using only the maxima lines where the energy contained in the whole scalogram is reapportioned to the maxima lines in the reconstruction. The reconstruction using

all the maxima lines in figure 5.27(a) is shown in figure 5.27(b). Figure 5.28 illustrates the filtering of the initial signal using the modulus maxima. This is done in an anticlockwise manner from figure 5.28(a) to 5.28(d). The maxima lines are thresholded at a frequency of 310Hz, shown in figure 5.28(b). All maxima lines which do not extend down from the higher bandpass frequencies to this threshold level are removed (figure 5.28(b)). Those which do extend down to and beyond the threshold are retained. Only the retained lines (figure 5.28(c)) are used to reconstruct the signal (figure 5.28(d)). The threshold is chosen to be lower than the ringdown artefact in the signal, hence the maxima lines from the ringdown artefact will fall below the Copyright © 2002 lOP Publishing Ltd.

input pulse

0.04

303m

-

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00

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(b)

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(e) Figure 5.26. Sonic echo velocity trace from pile head and associated scalogram. (a) Schematic of a foundation pile. A necking fault has been modelled approximately one third of the way down the pile. (b) Sonic echo velocity trace taken from the pile head. (c) Wavelet transform plot of the signal in (b). (Large positive components in black, large negative components in while.) Reprinted from Watson and Addison (2002), Mechanics Research Commullicatiolls (in press at lime of publication), with permission from Elsevier Science.

Copyright © 2002 lOP Publishing Ltd.

~ ~

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Figure 5.27. The reconstruction from only the scalogram modulus maxima lines. (a) Modulus maxima plot derived lines from the scalogram in previous figure. (b) Reconstructed trace using only the modulus maxima lines in (a). Reprinted from Watson and Addison (2002), Mechanics Research COlIIlJlunications (in press at time of publication), with permission from Elsevier Science.

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Figure 5.28. The partitioning of the modulus maxima lines for signal filtering. (a) Original data. (b) Discarded modulus maxima lines. (Threshold shown as horizontal dotted line.) (c) ReLained modulus maxima lines. (d) Filtered data. Reprinted from Watson and Addison (2002), Mechanics Research Communications (in press at time of publication), wiLh permission from Elsevier Science. CopyrighL © 2002 lOP Publishing Ltd.

threshold and be removed. In addition, noise, which also manifests itself as modulus maxima restricted to high frequency regions, is also removed from the signal. The resultant reconstructed trace shown in figure 5.28(d) illustrates how all ringdown has been eliminated whilst retaining the pertinent signal features. In addition, the retained features still contain their high frequency components and are not excessively smoothed, as would be the case if bandpass filtered using Fourier techniques. Figure 5.29 illustrates the use of the Morlet wavelet with low central frequencies in the analysis of a highly oscillatory sonic echo signal where the pile toe is not obvious in the time domain (Addison ef ai, 2002a). When using central frequencies, wo, less than 5 (io < 0.8) the complete Morlet wavelet given by equation (2.36) in chapter 2 must be used. Morlet wavelets with low central frequencies result in analyses that are more 'temporal' than 'spectral' in that they are better at locating short duration temporal features than those with higher values of Wo (refer back to section 2.12, chapter 2). This can be seen in figure 5.29, where the pile toe can be located in the wavelet transform scalogram plots only at lower values of woo The location of the pile toe is indicated both in the time signal and the lowest scalogram plot which corresponds to the complete Morlet wavelet with Wo = 1.5 (i.e. fo = 0.238). It is interesting to note that, although they do seem to be particularly useful for certain tasks, the literature contains surprisingly very little on the use of complex wavelets with few oscillations such as the complete Morlet wavelet of low central frequency or the complex Mexican hat. A variety of continuous wavelets were used by Abbate ef al(1997) in a study of the signal detection and noise reduction properties of the wavelet transform when used to elucidate ultrasonic pulse--echo traces in steel specimens. They used a combination of 'pruning' and soft thresholding to reduce the noise from the signals. Pruning simply sets wavelet coefficients outside a certain a-scale range to zero, i.e.

T(a,b)

=

{;(a'b)

for

a

for

(II

for

<

{II

< a < ([2 a> {/2

(5.3)

where the coefficients outside the range set by al and a, are thought most likely to come from noise. Pruning is effectively scale dependent thresholding within a band of limits al and a,. The soft thresholding employed by Abbate and co-workers had the usual form:

T(a, b) =

o { sgn[T(a, b)] x (IT(a,b)l- A)

for for

IT(a,b)1 < A IT(a,b)I;o: A

(5.4)

where A is the threshold. Figure 5.30(a) shows an input signal used in the ultrasonic testing of steel specimens. Figure 5.30(b) shows the same signal with added noise and its wavelet transform plot is given in figure 5.30(c). The reconstructed signal after pruning and thresholding is shown in figure 5.31 (a) with the filtered wavelet transform plot used for the reconstruction shown below in figure 5.31(b). Figure 5.32 contains the signal from a cast iron sample together with its wavelet-filtered version. A Morlet wavelet was employed and the filtering used both pruning and thresholding. Both echoes are clearly detected without any other contribution from the acoustic noise. Copyright © 2002 lOP Publishing Ltd.

pile toe location

o

2

4

6

8

10

12

14

16

time (ms)

Figure 5.29. A complete Morlet wavelet analysis of a sonic echo signal. The figure contains a sonic echo signal taken from a pile, together with a sequence of scalograms generated using a complete Morlet wavelet decomposition of the signal with the central frequency set to (from top to bottom) Wo = 5.5, 4.5, 3.5, 2.5 and 1.5. After Addison el 01 (2002a). Reproduced with kind permission of Academic Press Ltd.

Copyright © 2002 lOP Publishing Ltd.

flaw signal y (f) 1.0,-------,---------,

0.5

o r.------1IIIIV.....- - - -0.5 (a)

- 1.0 :------:c::::--:c=--:::-:-----:c=----:c=--:c:' o 500 1000 1500 2000 2500 3000 r----~--.~-~-~-__,

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(b)

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1500

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2000

2500

3000

2000

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-2 -3

-4

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500

1000

1500 time

Figure 5.30. \Val'elef analysis of a pulse echo reflection signal. (a) Original signal. (b) Signal in (a) with added acoustic noise generated by spherical voids in steel. (c) The wavelet transform contour plot of the signal in (b). After Abbate et al 1997 IEE£ Transactions 011 Ulrrasonics alld Frequency Conrrol 44(1) 14-26. © IEEE 1997.

Chen ef al (1996) have proposed a wavelet-based technique to improve the signal to noise ratio of ultrasonic inspection signals obtained from coarse-grained stainless steel specimens. Chen ef al (1999c, 2000) have detailed both wavelet and wavelet packet methods for denoising ultrasonic signals for the non-destructive evaluation

'" ~

~

(a)

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c

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15 8u

500

1000

1500

2000

2500

3000

2000

2500

3000

2 I 0 -I -2

o

-3

0

500

1000

1500 time

Figure 5.31. Filtered pulse echo reflection signal. (a) The reconstructed signal obtained from the filtered wavelet transform plot shown in (b). After Abbate et af 1997 IEEE 7i'(l1lsactions Oil Ultrasonics and Frequency C0ll1rol44(I) 14-26. © IEEE 1997.

Copyright © 2002 lOP Publishing Ltd.

40

r-~~~~~-----~~-,

20

o ~Mlil -20 (a)

-40

-~ e

L-_=,--~----,-,"-c--_,.,.-_,.J

o

1000

2000

3000

4000

SOOO

20

0 f--;Jlllr-----"I\",V/\r----I -20

(b)

time

Figure 5.32. Wavelet of a pulse echo reflection signal taken from a cast iron sample. (a) Ultrasonic signal. (b) Output after wavelet filtering clearly showing the two echoes. After Abbate et al 1997 IEEE Transactiol/s 01/ Ultrasonics and Frequency Control 44(1) 14-26. © IEEE 1997.

of steel samples with known defects. Cho et a/ (1996) employed Morlet wavelets to detect subsurface defects in steel test specimens from non-contact laser ultrasonic signals. Staszewski et a/ (1997) have described a wavelet-based signal processing

method to enhance defect detection in a carbon fibre composite plate interrogated using ultrasonic Lamb waves and incorporating an optical fibre receiver. Wu and

Chen (1999) have also used the Morlet wavelet to analyse non-contact laser ultrasonic signals from epoxy-bonded copper-aluminium layered specimens. Their study focused on the detection of unbonded regions. A Mexican hat wavelet was employed by Guilbaud and Audoin (1999) in the interrogation of laser-induced ultrasonic signals used to measure stiffness coefficients in a viscoelastic composite material.

They found that the reliability of their method may justify its use in the field of material behaviour characterization.

A number of other authors have developed wavelet-based tools to aid the interpretation of NOT signals. Shyu and Pai (1997) have performed impact--echo tests on free standing concrete cylinders using Daubechies wavelets. Tang and Shi (1997) used wavelet techniques to detect and classify a variety of welding defects from NOT signals. Doyle (1997) has employed a wavelet technique to identify the impact force on structures using a knowledge of the structure and its response to the force. Hamelin et a/ (1996) used wavelets to analyse eddy current signals. They analysed the complex and real parts of the signal separately in order to develop a classification scheme to inspect the transformed signals for the location of distinct maxima and associated phase. Pierri et a/ (1998) investigated the use of twodimensional Haar wavelet transforms in eddy current NOT, and Lingvall and Stepinski (2000) have described an automatic method for the detection and classification of cracks located in aircraft riveted lap-joints during eddy current inspection which employs Coiflet wavelets. Qi (2000) has described a wavelet-based method to analyse acoustic emission (AE) signals in a study of material fracture behaviour. Qi employed Daubechies wavelets to analyse experimental AE signals and found that Copyright © 2002 lOP Publishing Ltd.

wavelet-based techniques better approximate the relationship between the stress and stress intensity factor than do classical techniques. The AE behaviour of reinforced concrete beams tested under flexural loading was investigated by Yoon et al (2000) using both Fourier- and Morlet-based wavelet methods. They found that the Fourier spectra and wavelet transforms of the AE signals gave useful information about the relationship between the damage mechanisms of the concrete (e.g. microcracking, localized cracking, flexural cracking and shear bond cracking) and the AE response. An examination of the integrity of thin coated foil used in the food packaging industry was conducted by Futatsugi et al (1996) using both AE monitoring and microscopic observation. AE signals were omitted by the cracking of the surface coating (SiO x film) and also from its delamination from the foil. The threshold tensile strain necessary to cause the first fracture, estimated from the AE signals, agreed well with the strain determined using a laser microscope. An experimental study of crack detection in metallic structural elements using fourth-order Daubechies wavelets has been carried out by Biemans et al (1999). They instrumented a pre-cracked rectangular metal plate with piezoceramic sensors and subjected it to both static and dynamic tensile loading. A statistical measurement based on the logarithmic wavelet variance of the strain data was used as a damage index. This parameter provided an insight into the scale dependent changes in energy of the strain data from the piezoceramic sensors. Wang and Deng (1999) have used wavelets to probe the spatial profiles of damaged cantilevered beams under static and dynamic loading. Their results clearly show that the location of a structural crack in the beam can be pinpointed through the Haar wavelet transform coefficients. They found similar results using Morlet-based transforms. See also the related papers by Deng and Wang (1998), Liew and Wang (1998) and Quan et al (1999). Morlet wavelet transforms have been employed by Li and Berthelot (2000) to analyse pulse-echo signals from thick annular waveguides. They developed a local spectral-temporal wavelet energy measure by integrating the energy density scalogram over a box of limited extent in scale and location. They found this energy measure to be particularly good at localizing cracks in faulty annular components and tested it successfully on data from both an annular waveguide with a machined crack and on a partially annular component of the pitch shaft of an H-46 helicopter. Marwala (2000) has developed a 'committee of neural networks' technique which employs frequency response functions, modal properties and wavelet transform data simultaneously to identify damage in structures. The method was tested on synthetic data from three coupled oscillators and then used to identify the damage in seam-welded cylindrical shells. Gros et al (2000) have presented results concerning the fusion of images from multiple NDT sources gathered during the inspection of a composite material damaged by impact. The images were fused using a variety of techniques, including one based on Daubechies D8 wavelet transforms, in order to improve the defect detection and provide a more accurate measurement of defect dimensionality. Two-dimensional Morlet wavelets have been used by Li (2000) to detect partial fringe patterns generated in NDT interferometry induced by defects. The author found that the wavelet method was suitable for both holographic interferometry and electron speckle pattern interferometry. Chan et al (2000b) have Copyright © 2002 lOP Publishing Ltd.

used wavelet packet denoising within a digital speckle correlation method to detect defects in multilayer ceramic capacitors in surface-mounted printed circuit boards.

5.7

Surface characterization

The characterization of engineering surfaces is pertinent to a number of engineering fields providing quantitative information on the formation process of the surface, for example the manufacturing process used to form a machine component or the fracture process causing a rugged crack surface. Surface topography is one of the most important factors affecting the performance of manufactured components. It can be related to a number of pertinent engineering aspects such as wear, lubrication,

friction, corrosion, fatigue, coating, paintability, etc. Two-dimensional biorthogonal wavelet transforms have been used by Jiang e1 al (1999) to probe the surface topography of orthopaedic joint prostheses. They used three wavelet-based parameters to characterize the surface: roughness, waviness and form. The roughness of the surface was defined as the detailed surface found through the inverse transform of the thresholded wavclet coefficients within a band of the smallest scale indices. Similarly, the waviness of the surface was defined as the detailed surface found through the inverse transform of the thresholded wavelet coefficients within a band of the next smallest scale indices. The thresholding method employed in both cases simply limited the maximum absolute value of the coefficients to three times their standard deviation at each scale. This was carried out to disassociate the

overall surface topographic features from localized peaks, pits and scratches, as it was assumed that each detail coefficient belonging to the roughness or waviness follows a Gaussian distribution. (Hence, a coefficient is very unlikely to appear outside three standard deviations of the distribution.) The form of the surface was defined to be the original surface minus the detail signals over the roughness and waviness scales.

Figure 5.33 contains the original centre profile of a ceramic femoral head together with the multi resolution approximations of the profile after wavelet decomposition with the biorthogonal wavelet pair. The detail signals at scales I to 4 are shown on the right-hand column. These scales cover the roughness wavelengths and the addition of these four detailed signals gives the roughness profile shown at the top right-hand of the figure. Figure 5.34 shows three three-dimensional plots of the same surface separated into its roughness, waviness and form components.

Multiscalar topographical features such as peaks, pits and scratches on the surface were then defined by Jiang and his colleagues by hard thresholding the coefficients using two standard deviations of the coefficient amplitude at each scale as the threshold. An example of the multiscalar topographical features defined in this way is given in figure 5.35. Tn a later paper, Jiang ef al (2000) have provided details of a wavelet-based analysis of both rolled steel sheets and ceramic femoral heads. See also Chen ef al (I999d), who have developed a similar wavelet-based method to decompose surface data sets into a surface roughness component and a wavelet reference surface.

Chen e1 al (1995) have analysed surfaces produced by typical manufacturing processes using Daubechies wavelets, citing the advantage of the space-scale Copyright © 2002 lOP Publishing Ltd.

~m

~m

1~1z::sJ

0.70

o

0.0605

0.1210 0.1815

0.2420

the original centre profile

nun

0006~

0.003 0.000

-0.003 -0.006

o

0.0605 0.1210 0.1815 0.2420 the roughness centre profile nun (D 1+D2+D3+D4 )

j=1

j=2

D, j=3

j=4

A4=reference profile

0.0605

0.1210

0.1815

0.2420

the approximations

o

0.0605

0.1210 the details

0.1815

0.2420 mm

Figure 5.33. Multiscalar decomposition of a ceramic femoral head using a two-dimensional biorthogonal wavelet pair: centre profile. This material has been reproduced from the Proceedings of rhe Illstill/riol/ of Mechanical Engineers, Parr fI, Journal of Engineering ill Medicine 199921349-68, figure 6A, by Jiang et ai, by permission of the council of the Institution of Mechanical Engineers.

localization properties of the wavelet transform over other techniques. A number of continuous wavelets are considered by Lee et al (1998) in a study of the morphological characterization of engineering surfaces. They employed the Morlet and Mexican hat (second derivative of Gaussian) wavelets, as well as the less common Barrat wavelet and an eighth derivative of a Gaussian function, to investigate the potential applications of wavelet decomposition in assessing the muItiscale features of the engineered 0.10 0.05

o

o.o~j

I!m

Figure 5.34. Reconstrucled surface lopographies of a ceramic femoral head in different transmission bands: rough (left), wavy (centre) and form (right). This material has been reproduced from the Proceedings of the II/srittlrion of Mechanical Engineers, Part H, Jotlmal of Engineering in Medicine 1999 213 49-68, figure 7A, by Jiang et aI, by permission of the council of the Institution of Mechanical Engineers.

Copyright © 2002 lOP Publishing Ltd.

slope intensity image

Figure 5.35. Multiscalar topographical features (peaks, pits and scratches) in the equatorial region of the ceramic femoral head. This material has been reproduced from the Proceedings a/the /nstitlltion 0/ Mecl/{micaf Engilleers, Part /-1, Joumaf o/Engineering ill Medicine 1999 21349-68, figure 8A, by Jiang et af, by permission of the council of the Institution of Mechanical Engineers.

surface, considering both its manufacturing and functional aspects. Song el al (2000) have developed a technique for the inspection of surface mount devices in the electronic industry using modified Haar wavelets. Oogariu el al (1994) have described the light scattered by a slightly rough surface in terms of two-dimensional wavelet transforms, and Zhuang el al (1998) have developed a laser-based noncontact system for the inspection of pipe inner walls where wavelet-based decomposition of the reflected signal from the pipe wall is used to characterize the surface texture as form, error, wavincss, roughness and sporadic scratches. Jasper el al (1996) have cmployed adaptive wavelet bases to capture texture information and locate defects in woven fabrics from their images, and Lin and Xu (2000) have introduced a wavelet-based method to quantify the fuzziness of woven and knitted fabrics from their images. The fractal structure of fractured granite surface profiles is examined using wavelets by Simonsen el al (1998), and Addison and Watson (1997) have detailed the use of wavelet transforms to analyse rough surfaces modelled using fractional Brownian motions. In an analysis of the fractal properties of cracked concrete surfaces, Dougan el al (2000) have compared both wavelet and Fourier-based power spectral methods with traditional methods for determining fractal parameters. The multiscale characterization of pitting corrosion damage has been carried out by Frantziskonis el al (2000) using Oaubechies wavelets with four vanishing moments (i.e. 08s) to determine the Hurst exponent associated with the geometry of the corrosion pits. The submicron surface roughness of anisotropically etched silicon has been analysed using Meyer wavelets by Moktadir and Sato (2000) who found a scaling exponent for the surface close to 0.5. Srinivasan and Wood (1997) used Oaubechies 012 wavelet transforms as a tool to compute the relevant fractal parameters in a fractal-based approach to geometrical tolerancing (see also Tumer el ai, 1995). The authors focused on mechanical tolerances applied to physical-product features, such as machined parts and assemblies. They used the roundness of ball bearing elements as a design example. Chapter 7 contains more information concerning the use of wavelet Copyright © 2002 lOP Publishing Ltd.

transform-based methods for the analysis of both geophysical topographic features and surfaces exhibiting fractal structure.

5.8 5.8.1

Other applications in engineering and further resources Impacting

The impacting of detector tubes in boiling water nuclear reactors has been detected by Ritcz and Pazsit (1998) using Haar wavelets. They thresholded the Haar wavelet coefficients to remove noise and reconstructed the signals to produce a time series of the impacts. The threshold was set to four times the standard deviation of the coefficients whereby all coefficients below this level were set to zero. Reconstruction using only the remaining large coefficients gave an indication of the impact events in the signal. RilCZ and Pazsit investigated both modelled data and in situ measurements. The top trace in figure 5.36 shows a detector signal taken from the Swedish Barseback-l reactor where no noticeable vibrations were observed to occur. The fuel box vibration signal, after filtering out the low frequency detector string vibrations, is shown in the middle of the figure. This signal is then transformed and filtered using the Haar transform as described above. The reconstructed signal is shown in the bottom plot. A few intermittent spikes can be seen to occur over the signal length. The paucity of spikes in this wavelet-filtered signal can be contrasted to the 5 ,_~_ _~_d",e",tec",>;to""r""si,,n",a,,"1!=L,",PRM=,,-18~3,--_~ _ _~_---,

o -5

__ 1000 1500

L._~.,-_~

o

500

__ 2000 2500

~_~

......J..-_ _ 3000 3500

~.,-_

-J-~ __

4000

time fuel box vibration (after stop filtering)

0.1 r--~-':'::;:'=.:.c.:;==:;====---~----,

o -0.1

_ _~ _ ~_ _~ _ ~ _ ~ ~ _ . J

"---_~_~

o

500

1000

1500

2000

2500

3000

3500

4000

time fuel box vibration; severity =0.04069

0.1 r--~-"':':;:=::"':';==':;:::':=~=::':""--~----,

o H-4--+----+-+ -0.1

+tfI-Ht-f-lt---+--HI ----1+-..- ..

"---_~:_-:~;___;=-~=---:=~-

o

500

1000

1500

2000

2500

L..-

3000

. _ J - .._ _ ..__

3500

4000

time

Figure 5.36. Signal from detector tube experiencing no \'ibrations. The filtered detector signal (middle trace) is shown below the original signal (LOp). The reconstructed signal from the Ihresholded Haae coefficients is shown in the bottom trace. Reprinted from Rilcz and Pilzsit 1998 AIII/als of Nuclear Energy 25(6) 387--400. Copyright (1998), with permission from Elsevier Science. Copyright © 2002 lOP Publishing Ltd.

detector signal LPRM033

0.5~£U:~

_O:c::·=_=-.J.....-~_~_~ o 500

1000

1500

2000 2500 3000 time fuel box vibration (after stop filtering)

3500

4000

,--~-~.:-:.---~,'---'---e.---'---cc.,:.'---=~-~----,

0.5

o -0.5 0~--:5~00:--1000:':-c-15~00:--'2~000:-----:2~5OO~----:3-:000~-3:-:5~00:--4000""· Time fuel box vibration; severity = 0.765 _ _~_-, 0.5 ,--~ _ _.;cc:'-"c"-,-==:;:..:==.,==~

o -0.5

'----~-_,_'--~-~--~-~--~--'

o

500

1000

1500

2000 time

2500

3000

3500

4000

Figure 5.37. Signal from detector tube which is known to ha\'e experienced impacting. Reprinted from Racz and Pazsit 1998 Allllals of Nuclear Energy 25(6) 387--400. Copyright (1998), with permission from Elsevier Science.

high occurrence of spikes in the bottom plot of figure 5.37. This signal was taken from a dctcctor tubc which was found from subscqucnt inspcction to have cxperienced impacting. The authors determined a parameter, 5, for the severity of impacting based on a normalized value of the estimated impact rate. High values of the severity 5 indicate the occurrence of strong vibrations and the possibility of impacting. 5.8.2

Data compression

An investigation of the use of wavelet-based methods for the compression of vibration signals has becn carried out by Staszewski (I998b). Hc compared Fourier-based compression with wavelet-based compression (using Oaubechies 04 and 020 wavelets) for a variety of signals. Figurc 5.38 shows a periodic signal with its associated Fourier frequency spectrum and wavelet coefficients. We can see by looking at the Fourier and wavelet domain representations of the signal that the Fourier case is more compact. [n fact, we would expect Fourier representation to favour the concise representation of a periodic signaL Figure 5.39 shows the normalized mean square error (MSE) of the reconstructed signal as a function of the number of coefficients (both Fourier and wavelet) used in its reconstruction. This plot confirms what we can see by eye. That is, for a periodic signal, a much lower error results from using the same number of Fourier coefficients as wavelet coefficients. The nOlmalized MSE is defined as N

MSE(x) =

100", " - , L. (x; - x,)Na;.

Copyright © 2002 lOP Publishing Ltd.

;=]

(5.5)

1.0 "0 .g" 0.5

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where X; are the components of the original signal of length N, ,,; is the variance of the signal, x~ is the reconstructed signal using selected coefficients, and the factor of 100 gives MSE(x) as a percentage. The situation depicted in figures 5.38 and 5.39 changes when a transient signal is compressed. An example of such a signal is given in figure 5.40(a). We can see from figures 5.40(b) and (c) that the Fourier spectrum for this signal is relatively broad band and the dominant wavelet coefficients are relatively localized. For this transient signal, the mean square error plot indicates that far fewer wavelet coefficients are required to form a good approximation to the original signal (see figure 5.41). Staszewski went on to use wavelet compression as a method for feature dctection in two data sets: ultrasonic flaw detection signals and gear vibration data. He used a variety of methods to select the most appropriate coefficients for this task including: simple thresholding; a priori knowledge of the Copyright © 2002 lOP Publishing Ltd.

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system; genetic algorithms; and the temporal location of coefficients. Tanaka ef at (1997) havc also approached the problem of data compression of mechanical vibration data. They compressed a variety of signals taken from rotating machinery using Daubechies wavelets (D4 to D20) and detailed the signal distortion at various compression ratios. Desforges ef at (1998) have described their use of wavelet transforms to compress data for the practical application of probability density function (PDF) estimation. The original data sets were transformed and only a limited number of wavelet coefficients were retained and subsequent PDF estimates made using these values rather than the original data sets themselves. The retained coefficients were chosen using a genetic algorithm. They applied their combined data compression and PDF estimation approach to faulty gearbox signals and military target radar data. 5.8.3

Engines

Thomas ef at (1997) have employed a wavelet network to detect the occurrence of engine knock from vibration signals taken from spark-ignition engines. They used 40 Morlet wavelets of predetermined scales and locations to decompose the signal in an attempt to differentiate between three classifications: (I) absence of knock, (2) increasing knock, and (3) heavy knock. Rasping noises in automotive engine exhaust ducts due to abrupt accelerations have been characterized by Ayadi ef at (200 I) using STFT, CWT and wavelet packet decompositions of the internal tailpipe pressures. Liu and Ling (1999) have developed a modified matching pursuit method for machinery fault diagnosis which uses a mutual information measure to select the best wavelets from the dictionary, i.e. those which carry important information with

little redundancy. They applied their method to the detection of two diesel engine malfunctions-injection timing advance and 'blow-by'-and found that the technique performed better than principal component analysis which is widely used in practice. Sullivan ef at (1999) have reported on their examination of the turbulent velocities within the cylinder head of a spark-ignition engine. They employed Copyright © 2002 lOP Publishing Ltd.

continuous Mexican hat wavelet transforms to produce energy maps and investigated correlations within the flow. In addition, they used Daubechies 04 wavelets in a fast discrete wavelet transform method to subtract the mean from the instantaneous velocity field to give the turbulence velocities, comparing the resulting velocity-time series with ensemble-averaged and cyclic-averaged methods. See also the related paper by Ancimer el a/ (2000). Hoss el a/ (2000) have detailed a study of the stall inception behaviour of a jet engine which employed Daubechies wavelets and other analysis tools to interrogate pressure measurements taken within the engine's compressor system. They identified three different types of stall inception processes for undistorted inlet flow and suggested that an active avoidance control strategy may be devised using wavelet analysis and extended statistical evaluation.

5.8.4

Miscellaneous

Williams and Amaratunga (1994) have provided a comprehensive introduction to the use of discrete wavelet transforms in engineering which includes their role in data analysis and the solution of partial differential equations. Piezoelectric accelerometers have been used by Hale and Adhami (1998) to measure the highly nonstationary vibration data collected from helicopter missiles during manoeuvres. They analysed the signals using both wavelets and wavelet packets and found these methods to be excellent candidates for establishing vibration specifications for ensembles of nonstationary events. Le-Tien el a/ (1997) have detailed the wavelet analysis of a radar echo signal from a spinning rotor. Nygaard and Grue (2000) have described a wavelet-based method for the computation of wave properties and hydrodynamic forces on arrays of floating bodies. Sureshbabu and Farrell (1999) have developed a wavelet-based system identification method for nonlinear control and Patton and Marks (1996) have detailed a one-dimensional finite element based on Daubechies 018 wavelets. The application of wavelet methods to data analysis problems in electrical and chemical engineering is dealt with briefly towards the end of chapter 7.

Copyright © 2002 lOP Publishing Ltd.

Chapter 6

Medicine

6.1

Introduction

In this chapter we review the many areas of medical science where the wavelet trans-

form has made an impact. We begin with the ECG signal, where attempts have been made using wavelet methods to determine its characteristic points, compress it, detect abnormalities, characterize heart rate variability, and probe a variety of arrhythmias. The next section considers the wavelet as a potential diagnostic tool for neuroelectric waveforms: the EEG, evoked potentials and event-related potentials. Medical sounds are then examined, including the sound from the turbulence due to arterial blockage as well as the sound of the heart itself, respiratory sounds and the acoustic response of the ear. In addition, ultrasonic time-series signals are covered in this section (although not ultrasonic images which are left to the next section on medical images). The section on medical images considers the application of wavelet transform to ultrasonic, radiographic and optical images. There then follows a short section on the analysis of blood flow and blood pressure data before the final section containing a selection of other medical applications of the wavelet transform, including the wavelet analysis of DNA, the electromyograph, sleep apnoea syndrome, chronobiological rhythms, the fractal structure of foetal breathing rates, and more. This final section ends by providing details of other resources concerned with the application of the wavelet transform and related methods to medical signals.

6.2

The electrocardiogram

Muscular contraction is associated with electrical changes known as depolarization. The electrocardiogram (ECG) is a measure of this electrical activity associated with the heart. The ECG is measured at the body surface and results from electrical changes associated with activation first of the two small heart chambers, the atria, and then of the two larger heart chambers, the ventricles. The contraction of the atria manifests itself as the 'P' wave in the ECG and contraction of the ventricles produces the feature known as the 'QRS' complex. The subsequent return of the ventricular mass to a rest state-repolarization-produces the 'T' Copyright © 2002 lOP Publishing Ltd.

R

Q S Figure 6.1. A schematic of the EeG exhibiting normal sinus rhythm. Note that the shape of each feature can vary depending 011 the configuration of the ECG leads.

wave. Repolarization of the atria is, however, hidden within the dominant QRS complex. Figure 6.1 shows a schematic of the ECG waveform for normal sinus rhythm. Analysis of the local morphology of the ECG signal and its time-varying properties has produced a variety of elinical diagnostic tools. In this section we review the application of the wavelet transform to the interrogation of the ECG signal. 6.2.1

ECG timing, distortions and noise

Producing an algorithm for the detection of the P wave, QRS complex and T wave in an ECG is a difficult problem due to the time-varying morphology of the signal subject to physiological conditions and the presence of noise. Recently, a number of wavelet-based techniques have been proposed to detect these features. Senhadji el al (1995) compared the ability of wavelet transforms based on three different wavelets (Daubechies, spline and Morlel) to recognize and describe isolated cardiac beats. Sahambi el al (1997a,b) used a first-order derivative of the Gaussian function (figure 6.2(a» as a wavelet for the characterization of ECG waveforms. They used

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modulus maxima-based wavelet analysis to detect and measure various parts of the signal. specifically the location of the onset and offset of the QRS complex and P and T waves. Note that for an anti symmetric wavelet, such as the first derivative of the Gaussian, dominant peaks in the signal correspond to zero crossings in the wave-

let transform. (For symmetric wavelets, dominant peaks in the signal correspond to extrema in the transform plot.) One of the ECG signals they analysed, together with its wavelet transform at four consecutive scales, is shown in figure 6.2(b). The maxima and minima of the wavelet-transformed signal are used to determine the location and width of the QRS complex. This is shown in the lower two plots of figure 6.2(b) containing, respectively, the signal and its corresponding wavelet transform at the largest scale. The vertical lines above the ECG signal at the top of the plot show the location of the QRS complex determined from the zero crossings of the modulus maxima of the transformed signal. Figures 6.3 and 6.4 show the same analysis as figure 6.2, but this time baseline drift and high frequency noise have been added to the signal respectively. A schematic of the timing intervals is shown in figure 6.5(a), and figure 6.5(b) shows the values of the timing intervals of one beat of the ECG signal computed using the wavelet modulus maxima-based method of Sahambi and his co-workers. The measurements of these intervals give the relative position of the components in the ECG which are important in delineating the electrical activity of the heart. Improvements to the technique are described in Sahambi ef al (1998). An algorithm based on quadratic spline dyadic wavelet transforms and the modulus maxima method has been developed by Li ef al (1995) to detect the Copyright © 2002 lOP Publishing Ltd.

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characteristic points ofECG signals. The algorithm can distinguish the QRS complex from a host of other features in the signal, including high P and T waves, noise, baseline drift and signal artefacts. Sivannarayana and Reddy (1999) have proposed the use of both launch points and wavelet extrema to obtain amplitude and duration parameters from the ECG. Kadambe er al (1999) have given details of a waveletbased QRS complex detector which also incorporates a modulus maxima algorithm. Their interest was in the determination of heart rate and hence their study concentrated on detecting the temporal location of the R wave and measuring the R-R interval. Other work has been undertaken by Park er al (1998) using a wavelet adaptive filter to minimize the distortion of the ST segment due to baseline wanderings, and Tikkanen (1999) has evaluated the performance of different wavelet-based and wavelet packetbased thresholding methods for removing noise from the ECG. 6.2.2

Detection of abnormalities

Tuteur (1989) was one of the first proponents of the wavelet transform as an analysis tool for medical signal processing, using a complex Morlet wavelet to detect abnormalities in ECG signals. In particular, Tuteur was interested in an abnormality known as a ventricular late potential (VLP). This represents low-amplitude electrical activity due to delayed electrical conduction by the ventricle muscles. VLPs occur in the ECG after the QRS complex and are often masked by noise. They have been used as a marker to identify patients at risk from certain types of life threatening arrhythmias. Tuteur added a synthetic VLP-like segment of signal to ECG data from a child with a cardiac defect. Figure 6.6(a) shows the original cardiac signal and figure 6.6(b) shows the original signal together with the synthesized defect. Figure 6.7 shows the wavelet transforms of the signal at three a scales. The synthesized abnormality can be seen as a bulge on the middle peak at an a scale of 1/16. Wavelet energy scalograms were used by Meste el al (1994) as a method of highlighting VLPs and observing temporal and frequency variability in the ECG from beat to beat. Batista and English (1998)

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employed both the harmonic and closely related musical wavelets in the detection of VLPs. They performed a wavelet decomposition of the ST and TP segments of the ECG and compared the relative energies contained at each level in order to detect VLPs. They reported superior results using their technique over the Simson method, widely used in clinical practice for the detection of VLPs. Tn addition, they found that the reduction in spectral leakage of these wavelets provides better results than using Daubechies wavelets although there is a reduction in the associated time resolution.

Couderc el al (1996) employed the Morlct wavelet transform to analyse highresolution ECGs in post-myocardial infarction patients both with and without documented ventricular tachycardia. A discretization of ten wavelet scales covering the relevant range of the time--frequency plane allowed them to stratify the resulting time-frequency information concerning ECG abnormalities. In the group of myocardial infarction patients with documented ventricular tachycardia they found significantly increased high-frequency components corresponding to prolonged QRS durations and late potentials in the area 80-50 ms after QRS onset. They also applied their method to the intra-QRS abnormalities in patients with congenital long QT syndrome. Rakotomamonjy el al (1998) have detailed a method for detecting VLPs using Morlet wavelet preprocessed data as input to a fccdforward neural network. They tested the technique on simulated ECGs containing VLPs and a range of additive noise, and found a high degree of accuracy in classification, even for high levels of noise. Rakotomamonjy el al (1999) have also described a wavelet-based filtering method for signal-averaged ECGs used for the detection of late potentials. The detection of myocardial ischaemia in pigs using a wavelet-based entropy measure is described by Lemire el al (2000). They considered the morphology of the combined ST segment and T wave, performing a fast wavelet transform using spline wavelets. The Shannon entropy of the coefficients at each scale was determined for the combined ST segment-T wave at each beat. An increase in entropy was detected at certain scales due to coronary occlusion, which led the authors to suggest a threshold entropy value as an indicator of the occlusion state. The best scale for use as a marker corresponded to an approximate frequency band of 30--60 Hz. In a pilot Copyright © 2002 lOP Publishing Ltd.

study, Gramatikov el al (2000) used Morlet wavelet transforms to analyse the ECG recordings from patients with left and right coronary stenosis taken before and after angioplasty. They focused on the morphology of the QRS complex in wavelet space plotting both two-dimensional contour plots and three-dimensional representations of the transform magnitude and demonstrated the wavelet's ability to detect short-lasting events of low amplitude superimposed on large-scale deflections. The study found changes in the mid-frequency range which reflected the ECG's response to percutaneous transluminal coronary angiopiasty.

6.2.3

Heart rate variability

Rather than consider the morphology of the whole ECG signal, many researchers have focused on the temporal variability of the heartbeat. To do this, they monitor the timing interval between beats, taken between each R point on the QRS complex, and plot this R-R interval against time to give the heart rate. The minute fluctuations present in the R-R intervals have becn used for assessing the influencc of the autonomic nervous system on the heart rate. In addition, long-range correlations and power law scaling have been found through the analysis of heartbeat dynamics. Much of the current work concerning heart rate variability focuses on its use as a

Figure 6.8. Short term analysis of heart rate variability by adapted w3\'elet packets. (a) Recorded data and estimated trend for a normal subject. (b) Detrended HRV signal. The vertical lines indicate the onset of different procedures with controlled respiration al 6 and 12 breaths/min, and when the subject is in the upright position after passive tilt. After Wiklund er al 1997 IEEE Engineering il/

Medicine nnd Biology 16(5) 113-118. © IEEE 1997.

Copyright © 2002 lOP Publishing Ltd.

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marker for the prediction and diagnosis of heart disease and assessment of heart function.

Wiklund e1 al (1997) used adaptive wavelet transforms (wavelet packets and cosine packets) to analyse the regulation of heart rate variability (HRV) by the autonomic nervous system. Figure 6.8 contains an original HRV signal used by Wiklund and eo-workers in their study together with a detrended version. The resulting time-frequency plane decomposition of the detrended signal using wavelet packets is shown in Figure 6.9. Their results suggested that adapted wavelet transforms can be used to detect transient changes in the signal and characterize both tonic and reflex autonomic activity. Thurner e1 al (1998a) have employed both Oaubechies 010 and Haar wavelets in the analysis of human heartbeat intervals. They found that at distinct wavelet seales, corresponding to the interval 16-32 heartbeats, the scale-dependent standard deviations of the wavelet coefficients could differentiate between normal patients and those with heart failure. Significantly, they could do this with 100% accuracy for a standard 27-patient data set. Figure 6.10 shows a typical series of interbeat intervals against interval number together with its wavelet

coefficients at three scales obtained using the Oaubechies 010 wavelet. Figure 6.11 shows plots of coefficient-dependent standard deviations against seale for the 27patient data set. Notice how the plots for normal and heart-failure patients decouple 4 at scales 4 and 5 (i.e. 2 -2' = 16-32 heartbeats). Further development of the technique is detailed in a subsequent paper by Thurner e1 al (1998b). Ivanov el al (1996) have investigated the ECG signals from subjects with sleep apnoea. By sampling at an a scale equivalent to eight heartbeats, they performed a local smoothing of the high-frequency variations in the signal in order to probe patterns of duration in the interval 30-60 s. The authors used the data to characterize Copyright © 2002 lOP Publishing Ltd.

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16265). (Adjacent values of the interbeat interval are connected by straight lines to facilitate viewing.) Subslanliallrends are evident. (c) Three-dimensional representation of the wavelet coefficient Was a function of scale (1 ::; III :S: 10) and interval number i (/I has been rescaled to i), over a portion of the

data set using the Daubechies IO-tap analysing wavelet. (d) Wavelet coefficient at three scales (m = 2, 4 and 8) for the data set illustrated in (b). The trends in the original interbeat-interval time series are removed by the wavelet transformation. After Thurner et (1/ (1998a). Reproduced with the kind permission of the American Physical Society and the authors.

al (1998) have compared two Fourier methods (the discrete Fourier transform and the nonequispaced Fourier transform) of computing the Fourier coefficients used the nonstationary heartbeat behaviour and elucidate phase interactions. Bates

el

in the discrete harmonic wavelet transform analysis of heart rate variability. The

same group (Hilton ef ai, 1999) have used the discrete harmonic wavelet transform as well as the discrete Fourier transform to perform spectral analysis of the HRV signals associated with sleep apnoea/hyponoea syndrome (SAHS). They compared their spectral analysis of the HRV signals with the current screening method of pulse oximetry. Their results indicated that spectral analysis ofHRV seems to provide a better indicator ofSAHS than oximetry in non-REM sleep and a comparable indicator in REM sleep. Akay and Fischer (1997) compared a wavelet-based method to others in a study to determine the fractal nature of HRV signals. Specifically, they used the method to determine the Hurst exponent of the signal. (See also Fischer and Akay, 1996, 1998.) Ivanov ef al (1999) have reported on the multifractality Copyright © 2002 lOP Publishing Ltd.

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using the same number of interbeal intervals, also exhibit low variability. The outcome is similar for both analysing wavelets. After Thurner et al 1998a. Reproduced with the kind permissions of the American Physical Society and the authors.

found in the healthy human heart rate signal using a wavelet-based analysis method. Further, they reported the loss of multifractality for a life-threatening pathological condition, congestive heart failure. See also Havlin el 01 (2000) in this regard. (For more information on fractals and wavelets see chapter 7, section 7.2.) Zhang el 01 (1997) employed techniques from nonlinear dynamics (phase space reconstruction, correlation exponent and Lyapunov exponents) to investigate heart rate variability. They applied these methods to both the R-R interval time series and to a time series of the variability of the QRS complex. Both time series were determined using the wavelet coefficients obtained from a spline wavelet decomposition of the original ECG signal. Joho el 01 (1999) analysed heart rate and left ventricular pressure variability during coronary angioplasty in humans. They presented threedimensional, Morlet-based wavelet transform plots which showed clearly a low frequency response of both signals to coronary occlusion. They concluded that the regional myocardial ischemia elicited a profound sympathoexcitory response

followed by a gradual suppression over time. This they attributed to the vagal inhibitory reflex. 6.2.4

Cardiac arrhythmias

A number of wavelet-based techniques have been proposed for the detection, classification and analysis of arrhythmic ECG signals. Zhang el 01 (1999) have proposed a novel arrhythmia detection method, based on a wavelet network. Their system detects the bifurcation point in the ECG where normal sinus rhythm degenerates into a pathological arrhythmia. Govindan el 01 (1997) have detailed an algorithm for classifying bipolar electrograms from the right atrium of sheep into four groupsnormal sinus rhythm, atrial flutter, paroxysmal atrial fibrillation (AF) and chronic AF. In their method, they use a Oaubechies 06 wavelet to preprocess the ECG data prior to classification using an artificial neural network. Using a raised cosine Copyright © 2002 lOP Publishing Ltd.

wavelet transform, Khadra el al (1997) have undertaken a preliminary investigation of three arrhythmias-ventricular fibrillation (VF), ventricular tachycardia (VT) and atrial fibrillation (AF). They developed an algorithm based on the scale-dependent energy content of the wavelet decomposition to classify the arrhythmias, distinguishing them from each other and normal sinus rhythm. Englund el al (1998) studied the predictive value of wavelet decomposition of the signal-averaged ECG in identifying patients with hypertrophic cardiomyopathy at increased risk of malignant ventricular arrhythmias or sudden death. They concluded, however, that wavelet decomposition was of limited value in this type of analysis. Figure 6.12 contains three beats of a normal sinus rhythm of a pig heart together with its (Morlet) wavelet energy scalogram shown as both a contour plot

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showing sinus rhythm. (b) The corresponding energy scalogram of the temporal location against bandpass frequency of the wavelet. (c) The Ihree~dimensionallandscape plot of (b). After Addison er 01 2000 IEEE Engineering in Medicine olld Biology 19(5) 104-109.

Copyright © 2002 lOP Publishing Ltd.

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Figure 6.13. Wa\'elet transform of ECG exhibiting ventricular fibrillation. (a) Single channel ECG showing VF. (b) The corresponding three-dimensional energy scalogram of the temporal location against bandpass frequency of the wavelet. The arrow points to a region of coherent modulation of the 10 Hz band. After Addison el at 2000 IEEE Engilleerillg il/ Medicine alld Biology 19(5) 104-109. © IEEE 2000.

(figure 6.12(b» and a three-dimensional surface relief (figure 6.12(c)). Note that the logarithm of the energy is plotted in the figures as it allows for features with large differences in their energy to be made visible in the same plot. The QRS complex of the waveform manifests itself as the conical structures in figure 6.12(b). These converge to the high frequency components of the RS spike. The P and T waves are also labelled in the plot. In addition, the continuous band evident in the plot at a frequency of around 1.7 Hz corresponds to the beat frequency of the sinus rhythm. The three-dimensional morphology of the signal in wavelet space is shown in figure 6. 12(c). Figure 6. I3 shows a portion of a pig heart ECG exhibiting ventricular fibrillation. The three-dimensional morphology of the energy scalogram reveals the presence of organized structure contained within the VF signal as seen by the three undulations within the 10Hz band of the scalogram. This transient modulation of the 10Hz band is located by the arrow in the plot. Figure 6. I4 shows another portion of VF together with the two-dimensional contour plot of the energy scalogram. Distinct high frequency spiking of a periodic nature can be seen within the scalogram. This regular structure is not at all evident from the ECG trace nor is it evident using short time Fourier transform (STFT) analysis due to its fixed window width (see below). The spiking becomes more prominent with increasing downtime (duration ofVF). (It is worth noting that this periodic spiking has also been observed in segments of human VF-see figure 6.15.) A global view of the porcine VF signal in wavelet space is given in figure 6.16, which contains an energy scalogram for a 5 min

period of VF followed by a 2.5 min period of cardiopulmanory resuscitation (CPR). The onset of CPR is distinguished by the large amplitude horizontal band appearing at low frequency at 5 min. Distinct banding can be seen in the scalogram over the first 5 min: a high frequency band at around 10 Hz and two lower energy bands at lesser frequencies, labelled A, 8 and C. After the onset of CPR, an increase in the passband frequency of all three bands can be observed in the scalogram. The results depicted in figures 6.12 to 6. I6 show that VF, previously thought to represent disorganized and unstructured electrical activity of the heart, does in fact contain a rich underlying Copyright © 2002 lOP Publishing Ltd.

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structure hidden from traditional Fourier techniques (Addison ef ai, 2000; Watson ef ai, 2000). Figure 6.17 illustrates the shortcomings of traditional STFT analysis in detecting signal features of short duration. The figure contains a scalogram and a spectogram corresponding to the rhythmic ECG signal shown in figure 6. I7(a). The spectogram is generated from an STFT which has a 3.4 s Hanning window-typical for this type of analysis. The smearing and hence loss of local information across the spectogram over these timescales is evident in the plot. In a pilot study concerning the analysis of pressure traces and ECG corresponding to pig hearts exhibiting VF (Addison ef ai, 2002b) some evidence has been found to suggest that wavelet phase information may be used to interrogate the ECG for underlying low level mechanical activity in atria. Figure 6.18 shows the pressure in Copyright © 2002 lOP Publishing Ltd.

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the aorta and ECG corresponding to an episode of ventricular fibrillation in a pig heart. The ECG signal has a typical random or unstructured appearance. The aorta pressure trace, however, reveals regular low amplitude spikes. On opening the chest of this animal and observing the heart directly, it became apparent that the ventricles were fibrillating, but the atria were contracting independently in a coordinated

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manner. The irregular activity of the much larger ventricular muscle mass completely obscured this atrial activity in the standard ECG recording shown in the middle of figure 6.18. The wavelet energy scalogram for this signal is plotted below the ECG signal. (A Morlet wavelet was used in the study.) The high amplitude band at around 8-10 Hz is much more compact in extent in frequency Ihan that found for other traces where no atrial pulsing was apparent. Furthermore, there is some evidence of 'pulsing' in this band belween I and 2 Hz in the scalogram. This is confirmed in figure 6.19 where the location of zero wavelet phase is plotted over a

sharI range of the bandpass frequencies, between 1.1 and 1.5 Hz. Below this zerophase plot is the pressure tracing. The phase plot exhibits a strikingly regular pattern Copyright © 2002 lOP Publishing Ltd.

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with the zero-phase lines aligning themselves remarkably well with the atrial pulsing of the pressure trace. Atrial fibrillation (AF) is an arrhythmia associated with the asynchronous contraction of the atrial muscle fibres. It is the most prevalent cardiac arrhythmia

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Copyright © 2002 lOP Publishing Ltd.

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in the western world, and is associated with significant morbidity. Figures 6.20 and 6.21 illustrate a technique for the elucidation of AF from within an ECG signal using a modulus maxima denoising technique (Watson ef ai, 2001). Figure 6.20 shows the wavelet transfoml decomposition of a 2 s segment of ECG from a patient with atrial fibrillation. Below the trace is a scalogram plot, obtained using a Mexican hat-based wavelet transform. This yields high temporal resolution in the wavelet domain, but generates a very large data set. The modulus maxima of the scalogram are plotted below the scalogram. As can be seen from the figure, dominant modulus maxima lines at the scale of 10Hz and below are almost solely associated with the coherent QRS and T structures. Therefore the modulus maxima lines at this scale with a high proportion of the total energy within this scale are selected. The selected modulus maxima lines arc then followed across scales and subtracted to leave a residual signal associated with both system noise and, morc importantly, atrial activity. An inverse transform, performed separately on both sets of retained maxima lines, recovers the Copyright © 2002 lOP Publishing Ltd.

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partitioned signals. This time-frequency partitioning of the signal results in two components: one (1) containing combined low and high frequency components that correspond to large scale features in the signal, and a second (2) containing the remaining high frequency components that correspond to small scale AF features and noise. Most applications are concerned with signal denoising and hence the retention of component (1). This application, however, is concerned with the removal of large amplitude features to allow examination of the lower amplitude AF components of the signal, and hence component (2) is retained for analysis. Figure 6.21 contains a 7 s segment of EeG taken during a pilot study of patients with AF. The signal has been partitioned using the modulus maxima technique described above where the modulus maxima have been separated into large and small scale features. An enlarged part of the signal is given in the lower three plots in the figure. The middle plot contains the partitioned signal with the QRS complex and T wave filtered Copyright © 2002 lOP Publishing Ltd.

out, revealing regular, coherent features that appear at a frequency of approximately 400 beats per minute, often seen during invasive studies of atrial activity in patients with AF. The lower plot contains the partition with the filtered-out QRS and T waves. Although a relatively simple modulus maxima technique was used, whereby the modulus maximum lines were simply partitioned into two subsets, the ability of the technique to separate the signal into QRS and Twaves and underlying AF is evident from the preliminary results. 6.2.5

ECG data compression

ECG signals are collected both over long periods of time and at high resolution. This creates substantial volumes of data. Data compression seeks to reduce the number of bits of information required to transmit or store digitized ECG signals without significant loss of signal information. An early paper by Crowe et al (1992) suggests the wavelet transform as a method for compressing both ECG and heart rate variability data sets. Using discrete orthonormal wavelet transforms and Daubechies DIO wavelets, Chen et al (1993) compressed ECG data sets resulting in compression ratios up to 22.9: 1 while retaining clinically acceptable signal quality. Thakor et al (1993a) compared two methods of data reduction on a dyadic scale for normal and abnormal cardiac rhythms, detailing the errors associated with increasing data reduction ratios. Karczewicz and Gabbouj (1997) have proposed a novel compression scheme for ECG data based on B-spline basis functions. Popescu et al (1999) have developed a multiresolution distributed filtering data reduction method for high resolution ECG signals (RRECGs) used in the assessment of ventricular tachycardia risk in post-myocardial infarction patients. The method detects small amplitude late potentials, which are established arrhythmogenic markers in this group of patients. The authors found the method superior to hard and soft wavelet thresholding techniques as well as other established non-wavelet methods. A comparison of the performance of the many ECG compression methods-wavelets and other-can be found in the paper by Cardenas-Barrera and Lorenzo-Ginori (1999). More recent data compression schemes for the ECG include the method using non-orthogonal wavelet transforms by Ahmed et al (2000) and the set partitioning in hierarchical trees (SPIRT) algorithm employed by Lu et al (2000).

6.3

Neuroelectric waveforms

The electroencephalogram (EEG) signal is obtained from a set of electrodes which are usually placed on the scalp. In some cases, however, specially designed subdural electrodes are surgically implanted below the skull to monitor electrical activity obscured by the skull. The electrical potentials picked up by the electrodes-the EEG signalreflects brain electrical activity owing to both intrinsic dynamics and responses to external stimuli. Experimental external stimuli can take the form of evoked potentials (EPs), which are in general well defined sensory inputs such as sounds, flashes, smells or touches, or event-related potentials (ERPs) where experiments are set up to probe higher cognitive functions such as those associated with memory function or Copyright © 2002 lOP Publishing Ltd.

mechanical response. A good place to begin this section is with the comprehensive introduction to the wavelet analysis of neuroelectric waveforms given by Samar

et al (1999). They provide a concept-driven (minimal mathematics) account of the use of wavelet techniques in the analysis of EEG and event-related potential waveforms. Although the rest of this section deals solely with EEG data and associated EPs and ERPs, it should be noted that some work has been carried out on the use of wavelet-based methods in the analysis of magnetoencephalographs (MEGs) which measure small magnetic fields induced by the electrical activity of the brain. See for example the papers by Lukka et al (2000) who used wavelet packets within a stimulus classification algorithm, Nikouline (2000) who studied somatosensory evoked responses using the Morlet wavelet, or Kneif et al (2000) who investigated the perception of coherent and non-coherent auditory objects using Morlet wavelets. 6.3.1

Evoked potentials and event-related potentials

Ademoglu et al (1997) investigated the transient response ofEEG signals to a set of brief visual stimuli. The authors studied a specific class of EP due to visual stimulipattern-reversal visual evoked potentials (PRYEPs)-in an attempt to aid the clinical diagnosis of dementia. Figure 6.22(a) contains plots of 24 normal PRYEPs and 16 from patients with dementia. Using a quadratic B-spline wavelet (figure 6.22(b», the researchers decomposed each PRYEP signal containing 512 data points into six coefficient scales (/11 = 1-6) plus a residual component. That is, the multiresolution analysis was halted at scale index /11 = 6. Thus the transform vector contains eight approximation coefficients which contain the information from scale indices 7-9.

The authors found that the residual scale coefficients had consistent sign changes for the normal cases (top of figure 6.23). Reconstructing the waveforms using only these residual scale coefficients produces identifiable, overlapping waveforms for the normal cases (middle of figure 6.17). The lower plot of figure 6.23 shows the synthesized signals, reconstructed from the residual scale, for the PRYEPs obtained from patients with dementia where no regular pattern is evident. Huang et al (1999a) have detailed an investigation of auditory evoked potentials (AEPs) as a measure of the depth of anaesthesia in dogs. These AEPs consist of a series of waves that represent processes of transduction, transmission and processing

of auditory information from the cochlea to the brain stem. They developed an automated monitoring system to control the delivery of intravenous anaesthetic based on a neural network classification of significant wavelet coefficients determined from the decomposed signal using Oaubechies 020 wavelets. A number of other research workers have examined EPs in the EEG. For example, Bartnik et al (1992) proposed a method for extracting single EPs from background EEG using a multi resolution framework based on cubic spline wavelets. Thakor et al (I 993b) used wavelet transforms to investigate EPs from anaesthetized cats, Betrand et al (1994) examined middle latency auditory EPs using discrete wavelets and Lim et al (1995) have used wavelets to examine respiratory-related EPs in human subjects. Tzelepi et al (2000) have described the findings of a study concerning scalp-recorded visual evoked potentials (YEPs) in humans where activity in the alpha, beta and gamma range was quantified over the

first 200 ms of the YEP using a discrete CoiRet wavelet-based decomposition of the Copyright © 2002 lOP Publishing Ltd.

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signal. Siobounov et at (2000) have also probed oscillatory brain activity in response to visual stimuli. They employed a continuous Morlet wavelet transform decomposition of the signal and concentrated on the Gamma band activity (30-50 Hz) of the EEG during visual recognition of non-stable postures of a computer-generated 'virtual person'. Raz ef at (1999) have developed a wavelet packet model of EPs and Dear and Hart (1999a,b) have reported evidence that stimulus-related synchronized neuronal discharges in bats are structured to closely resemble some members of the Symmlet wavelet packet family. Basar et at (1999) used wavelet techniques to investigate the functional significance of evoked resonance phenomena in the brain, confirming (and enhancing) previous findings using Fourier methods. Basar ef at (2001) have presented a comprehensive report on new strategies for the wavelet analysis of event-related Copyright © 2002 lOP Publishing Ltd.

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44(9) 881-890

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oscillations. Using B-spline wavelels, Demiralp er al (1999) studied the wavelet transform decomposition of event-related potentials elicited by human subjects in response to auditory stimuli to assess differences in cognitive information processing. The techniques developed in this work were later applied to the analysis of oddball P300 data in a later study (Demiralp, 2001). Devrim ef al (\999) have analysed nearthreshold and suprathreshold visual ERPs in order to determine the generation mechanism of the P300 wave. Quiroga (2000) has presented a wavelet-based denoising method, using quadratic biorthogonal B-spline wavelets, for the elucidation of single stimulus EPs. Both auditory and visual evoked potentials were investigated in the study. Zygierewicz ef al (1998) used matching pursuits to extract and quantify EEG frequency following responses produced in the human primary somatosensory cortex by 33 Hz vibrotactile stimulation of the right index fingertip in a single subject. [n a later study, the same group (Zygierewicz ef ai, 1999) employed matching pursuit time-frequency decomposition in a parametric study of EEG structures during sleep. See also the paper by Durka and Blinowska (200 I) which gives a general overview of the matching pursuit method applied to the EEG. Effern ef al (2000) have applied their new wavelet-based method for denoising transient dynamical signals to both modelled test data and experimentally collected event-related potentials. Their method was constructed for short and transient time sequences using circular state space embedding. Sutoh er al (2000) have used Morlet wavelets in a study of event-related desynchronization during an auditory oddball task. Copyright © 2002 lOP Publishing Ltd.

6.3.2

Epileptic seizures and epileptogenic foci

Various wavelet transforms have been compared by Schiff et al (I 994a) to characterize epileptic seizures exhibited in the EEG. Figure 6.24 shows an EEG signal (figure 6.24(a)) with six associated transform plots generated using a variety of wavelet transform methods, from a continuous Mexican hat transform calculated at each time step (figure 6.24(b» to a discrete B-spline wavelet critically sampled using cubic B-spline wavelets and multiresolution framework (figure 6.24(g». Schiff and his colleagues found that the use of spline techniques to speed up computation did not impair feature extraction from the signal. Tn addition, they pointed out that, since they do not require a fixed length data window, wavelet transforms provide an 8 rr--.-r-.-. - n n

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improved method for spike detection in the data compared with windowed Fourier analysis. Blanco ef al (1996) analysed the EEG traces from two patients exhibiting epileptic seizures. They compared the wavelet analysis of the signals with the more traditional Gabor transformation (i.e. a short term Fourier transform with Gaussian window). They found that although the Gabor transform provided a good global average description of the signal, the (cubic spline) wavelet transform provided a more accurate temporal localization as well as a good detection of short events. They proposed the wavelet transform as a potentially useful tool for quantifying and visualizing the time evolution of the frequency content of the EEG. In later work, Blanco ef al (1998) further analysed epileptic seizures using EEG data. The EEG data taken during tonic-clonic epileptic seizures was inspected using wavelet packets to filter the signal noise present due to skeletal muscle activity. They showed that it is possible to obtain useful dynamical parameters and hidden frequency information from these signals, which are usually neglected by physicians due to the excessive noise present within them. Sirne ef al (1999) have proposed a data-reduction algorithm for long term EEGs which selects segments of the EEG in which a transient is detected. Only these segments are presented to the clinician for review. Akay and Daubenspeck (1999) have attempted to separate contaminating facial muscle electromyographic (EMG) activity from EEG signals using matching pursuit analysis. Sun ef al (2000) performed soft thresholding ofbiorthogonal wavelet transform coefficients in a method to preprocess EEG data containing epileptic seizure events. Using their technique they removed sharp spikes and low amplitude slow waves prior to displaying the time-freq uency properties of the signal as a pseudo-Wigner distribution. Petrosian ef al (2000) have applied recurrent neural networks combined with signal wavelet decomposition in an attempt to predict the onset of epileptic seizures in intra- and cxtracranial EEGs and Gamero ef al (1997) have used the EEG displaying an epileptic seizure to illustrate their discussion on multiresolution-based information measures for dynamical signals. A study of temporal lobe epilepsy was carried out by Casdagli ef al (1996) to determine whether wavelet analysis had anything to offer the detection of the origines) of the epileptic seizure within the brain (the epileptogenic foci). They used wavelet transforms to analyse nonlinearitites within EEG data taken from electrodes surgically implanted at various locations in the brain. In addition to wavelet analysis incorporating the Daubechies 012 wavelet, they employed a variety of other nonlinear time series analysis techniques. They emphasized the compatibility of the wavelet transform as a method of characterization for the spiking activity in the electrode signals due to its localization in both frequency and time. Figure 6.25 shows four of the electrode signals (RST4, LTD3, LSTI, ROF2) together with their corresponding wavelet coefficients at the second finest resolutions. Figure 6.26 shows the temporal evolution of the kurtosis of the wavelet coefficients at the second scale kw(2). This is done over an 85 min period computed for two of the electrode signals, LTD3 and ROF2, and shown in figures 6.26(a) and (b) and figures 6.26(c) and (d) respectively. Significant spiking activity is detected at the seizure location (61 min) by the ROF2 electrode. The lower plots ((e) and (f» show a map of spiking activity where values of kw(2) are plotted for all 28 electrodes on a grey scale from 2 to 10. It can be seen from the plots that the signals exhibiting Copyright © 2002 lOP Publishing Ltd.

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continuous spiking activity are LST3, LTD I ,3,5 and RST I,2,3,4. The authors state that these are a characteristic of the nonlinearities within the signal. Another study

seeking to locate the epileptogenic foci by SchilT ef al (1994b) employed Mexican hat wavelet transforms to analyse EEG signals exhibiting epileptic seizures. They performed both a one-dimensional analysis of single EEG signals and a twodimensional wavelet analysis on a coarse two-dimensional array of EEG signals

obtained from a 5 x 5 subdural array of electrodes. They emphasized that the wavelet transform overcomes the requirement of using the fixed window lengths required for

windowed Fourier transforms (the length of which is subject to observer bias). In order to clarify which electrodes were associated with the seizure onset, the wavelet

coefficients were filtered using a hard thresholding technique. Mizuno-Matsumoto ef al (1999) have used Haar wavelet filtering of electrocorticographic signals prior to a cross-correlation analysis to determine both the foci and nature of propagation of epileptiform discharges. They found evidence for two types of focus in their work. Franaszczuk ef al (1998) performed time-frequency analysis of intercranial EEG recordings during seizure using the matching pursuit method. They found the matching pursuit method to be a valuable tool in analysing dynamic seizure activity. Copyright © 2002 lOP Publishing Ltd.

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Figure 6.26. Dependence of the kurtosis of wavelet coefficients kw(2) on time. (a) Solid line: electrode LTD3. Dotted line: surrogate data. (b) Blowup orCa) in the vicinity of the seizure. (c) Same as (a) for electrode ROF1. Cd) Same as (b) for electrode ROF2. (e) All 28 electrodes. Magnitudes of kll'(2) corresponding to the grey scale are illustrated by the bar to the left of (he figure. (f) Blowup or(e) in the vicinity of the seizure. Reprinted from Casdagli et aJ 1996 Physico D 99 381-399. Copyright 1996, with permission from Elsevier Science.

Figure 6.27 shows selected segments of the seizure obtained from one of the intercranial contacts used in the study. Figure 6.28(a) shows the time-frequency energy distribution of an entire seizure acquired by the deepest depth electrode contact (HI) located near the region of seizure onset. Figures 6.28(b) and 6.28(c) show the time-frequency distribution in three-dimensional format with respectively 296 and 6000 of the matching waveforms shown. 6.3.3

Classification of the EEG using artificial neural networks

A study by Kalayci and Ozdamar (1995) used the wavelet transform as a preprocessing tool for neural network analysis of EEG signals, i.e. the neural network was trained on the wavelet coefficients rather than the original signal. They found that the size of the input data to the artificial neural network could be drastically reduced without significantly compromising its performance. They employed both the Oaubechies 04 and 020 wavelets and found that the proper selection of scale (or combination of scales) to use as the input was critically important. A schematic diagram of the methodology is shown in figure 6.29. The authors found over 90% accuracy in the detection of EEG spikes using only eight inputs from scale 3 alone. Copyright © 2002 lOP Publishing Ltd.

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Figure 6.27. Matching pursuit algorithm applied to seizures originating from the mesial temporal lobe. Selected segments of a complex partial seizure origimning from the mesial tempoml structures and recorded with combined depth electrodes and a 32 conlact subdural grid array. Two six-depth contact deep electrode arrays pass through the grid and with the deepest conlacts AI and HI, H2 residing in the amygdala and hippocampus, and the most superficial contacts (e.g. A6, H6) al the lateral temporal neocortex. The seizure begins nearest the deepest hippocampus contacts HI, H2 and subsequently spreads to regionally involve the temporal lobe. To facilitate illustration, only three depth electrode contacts (those nearest the seizure onset) and three representative subdural grid contacts (two (TI7, T23) from the anterior region, one (T32) from the posterior region of the 32 contact grid) are shown. Panel (a) shows 20 s of the intercranial EEG that includes the later portions of the period of seizure initiation (INI; the low·voltage fast activity was preceded by some periodic spiking as shown) and the period of transitional rhythmic activity (TRA, 145). (b), continuous with (a), shows 20 s of intercranial EEG showing the initial 20 s of organized rhythmic

Copyright © 2002 lOP Publishing Ltd.

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Figure 6.28. Matching pursuit algorithm applied to seizures originating from the mesial temporal lobe. (a) Time-frequency energy distribution of the entire seizure (shown in the previous figure) recorded by conlact HI, the deepest electrode contact, located near the region of seizure onset. The intercranial EEG recording is shown below the time-frequency plot. Only the first 296 matching waveforms, representing 90% of the total energy, are shown. The periods of seizure initiation (INI), transitional rhythmic activity (TRA), organized rhythmic activity (ORA) and intermittent bursting (IBA) are marked. (b) Time-frequency energy distribution of the entire seizure recorder by contact HI. This contains the same information as (a) now plotted in three dimensions. (c) Another three-dimensional plot of the seizure. Here 6000 matching waveforms are shown representing 99.996% of the total energy of the signal. The vertical axis represents the square root of energy rising out of the time-frequency plane. Reprinted from Franaszczuk er 0/1998 Electroellcephalography alld Clillical Neurophysiology 106 513-521. Copyright 1998, with permission from Elsevier Science.

This was comparable to using 20 inputs from the original EEG signal, representing over 50% reduction in the input size. Neural networks trained on wavelet coefficients were also employed with some degree of success by Hazarika ef al (1997) to classify EEG signals from normal patients and those with a diagnosis of schizophrenia. Again, the wavelet transform was used as a preprocessing tool for the data sets before the neural networks learned the data. Although good at differentiating between the EEGs from normal patients and those with schizophrenia, the authors found that the technique was quite poor at identifying signals from patients with obsessive compulsive disorders. Heinrich ef al (1999) have developed a wavelet network to analyse single ERP responses in a study concerning attention deficit hyperactivity disorder (ADHD) in children. They studied auditory evoked potentials in a group of 25 ADHD boys, comparing them with a control group of the same number. They found group-specific differences using their wavelet network which could not be detected using traditional activity (ORA). Panel (c) shows 20s of intercranial EEG late in the seizure beginning 40s after (b), and illustrates the pattern of intermittent bursting activity (IRA) frequently seen at the conclusion of a seizure. Reprinted from Franaszczuk er 0/1998 Electroellceplwlography and Clinical Neurophysiology 106 513-521. Copyright 1998, with permission from Elsevier Science. Copyright © 2002 lOP Publishing Ltd.

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Ozdamar 1995 IEEE Medicine in Engineering and Biology 14(2) 160-166.

© IEEE 1995.

extracted latency and amplitude parameters. They concluded that the wavelet network single sweep analysis is a sensitive tool for clinical ERP studies which should be applied in addition to the investigation of averaged responses.

6.4

Pathological sounds, ultrasounds and vibrations

This section deals with the wavelet analysis of sound signals of clinical relevance. both emitted sounds (e.g. turbulent blood sounds. lung sounds and otoacoustic emissions) and reflected sounds (Doppler ultrasound time series). Copyrighl © 2002 lOP Publishing Ltd.

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Figure 6.30. Recordings of flow sounds for the no occlusion case before the injection of the drug. (a) Top panel: flow. Second panel from top: sounds. (b) Wavelet power. Note that in (b), the number 2 was dropped from the detail function D2j.f. After Akay et al 1994 IEEE Transactiolls 011 Biomedical Engineering 41(10) 921-928. © IEEE 1994.

6.4.1

Blood flow sounds

The turbulent sounds generated by femoral artery stenosis in dogs have been analysed by Akay el al (1994) using both wavelet transforms and short term Fourier transforms. They compared signals from the unblocked case (0% occlusion) with 72% and 85% occlusion. Figure 6,30(a) shows the recording of blood flow rate (top plot) and the flow sounds (second top plot) taken before the arterial blockage. Figure 6.30(b) shows the wavelet power plotted at the four smallest scales, where the wavelet power plot is simply the square of the detail (wavelet) coefficients plotted against time. The smallest scales correspond to wavelet scale indices of 111 = 1,2,3 and 4, which in turn correspond to frequency bands of 1000-500, 500-250, 250-125 and 125-62.5 Hz, all well above the dominant signal frequency of the flow rate of around 2 Hz. It is at these high frequencies that the fluid turbulence in the blood generated by the blockages can be detected. Figure 6.31 shows corresponding plots for the 85% occlusion case. By comparing this figure with the previous one (0% occlusion) the increase in power associated with the higher frequencies can be seen in the wavelet power plots, The authors also went on to investigate the effect of the introduction of a vasodilator drug in the performance of this acoustic approach to the diagnosis of arterial blockage. They found that the power corresponding to the first two wavelet frequency bandwidths (100-250 Hz) associated with arterial stenoses increased significantly after the injection of the vasodilator drug papaverine. They concluded that the diagnostic performance of diastolic heart sounds associated with occluded arteries can be improved using vasodilator drugs. In another study, Aydin el al (1999) analysed a large number of embolic signals, obtained using a transcranial Doppler ultrasound system, from patients with sympto· matic carotid stenosis. They employed the Morlet wavelet to detect short duration Copyright © 2002 lOP Publishing Ltd.

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Figure 6.31. Recordings of flow sounds for the 85% occlusion case before the injection of the drug. (a) Top panel: flow. Second panel from top: sounds. (b) Wavelet power. Note thaI in (b), the number 2 was dropped from the detail function D 2j f. Arter Akay el af 1994/£££ Transactiolls 011 Biomedical Engineering 41(10) 921-928. © IEEE 1994.

transients in the signal associated with emboli passing through the sample volume. They found that the method improved both the temporal resolution and temporal localization of the detected events when compared with short term Fourier techniques. Devuyst el a/ (2000) have used matching pursuits to detect circulating microemboli within cerebral arteries using transcranial Doppler ultrasound. Their method allowed them to discriminate between solid and gaseous brain microcmboli and artefacts. See also the paper by Krongold el a/ (1999) concerning wavelet-

based detection of microemboli in flowing blood, and the ultrasonic Doppler signal work by Matani el a/ (1996) described briefly later in section 6.5 concerning blood flow and blood pressure. 6.4.2

Hearl sounds and heart rates

Chan el a/ (1997) have used the wavelet transform to detect venous air embolism (YAE) during surgery. Due to its life-threatening nature, the fast detection of YAE is essential to ensure prompt clinical treatment. They used a quadratic spline wavelet

(figure 6.32) to analyse Doppler heart sounds (DHS) from dogs. A trace of the heart sound signal is plotted at the top of figure 6.33. The arrow indicates the point at which 0.02 ml of air was injected. Wavelet transforms of the time series are plotted below the original signal for scale indices 111 = I, 2 and 3. (Note that the authors use j for scale index.) A distinct increase in the wavelet coefficients at the first scale is obvious from the plot. A quantitative assessment was made by plotting the normalized power of single heartbeats at each scale. Figure 6.34 shows the normalized power plotted against heartbeat for 22 beats. The largest peak corresponds to the first scale and shows its usefulness as an indicator for YAE. The authors also sought a relationship between the sum of the normalized power of the heartbeats above a Copyright © 2002 lOP Publishing Ltd.

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set threshold, which they called the cumulative embolic power (CEP), and the volume of air injected. They stated that such a relationship could prove important to the anaesthetist, allowing him or her to act only when a clinically significant volume of air embolism is present. The relationship between CEP and air injected is shown in figure 6.35. The left-hand plots correspond to scale index I and the right-hand plots to scale index 2. The authors concluded that the wavelet transform of the heart sound signal can provide both fast detection of VEA and an accurate estimation of the embolic air present. The short paper by Bentley el af (\998) describes the use of both continuous and discrete wavelet transforms, among a range of techniques, to classify the sounds from

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Figure 6.33. A typical DHS signal containing seven heartbeats and its WT at different scales (j = 1~3). With 0.02 ml or air injected, the embolic heartbeat (marked by the arrow) \Vas confirmed by an experienced anaesthetist by listening to the DHS signal. Arter Chan et a/1997 IEEE Transactiolls 011 Biomedical Engineering 44(4) 237-246. © IEEE 1997. Copyright © 2002 lOP Publishing Ltd.

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both native and replacement prosthetic heart valves. They employed the Daubechies D20 wavelet to extract salient features from the recorded heart sounds for the determination of valve condition. Zhang ef al (1998a) have employed the matching pursuit method of Mallat and Zhang to both analyse and synthesize phonocardiograms (PCGs). They showed that the matching pursuit method acts as a powerful filter for the removal of Gaussian noise. They use the method to develop a time-frequency scaling technique to enhance these audio signals (Zhang ef ai, 1998b) in order to enhance the diagnosis of heart and heart valve disease. Sava ef al (1998) have also used matching pursuits to determine the important coherent transient components within phonocardiographic signals. They applied the technique to PCG recordings from patients with aortic bioprosthetic valves in order to detect specific events found within the cardiac cycle. They found their method outperforms other methods and is particularly good for the analysis ofPCG signals from patients with malfunctioning bioprostheses. Masson and Rieu (1998) have analysed signals from artificial heart valves using Auscher's wavelet in a study which aimed at identifying valve noise associated with patient discomfort. Simultaneous signals were collected using both a hydrophone and accelerometer for four different heart valve protheses-three mechanical and one biological valve-in an ill vitro study using a cardiovascular simulator. They found that, unlike the bioprosthesis, the mechanical valves produced high sound levels within octave bandwidths centred on 64 Hz up to 512 Hz. Sandham ef al (1998) have analysed seismocardiograms (SCGs) from a male patient, pertaining to three different physiological conditions: rest, isometric exercise and hyperventilation. They found statistically significant power changes at the lower wavelet scales which could be used to determine the physiological condition. In contrast, no discernible differences were observed in the signal in the time domain. In order to probe the nonlinear dynamics of foetal heart rate signals, Papadimitriou and Bezerianos (1999) used a modulus maxima technique to denoise Dopplcr ultrasound signals. They found a significant improvemcnt in the 'chaoticity' detectable in the signal after denoising with their method. Kimura ef al (1998) have also employed wavelet transforms to study the nonlinear dynamics of the foetal heart rate derived from Doppler ultrasound signals reflected from the moving Copyright © 2002 lOP Publishing Ltd.

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Figure 6.35. Relationship between the CEP and the volume of injected air at two different scale index (j) values. (a) Linear plot of the mean normalized CEP (j = I) versus the volume of injected air (r = 0.83). (b) Linear plot of the mean normalized CEP (j = 2) versus the volume of air (,. = 0.81). Each point shows the mean and standard deviation (error bar) of 12 incidences of air injections obtained in four dogs. Significant increase: ~ P < 0.05, # P < 0.01 and * P < 0.0005

compared with control. (c) Linear regression in log-log scale of (a) (y = 0.661' + 4.06, ,.2 = 0.98). (d) Linear regression in log-log scale of (b) (y = 0.94,· + 4.35, ,) = 0.96). (e) Log-log plot of the normalized CEP (j = 1) versus the volume of injected air for dog 3 (y = 0.51r + 3.78, ,.2 = 0.98). (f) Log-log plot of the normalized CEP (j = 2) versus the volume of injected air for dog 3 (y = 0.811' + 3.84, 1'2 = 0.95). Natural logarithms are used for all the log-log plots. Data of (e) and (f) represent averages of three air injections in dog 3. After Chan el a/1997 IEEE Transactions 011 Biomedical Engineering 44(4) 237-246. © IEEE 1997.

foetal heart valves. They propose their method as a novel quantitative index of foetal monitoring to diagnose foetal acidemia. 6.4.3

Lung sounds

Hadjileontiadis and Panas (1997) used wavelet analysis to separate discontinuous adventitious sounds (DASs) from vesicular sounds (VSs) in pulmonary acoustic signals. Adventitious sounds are divided into two categories: continuous (wheezes and rhonchi) and discontinuous (crackles and squawks), and indicate an underlying physiological malfunction. The algorithm of Hadjileontiadis and Panas combines multiresolution analysis with hard thresholding to provide a technique to partition Copyright © 2002 lOP Publishing Ltd.

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Figure 6.36. Wavelet fillering of lung sound time series exhibiting fine crackles. (a) A time section of 0.8192 s of fine crackles recorded from a patient with pulmonary fibrosis (case C I) considered as an input. (b) The nonstationary output of the wavelet filter, DAS. (c) The stationary output of the wavelet filter, YS. After Hadjileontiadis and Panas 1997 IEEE 7i"clI1sactiol/s 011 Biomedical Ellgineering 44(12) 1269-1281. © IEEE 1997.

OASs from VSs. Figure 6.36 shows a segment of pulmonary acoustic signal from a patient with pulmonary fibrosis containing fine crackles. The signal was decomposed using Oaubechies 08 wavelets, and then separately reconstructed OAS and VS signals are shown below the original signal in figures 6.36(b) and (c) respectively. Figure 6.37 shows a similar decomposition of a signal with squawks recorded from a palient with interstitial fibrosis. Squawks arc short inspiratory wheezes, of longer duration than crackles, which are heard in association with crackles. Both figures illustrate the separation achieved between OASs and VSs in the signal. See also the paper by Sankur el al (1996) who employed a wavelet-based detector to discriminate crackles in pathological respiratory sounds. They employed a Oaubeehies 06 wavelet, which has a similar shape to the crackle waveform, and found their method to be superior to two existing crackle detection methods. In another study, Charleston el al (1997) employed discrete wavelet transforms in a method to remove heart sounds from acquired respiratory signals. Heart sounds often represent severe disturbing interference and their removal from respiratory signals before clinical analysis is desirable. The interference cancellation scheme presented by the authors provides estimates of the location within the signal of Ihe interfering heart signals and then separates these heart sounds from the original signal. 6.4.4

Acoustic response

Otoacoustic emissions (OAEs) are acoustic signals emitted by the cochlea, either occurring spontaneously or in response to an acoustic stimulus, and reflect the Copyright © 2002 lOP Publishing Ltd.

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Figure 6.37. \Va\'elet filtering of lung sound time series exhibiting squawks. (a) A time section of 0.4096s of squawks recorded from a patient with interstitial fibrosis (case C12) considered as an input. (b) The nonstationary output of the wavelet filter, DAS. (c) The stationary output of the wavelet filter, VS. After Hadjileontiadis and Panas 1997 IEEE Trallsactions 011 Biomedical Ellgilleering 44(12) 1269-1281.

© IEEE

1997.

active processes that are involved in the transduction of mechanical energy into

eleclrical energy. Their form is related to the status of the cochlea and can be used to monitor cochlear functionality in patients exposed to prolonged noise and/or olotoxic agents. Tognola el af (1998) have studied the acoustic response of the cochlea AOJORI - 80 dB SPL

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Figure 6.38. Click emked otoacoustic emissions. CEOAEs from (a) a normal hearing adult and (b) a full-term baby. To reduce the influence of the stimulus artefact, responses have been windowed 2.5{20 ms post-stimulus time. In each row, two replicate recordings from the same ear (A and B replicate recordings in ILO equipment) are superimposed. Numbers on the left of each panel are the reproducibility values (in percentage points) between the two replicates. After Tognola et al 1998 IEEE Trallsactions 011 Biomedical Ellgineering 45(6) 686--697. © IEEE 1998. Copyright © 2002 lOP Publishing Ltd.

to acoustic stimuli of brief duration-specifically clicks of about 100 IlS duration. The time-frequency properties of these click-evokcd otoacoustic emissions (CEOAEs) have a close relationship with cochlear mechanisms. The authors compared various time-frequency analysis methods-the short time Fourier transform, wavelet trans-

forms, the Wigner distribution-and two smoothed Wigner-Ville distributions: the pseudo-smoothed Wigner distribution and the Choi-Williams distribution. They found that, although there was no optimal method in an absolute sense, the wavelet transform method offered the best compromise between time-frequency resolution and the attenuation of interference terms. Two examples of CEOAEs are shown in figure 6.38: one for an adult and one for a full term neonate. The OAE response of the neonate exhibits a typical sustained burstlike behaviour up to 20 ms, whereas the adult OAE shows clear frequency dispersion (i.e. reduction in high frequency components in time). The wavelet scalograms corresponding to the CEOAEs in figure 6.38 are shown in figure 6.39. The frequency dispersion of the adult signal is evident in the scalogram plot where low frequency components have a longer subject A030RI - 80 dB SPL

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Figure 6.39. Time--frequency energy densities of CEOAEs. (a) Time-frequency distribution (energy density, normalized arbitrary units) oCa CEOAE at 80-dB SPL of subject AOJORI (normal hearing adult). (b) Time-frequency distribution (energy density, normalized arbitrary units) of a CEOAE at 80-dB SPL of subject N360LO (full-term neonate). Arter Tognola ef (II 1998 1£££ Transactiolls 0/1 Biomedical Engineering 45(6) 686-697. © IEEE 1998.

Copyright © 2002 lOP Publishing Ltd.

subject P300P4 - 83 dB SPL

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5 2.5 Figure 6.40. Time-frequency distribution (energy densit)', normalized arbitrary units) of a CEOAE at 83dB SPL of subject P300P4 (suffering from noise-induced hearing loss). After Tognola er a/ 1998 IEEE Transactiolls 011 Biomedical c"'ngiJleerillg 45(6) 686-697. © IEEE 1998.

duration in the scalogram plot and reach maximal amplitude at longer latencies than the high frequency components. Figure 6.40 shows a scalogram for an adult suffering from noise-induced hearing loss. This hearing-impaired patient had hearing loss greater than 30 dB above a frequency of 2.5 kHz. The lack of OAE response at frequencies above this 2.5 kHz threshold is evident in the scalogram plot associated with this patient. Zheng el at (1999a) employed Morlet wavelets in a comparative study of modelled and clinically collected OAEs. Wavelet decomposition was used as it was considered suitable for detecting the mixed frequency components present in both data sets. Their results indicated that the modelled OAEs were similar to the clinically detected ones, not only in the time domain waveform but also in the frequencylatency relationship. Molenaar el at (2000) have also employed wavelet-based methods in a novel study of OAEs in which both click stimuli and wide band noise bursts were presented simultaneously. Heneghan el at (1994) have used the Morlet wavelet to analyse the motion of the hair cells in the inner ear in response to acoustic signals applied at the ear canal. Yang el al (1992) and Wang and Shamma (1995) have produced a model for the processing of acoustic signals within the auditory system in which the spatio-temporal pattern of displacements along the basilar membrane of the cochlea due to the acoustic signal entering the ear may be considered an affine wavelet transform. Complex wavelets have been employed by Wang and Shamma (1994) to analyse a one-dimensional acoustic signal within a model of the auditory functions in the primary cortex.

6,5

Blood flow and blood pressure

An attempt to shed light on the cardiovasular control mechanisms using waveletbased spectral methods has been made by Bracic and Stefanovska (1998). They Copyright © 2002 lOP Publishing Ltd.

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(top pIOI), local maxima (middle plot) and time-averaged values (lower plot). Arter Bracic and Stefanovska (1998). Reproduced with the kind permission of the Society of Mathematical Biology.

measured the peripheral blood flow in human skin over 20 min periods using laser Doppler flowmetry (LDF). Subsequent decomposition of these signals using the Morlet-based wavelet transform to form energy density scalograms (figure 6.41) revealed five characteristic frequency peaks. These local maxima in the waveletbased energy spectrum, they hypothesized, can be attributed to the heart rate (I Hz), respiratory activity (0.3 Hz), blood-pressure regulation (0.1 Hz), neurogenic (0.04 Hz) and metabolic activity (0.01 Hz). They proposed a variety of statistical Copyright © 2002 lOP Publishing Ltd.

measures to characterize their wavelet-based power spectra and used them to reveal differences in the dynamics of the blood flow between two distinct groups: a control group of healthy young subjects and a group of athletes. They showed that the increased blood flow in the trained subjects resulted from both the greater stroke volume and increased compliance of the peripheral vessels. In a related article Kvernmo et al (1998) compared the wavelet-based spectral analysis of these signals before and after exercise. In another related study, Kvernmo et al (1999) used their methods to determine the effect of vasodilators (endothelium dependent and endothelium independent) on the oscillatory components present in these human cutaneous blood perfusion signals. Humeau et al (2000) have presented a model of LDF signals produced when an arterial occlusion is removed. In such cases the LDF signal increases then returns to its initial value; this phenomenon is known as reactive hyperaemia and its study is important in the evaluation of the functional aspects of arterial blood flow. Humeau and his co-workers obtained their model parameters from experimental LDF signals obtained from a healthy subject after a 2 min vascular occlusion. These signals were denoised, prior to parameter estimation, using fourth-order Symmlet wavelets in a multiresolution decomposition. Reconstruction of the denoised signal was performed only with those wavelet coefficients from scales which were noise free. Matani et al (1996) compared short time Fourier transforms with wavelet transforms in the analysis of ultrasonic Doppler signals from both simulated and experimental cardiac blood flows. They found that although both techniques could reproduce the slow changes in velocity, only the wavelet transform could reproduce the fast changes in velocity with sufficient resolution. Hence they concluded that the wavelet transform would be more useful in the analysis of blood flow disorders such as regurgitation. A method based on wavelet transform analysis was presented by Sato et al (1996) for the non-invasive determination of the left ventricular (LV) end diastolic pressure in an isolated canine preparation. The determination of this pressure is clinically significant for the assessment of ventricular function. Honda et al (1998) extended this work to humans and developed a non-invasive tool which produced approximate estimates of LV pressure. Their method used wavelet decomposition of small amplitude LV vibrations to determine the oscillation frequency of the LV wall. Using this information, together with the internal radius and wall thickness, they were able to estimate LV pressures around end-diastole. Karrakchou et al (1995) have detailed a wavelet packet method for the analysis of post-occlusion pressure transients used in the determination of pulmonary microvascular pressure, important in treating lung oedema. Marrone et al (1999a) have performed Haar wavelet analysis on blood pressure waves in a study of vasovagal syncope, a rapid and reversing loss of consciousness due to a reduction of cerebral blood flow caused by dysfunction of the cardiovascular control. They interrogated the standard deviations and multifractal characteristics of the wavelet coefficients obtained from a time-series of pressure maxima and concluded their results indicate a tentative, diagnostic methodology for this syndrome. Ookawara and Ogawa (2000) have studied the behaviour of Newtonian and non-Newtonian (bloodlike) fluid within a straight pipe with a sudden contraction as a model for arterial stenosis. They found that the introduction Copyright © 2002 lOP Publishing Ltd.

of pulsations to the now enhances the recovery of the velocity profile downstream of the stenosis and their wavelet analysis revealed that the structure of the momentum transfer within the two nuids was substantially different.

6.6

Medical imaging

There has been much research concerning the application of wavelet methods to the denoising, visual enhancement and compression of medical images. A wide variety of medical images, including MRI, PET, SPECT, CT, mammograms, ultrasonic and optical images, have been considered by numerous research groups. In this section, only a very brief selection of examples is given of the application of wavelet methods to the elucidation of medical images. 6.6.1

Ultrasonic images

A method has been developed by Setarehdan and Soraghan (1998) for the measurement and assessment of left ventricular (LV) performance from echocardiographical images. Their spline-based wavelet method detects the edges of the LV endocardium. It defines the edge in terms of the global maxima of the wavelet transform of intensity profiles taken along radial search lines centred within the LV endocardium. Wavelet image analysis has also been used by Mojsilovic el al (1997) to decompose ultrasonic images of the heart in order to determine the success of thrombolitic therapy after acute myocardial infarction. Lee (1996) used a combination of wavelet transforms and genetic algorithms to detect the edge of tumours in ultrasonic images. Lee's algorithm incorporatcs a radial searching routine, whereby the image grey scales are determined along radial lines from within the tumour. Figure 6.42 shows the spatial variation in the grey scale along one such radial search line, together with its wavelet decomposition at three scales. The wavelet decompositions are used to detect the tumour edge shown in figure 6.43. Fu el al (2000) have detailed a discrete (Daubechies D6) wavelet-based method for the enhancement of gastric sonogram images and to compensate for information loss due to histogram equalization. 6.6.2

Magnetic resonance imaging, computed tomography and other radiographic images

In an early paper, Healy and Weaver (1992) reported on the use of wavelets to facilitate magnetic resonance imaging and stated that wavelet transforms offer tangible benefits over traditional Fourier-based imaging (see also Lu el ai, 1992). Xu el al (1994) introduced a spatially selective noise reduction method for magnetic resonance (MR) images. Tn their method, edges and other significant image features are identified from their strong correlation across scales in the wavelet domain and noise is identified as having weak correlation across scales. The correlation is determined from the simple multiplication of the dyadic transform coefficients across a limited number of scales. Due to their localization in the wavelet domain, image features remain sharp after filtering. They illustrated their method on both a Copyright © 2002 lOP Publishing Ltd.

Figure 6.42. Example profile and its three-scale WT. (a) Example profile. (b) Three-scale WT. After Lee (1996). Reproduced with kind permission of the lEE.

(a)

(b)

Figure 6.43. Contour extraction performance. (a) Image where contour of tumour has been detected by conventional radial search with edge detector. (b) Result using proposed GA-based algorithm. After Lee (1996). Reproduced with kind permission of the lEE. Copyright © 2002 lOP Publishing Ltd.

(a)

(b)

(c)

Figure 6.44. MR image from an axial head scan of a \'olunteer (SNR = 18dB). Ca) Before twodimensional filtering; (b) after two-dimensional wavelet filtering; (e) after two-dimensional Wiener filtering. After Xu et al1994 IEEE ]i-clI/sactiolls 011 Image Processing 3(6) 747-758. © IEEE 1994.

synthetic image and an MR image from an axial head scan. Figure 6.44 shows a noisy M R image from an axial head scan before and after filtering. The figure also compares wavelet filtering with Wiener filtering. It was found that wavelet filtering preserved better high frequency data around the image edges when compared with Wiener filtcring. The authors stated that, although their techniquc is slightly less accuratc than others, it is more straightforward, easier to implement and significantly more robust. Wood and Johnson (1999) have outlined a wavelet packet-based method for denoising MR images containing non-Gaussian (Rician) noise. An image compression method for both computed tomography and magnetic resonance has been proposed by Wang and Huang (1996) that uses a separable, non-uniform three-dimensional wavelet transform. The transform uses sets of image slices as a three-dimensional data set, rather than the working on each slice individually (two-dimensional method). This removes interslice redundancy. They found that their technique produces a 70% Copyright © 2002 lOP Publishing Ltd.

increase in data compression over that for

two-dimensional CT image sets and 35% increase for MR image sets. Knoll el al (1999) have detailed a method for localized multiscale contour parametrization of the prostate in both digital CT scans and digitized transrectal ultrasound images. Their automated method enables the prostate to be differentiated from adjacent tissues in low contrast, noisy images. Zheng ef al (2000) tested the effect of the digital compression of CT images on the detection of coronary artery calcifications (CACs). They found that images compressed up to 20: I using both lPEG and wavelet algorithms were acceptable for primary diagnosis of CACs by experienced radiologists. Other work concerning the use of wavelet transforms in the analysis of M Rand CT images include the wavelet-based approaches to tomographic image reconstruction described by Kolaczyk (1996) and Bhatia ef al (1996), the computer aided

support for the localization of pathological tissues such as brain tumours in MRI and CT images by Busch (1997), the new M R image guidance dynamic tracking method proposed by Wendt el al (1998), the denoising and contrast enhancement method for MR images by Alexander ef al (2000) and the wavelet-based feature extraction method within the neural network algorithm for the automatic windowing of M R images by Lai and Fang (2000). In order to improve the visualization of breast pathology, Laine ef al (1994) developed a wavelet-based feature enhancement method to make morc obvious unseen or barely visible features of significance in mammograms. Dhawan el al (1996) employed wavelet packet transform decomposition of grey level mammographic images to represent the local texture of the digitized microcalcification areas associated with the tumour cells. Li ef al (1997) have compared a fractal model of microcalcifications in digital mammograms with both a wavelet-based method and a morphological operations approach. Heine ef al (1999) developed multiresolution wavelct methods to perform statistical analysis of highly correlated non-Gaussian random fields and applied them to the analysis of digital mammographic images. A modulus maxima-based method has been developed by Bruce and Adhami (1999) to classify mammographic mass shapes. A new wavelet-based compression scheme for use in coronary angiographic image compression has been developed by Munteanu ef al (1999). Millet ef al (2000) have described a wavelet-

based filtering method for dynamic PET (positron emission tomography) data which improves signal-ta-noise ratios without loss of spatial resolution. See also

the paper by Bruckmann and Uhl (2000) which considers medical image compression techniques for telemedical and archival applications. 6.6.3

Optical imaging

Carmona ef al (1995) have developed a procedure for the analysis of brain images using dyadic wavelets and modulus maxima thresholding. Figure 6.45 shows a typical brain image (top left) together with the locations of the extrema (modulus maxima) of the wavelet coefficients at various a-scales. Their method allows a multiresolution representation of the image by discarding those wavelet extrema greater than the 80th percentile of all extrema values. By doing so they filter out the high gradients produced by the walls of the blood vessels. They go on to tackle the problem of the large vibratory movement over time of the blood vessels in such images. Copyright © 2002 lOP Publishing Ltd.

,

Figure 6.45. Example of a partial reconstruction from a selective set of extrema of the wal'elet transform. (Left) Top: a typical image. Bottom: image reconstructed from the values of the wavelet transform at the locations of the local modulus maxima which are smaller than their 80th percentiles and from the values of the coarse scale image (shown at the bottom of the right column). Notice that the blood vessels have been removed and replaced by Jow-frequency surfaces (compare with original image). (Right) The scale increases from lop to bottom. Positions of the local modulus maxima al the scales s = i, 22 , 23 ,2 4 after removing the local maxima with the modulus larger than the 80th percentile of the maxima sizes at each scale. Note that the modulus maxima near the boundaries of the blood vessels are removed. The larger the value of a pixel, the brighter it shows. Boltom: coarse scale approximation at scale s = 2 1 , After Carmona et al 1995 IEEE TJ'{//Isactiolls 011 Medical Imaging 14(3) 556-564. © IEEE 1995.

Witkowski el al (1998) used wavelet denoising of image pixels to reduce the noise present within CCO camera images of both ill vilro frog and dog hearts in a study concerning the nature of ventricular fibrillation. Using voltage sensitive dyes to convert transmembrane potentials into optical signals, the researchers were able to visualize the spatial propagation of wavefronts during ventricular fibrillation. Van de Wouwer el al (2000) have described an automatic elassification scheme for the diagnosis and grading of invasive breast cancer from digitized microscopic images of cell nuclei. The scheme incorporates a number of image parameters including wavelet energy-based texture parameters from both the low resolution and detail images. Copyright © 2002 lOP Publishing Ltd.

6.7

Other applications in medicine

6.7.1

Electromyographic signals

Electromyographic (EMG) signals represent the electrical activity of muscle during contraction. A small number of authors have used wavelet methods to gain an insight into these signals. EMG signals were probed using both the Mexican hat and Morlet wavelets by Laterza and Olmo (1997). They showed that because of its similarity to the constituent electrical impulses within the signal, the Mexican hat wavelet produced a better decomposition of the signal when compared with the Morlet wavelet, especially in the presence of noise. The maximum value of the Mexican hat transform was four times larger than that for the Morlet-based transform due to its better matching with the shape of the signal feature under investigation-the motor unit action potential (MUAP). Fang et af (1999) have developed a waveletbased tool for the identification of single motor unit (SMU) potentials within EMG signals. They use wavelet-based methods both for denoising the signal via thresholding and the subsequent identification of SMU spikes in the signal. The reduction of motion artefacts from surface EMG signals has been investigated by Conforto et af (1999) using a number of methods including a novel wavelet-based technique. The four methods they compared were: (1) high pass filtering, (2) a moving average procedure, (3) a median average procedure and (4) a wavelet-based procedure they developed themselves. They designed an experimental protocol which allowed them to evaluate the performance of each method with respect to a number of criteria and found the wavelet method performed better both with regard to timing detection and reduced distortion of clinically useful signal content. A number of other papers have appeared recently reporting on studies involving the wavelet analysis of the EMG, including the scheme by Karlsson et af (1999) which uses wavelet packets combined with a wavelet thresholding method for the spectral analysis of surface EMG signals, the study by the same group (Karlsson, 2000) of a variety of nonstationary signal analysis methods (including the continuous wavelet transform, for the interrogation of the myoelectric signal during dynamic contraction), the study of motor unit action potentials using a variety of discrete wavelets by Pattichis and Pattichis (1999), and the analysis of uterine EMG by Khalil and Duchene (2000).

6.7.2

Sleep apnoea

Sleep apnoea is a phenomenon characterized by prolonged interruptions of normal respiration during sleep, caused by the collapse of the upper airway. We have already come across a couple of studies concerning sleep apnoea in section 6.2.3 in relation to the study of heart rate variability. Using a spline function wavelet, Figliola and Serrano (1997) used the distribution of the energy of discrete wavelet coefficients across scales to characterize three physiological time series associated with sleep apnoea: heart rate, lung volume variation and blood oxygen saturation. They differentiated between three physiological states-'pre-apnoea', 'periodic breathing' and 'regular'-using wavelet scale-dependent energy distributions which they quantified Copyright © 2002 lOP Publishing Ltd.

using an information cost function (IeF) (essentially the Shannon entropy of the normalized energies-see chapter 4, section 4.2.4). They were able to show that the ICF decreases from the pre-apnoea to the apnoea state for all three time series. Another study of sleep apnoea by Kermit el al (2000) used a Haar wavelet decomposition of airflow signals, where the resulting wavelet coefficients were fed into a neural network-like classification algorithm which determined whether the signal represented apnoea events, hyper-apnoea events or normal airflows. (Hyper-apnoea is the condition where an irregular breathing pattern occurs rather than a complete cessation of breathing.) It was found by Kermit and his co-workers that apnoea events were easier to detcct than hyper-apnoea events. 6.7.3

DNA

A number of investigators have detailed the use of the wavelet transform in the analysis of the correlations that exist within DNA sequences. Ameodo el al (1996, 1998a,b) have probed the multifractal nature of DNA sequences using the wavelet transform modulus maxima method to analyse mapped representations of the sequence. Altaiski el al (1996) have also provided evidence of the multifractal nature of DNA sequences. Tsonis el al (1996) investigated the localized structure of DNA sequences using wavelet transforms and managed to decompose seemingly homogeneous regions in non-coding sequences into distinct subregions with their own repetition and construction rules. (More information on fractals and wavelets is given in chapter 7, section 7.2.) 6.7.4

Miscellaneous

Dickhaus and Heinrich (1996) have presented the components of a system for the classification of biosignals using 'wavelet networks'. These combine the feature extraction and selection properties of the wavelet transform within the decision capabilities of the artificial neural network. Maksimovic and Popovic (1999) have used both classical neural networks and wavelet networks to classify functional movements in humans with spinal cord injuries. They found that a combination of both techniques provides a suitable method for this kind of geometric feature analysis. Wavelet analysis is detailed, together with a variety of other techniques, in a comprehensive review of pattern recognition approaches in magnetic resonance spectroscopy by EI-Deredy (1997). Other applications of wavelet transform methods in medicine include: the investigations of the effect of morphine on foetal breathing rates using matching pursuits by Akay and Szeto (1995) and the fractal properties of foetal breathing rates by Akay and Mulder (1998); the analysis of sleep microstrucure through the wavelet decomposition of the polysomnogram by Kubicki and Herrmann (1996); the interrogation of neuronal activities of the recticular formation of the lower brainstem and the nucleaus tractus solitarii from an anaesthetized dog by Lambertz el al (2000); a wavelet-based method for the detection of ultrasound scattering centres in the liver by Tang and Abeyratne (2000); a method to detect respiratory and cardiac rhythm disorders from pressure measurements from an inflated wrist cuff by Dupuis and Eugene (2000); and the analysis by Chan el al Copyright © 2002 lOP Publishing Ltd.

(2000a) of chronobiological rhythms through the response of the locomotion of mice to a change in the light-dark cycle. 6.7.5

Further resources

The special issue of the journal Engineering in Medicine and Biology, edited by Akay (1995), contains a collection of papers covering a wide range of biomedical signals (EMG, EEG, EeG, breathing rates and pulmonary capillary pressures) analysed using a variety of time-frequency methods including wavelet transforms. In a later paper, Akay (1997) provides a brief summary of selected areas of biomedical research which have benefited from wavelet analysis, including the detection of coronary heart disease, turbulent blood flows, irregular heartbeats, the effect of alcohol on foetal breathing, hearing aids, mammography and medical image compression. In their paper, Unser and Alrdoubi (1996) present a much more comprehensive overview of wavelet applications in biomedicine. They include details of wavelet properties that they consider most important for biomedical applications and provide over one hundred references. The book edited by Aldroubi and Unser (1996) contains many detailed papers concerning the application of the wavelet transform to medical imaging, medical signal processing and biological models. Another collection of papers, edited by Akay (1998), presents a comprehensive review of the application of various time-frequency methods to a wide range of biomedical signals. Wavelet methods feature heavily in this text.

Copyright © 2002 lOP Publishing Ltd.

Chapter 7

Fractals, finance, geophysics and other areas 7.1

Introduction

In this last chapter we cover a variety of subject areas in briefer detail than the preceding chapters. Most of the chapter is devoted to three main topics-fractals, finance and geophysics. First we will look at how we can use wavelet transforms to characterize the scaling properties of self-similar fractal and multifractal objects. After this, we will consider the emerging role of wavelet analysis in financial analysis. The largest section of the chapter is then devoted to geophysics-where wavelet transform analysis began with the analysis of seismic signals. Finally, in the last section we take a brief tour of a selection of other areas where wavelet analysis has made an impact but has not been covered within the rest of the book.

7.2

Fractals

Fractals are objects which display self-similarity over scales. These objects can be exactly self-similar, as in figure 7.I(a), where the exact form of the object is repeated at smaller and smaller scales, or they can be statistically self-similar (figure 7.1 (b)), where the statistical properties of the object are consistent across scales (Addison, 1997). Many natural fractals are statistically self-similar, e.g. coastlines, cracking, tree branching, stock market indices, permeabilities in the subsurface, the distribution

of galaxies, and so on. Previous chapters have already touched upon some natural phenomena whose fractal properties have been interrogated using wavelet methods:

e.g. fluid turbulence (chapter 4), engineering surface characterization and chaotic altractors (chapter 5), heart rate variability, breathing rate variability, DNA sequences and mammographic images (chapter 6). All natural fractals exhibit selfsimilarity only over a finite range of scales and hence, unlike their regular mathema-

tical counterparts, their fractal description eventually breaks down. They do, however, exhibit these fractal properties over a sufficiently large range of scales to allow fractal geometric methods to be usefully employed in their description. The property of self-similarity across scales makes wavelet transform analysis a natural

candidate for the interrogation of such objects. Copyright © 2002 lOP Publishing Ltd.

self similarity everywhere

100 75

en

Y 50 sta

25

o

25,---------,-------,

~~

If ~

y

0 I--~'hj:

I~~

Ii"

-25 -50 -50-25

0

25

50 75 100 125 150 x

x

(bJ Figure 7.1. Exactly self-similar and statistically self-similar fractals. (a) The Sierpinski gasket: a fraclal object which is exactly self-similar over all scales. (Each of the circles contains self-similar parts of the whole gasket at different scales.) (b) The two-dimensional trajectory of ordinary Brownian motion. (The righi-hand plot contains lhe first 1/l6th of the trajectory of the left-hand plot enlarged to maintain the same degree of resolution between each pIOL) After Addison 1997 Frocrafs (/"d Chaos: All IlIusrrared Course, P S Addison 1997, Institute of Physics Publishing, Bristol and Philadelphia.

7.2.1

Exactly self-similar fractals

Figure 7.2(a) shows anolher common exactly self-similar fraclal, the Iriadic Cantor set. The construction method is shown in the plot where, at each step in the generation of the fraclal set, Ihe middle Ihird is removed from the remaining line segments. The iteration process begins on the unit line and proceeds ad bifinitum to construct the set. Copyright © 2002 lOP Publishing Ltd.

o

1

step

unit interval

k=Q

'initiator'

k=!

'generator'

k=2 k=3 k=4 k=5

Cantor

k=oo

set

3 times magnification of the left hand third

of the original Cantor se~ Cantor sel

9 times magnification

------'----~

sel

Ca)

(b)

Cantor

1

'"

Figure 7.2. Wavelet analysis of the triadic Cantor set. (a) Construction of the triadic Cantor sel. (b) Transform plots for the triadic Cantor set. Nole that (sgn(T)IT(a,b)II/2) is plotted against In(a) and b. (a) After Addison 1997 Fractals and Chaos; All Illustrated Course, P S Addison 1997, Institute of Physics Publishing, Bristol and Philadelphia. (b) From Ameodo A el af, Wavelet transform analysis of invariant measures of some dynamical systems, in WCII'elels, Combes J M, Grossmann A and Tchamitchian P (eds.) Springer-Verlag 1989 pp 182-196. Copyright SpringerVerlag 1989. Reprinted with kind permission of the authors and publisher.

Copyright © 2002 lOP Publishing Ltd.

Figure 7.3. The wavelet transform of a regular snowflake. The snowflake is shown in (a). The scale parameter a is successively divided by the same factor 1= 3: (b) a = a~, (c) a = a~ 13, (d) a = a~/32. T((I,b) is expressed in jog(5)/log(3) units in order to reveal the self·similarity of the geometry of the snowflakes. (See Argolll el (II (1989) for more details.) Reprinted from Argoul el al 1989 Physics Letters A 135(6{7) 327-336. Copyright 1989, with permission from Elsevier Science.

Figure 7.2(b) shows a three-dimensional plot of a Mexican hat wavelet transform for the triadic Cantor set. The branching structure of the set is easily seen in the transform plot (Arneodo et ai, 1989). Figure 7.3 contains a regular snowflake fractal generated in the plane, together with transform plots at three a scales using the radial (twodimensional) Mexican hat wavelet (Argoul et ai, 1989). Contour lines, set at an arbitrary value, show the construction rule of the fractal snowflake in figure 7.4. As the scale of the wavelet tends to zero, the transform plot approximates more and more the snowflake itself. See also Antoine et al (1997) who analysed a fractal Koch curve in their paper concerning the characterization of shapes using the information gained from the maxima lines (both modulus maxima and ridges) of the continuous wavelet transform. The Sierpinski gasket, Cantor set, snowflake fractal and the Koch curve are exactly self-similar. However, very few examples of exact self-similarity are to be found in nature (e.g. some fern shapes exhibit nearly exact self-similarity over a few scales). Most natural fractals exhibit stochastic selfsimilarity and consequently most research concerning fractal objects and processes in nature focuses on stochastic fractals. Copyright © 2002 lOP Publishing Ltd.

a=a*

a*/3

by

Ca)

(h)

bx

." (e)

(d)

Figure 7.4. lsocontour lines of the w3\'elet transform of the snowflake fractal. The isocontour line T(a,b)/c/°g(Sl/IOg(3) = k (k arbitrarily chosen) for different values of the scale parameter. (a) a = (I*, (b) (I = 0+/3, (c) a = (1+ /3 2 , (d) a = (1+ /3 3 . Reprinted from Argoll! eI 0/ 1989 Physics Lerrers A 135(6/7) 327-336. Copyright 1989, with permission from Elsevier Science.

7.2.2

Stochastic fractals

Figure 7.5 shows a sequence of shots for a diffusion limited aggregation (DLA) cluster and associated wavelet transforms. DLAs are essentially stochastic snowflakes 'grown' using a numerical technique which allows particles to wander randomly about the plane (in a Brownian motion) until they encounter the DLA aggregate, sticking to it on impact (Addison, 1997). DLA has been used to model a variety of physical phenomena including bacterial colonies, viscous fingering and electrochemical deposition. See also an extension to this type of work contained in the paper by Pei et at (1995). One area where wavelet transform analysis has been used extensively is in the determination of the scaling properties of fractional Brownian motion (fBm) (and its derivative, fractional Gaussian noise, fGn). Over recent years, these correlated random functions, first proposed by Mandelbrot and Van Ness (1968) as a generalization of Brownian motion, have been suggested as models for a whole range of natural phenomena, including DNA sequences, geometrical tolerancing in mechanical design, risk analysis, polymer models, landscape surfaces, image textures, the dynamics of nerve growth, crack profiles, permeability fields in porous media and non-Fickian diffusive processes (Addison and Ndumu, 1999). The 'smoothness' of the fBm function increases with the Hurst exponent H, which can vary in the range 0< H < 1. Three examples of fBm, denoted BH(t), are shown in figure 7.6 for different H. Fractional Brownian motions are nonstationary random processes Copyright © 2002 lOP Publishing Ltd.

T ,(a,b) a L60

(b)

r

Figure 7.5. Wavelet transforms of a DLA cluster. The DLA cluster is shown in (a). The scale parameter a is successively divided by the same faClor 6 = 1.55: (b) a = a", (c) a = a~ /6, (d) a = (//62, (e) a = a* /8 3 , (f) a = a" Itt. T(a, b) is expressed in a1.60 units in order 10 reveal the self-similarity of the geometry of the DLA clusters. (See Argoul er af (1989) for more details.) Reprinted from Argoul el a/1989 Physics Leffers A 135(6/7) 327-336. Copyright 1989, with permission from Elsevier Science.

where the standard deviation, a, of the rEm trace deviations 6.B I1 (figure 7.6(c» taken over a sliding window of length s scales as

(7.1 ) H > 0.5 corresponds to persistent rEm where the trace has a tendency to persist in its progression in the direction in which it was moving, H < 0.5 corresponds to anti persistent fSm where the trace has a tendency to turn back upon itself, and H = 0.5 corresponds to regular Brownian motion where the trace is free to move in either direction from step to step (for more information see Mandelbrot, 1982; Addison, Copyright © 2002 lOP Publishing Ltd.

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Figure 7.6. Fractional Brownian motion traces at ,'arious Hurst exponents. (a) rBm Irace, H = 0.2, antipersistenl. (b) fBm Irace, fI = 0.5, neutrally persistent (regular Brownian motion). (e) rBm trace, N = 0.8, persistent. After Addison 1997 Fractals alld Chaos: All JIlustrared Course, P S Addison 1997, Institute of Physics Publishing, Bristol and Philadelphia.

1997). The fractal dimension of an fBm trace function can be found from the Hurst exponent through the simple relationship D=2-H

(7.2)

There are many methods used in practice to determine either H or D for experimental data suspected of fBm scaling. One common method uses the Fourier power spectrum P,,(f), which for an fBm, should scale as

PF ([)

rxr(2H+l)

(7.3)

Hence, a logarithmic plot of power against frequency allows H to be determined from the slope of the spectrum. Wavelet power spectra, Pw ([), both continuous and discrete, also exhibit this scaling and can therefore also be used to determine H (and hence D if required). Figure 7.7 shows a single realization of an fBm trace with H = 0.6, together with both its Fourier and wavelet power spectra (refer back to chapter 2, section 2.9). A continuous Mexican hat wavelet was used in the decomposition. The slope corresponding to -2.2 (i.e. H = 0.6) is plotted on the graph for comparison. We can sce from the plot that the wavelet spectrum is much smoother due to the finite bandwidth of the spectral components associated with the wavelets. This can have an advantage when trying to abstract the fractal information from such Copyright © 2002 lOP Publishing Ltd.

(a)

1.10 7 1.10 6

slope =-(1+2H)

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.....l:l

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100 10 1 1.10-4

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1.10-3 0.01 0.1 frequency (arbitrary units)

1

Figure 7.7. Fourier and wavelet spectra of fractional Brownian motion. (a) A single realization of an fBm trace function H = 0.6. (b) Fourier (dashed line) and wavelet (smooth line) power spectra of the trace in (a).

data sets when only a limited number of data sets are available-too few to smooth the Fourier spectrum through ensemble averaging. We can compute the fractal scaling characteristics of a data set suspected of exhibiting fBm scaling from its wavelet decomposition: continuous or discrete. Simonsen et al (1998) used a discrete wavelet coefficient-based method to analyse two real data sets: fracture surface profiles and stock market indices. They showed that the mean absolute discrete wavelet coefficient value scales as ( I T m,n I

hn

ex:

a~ +!)

(7.4)

where 2M

- m

2:

(ITm,n I) m = Copyright © 2002 lOP Publishing Ltd.

_l

ITm,nl

_n=-;--=-M-=---m--

(7.5)

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100

a

Figure 7.8. The Awe function versus scale for various choices of \Val'elet order (Daubechies family) for self-affine profiles with Hurst exponent H = 0.7. Wavelet order indicated in the figure. The number of samples used was N

=

100. The length of the profiles was L

=

4096. Note lhal the data are rescaled

so lhal the largest scale is equal 10 unity (100). The extracted Hurst exponents were J-I = 0.68 ± 0.01 for Daubechies 04 and J-I = 0.70 ± 0.01 in all other cases. The curves are shined relative to DI2 for clarity. After Simonsen ef Society and the authors.

(//

(1998). Reproduced with the kind permission of the American Physical

is the mean absolute value of the wavelet coefficients at scale a,w The authors used dyadic grid Daubechies wavelets, hence am ex: 2/11, and also set the maximum a scale to unity. The authors called this measure the average wavelet coefficient (AWC)

function. Figure 7.8 shows the logarithmic plots of the A we against wavelet scale for an fBm profile computed using a variety of Daubechies wavelets. The plots are obtained using the average values of 100 realizations of f8m, each 4096 data points in length and with a Hurst exponent H = 0.7. There is very good agreement between the Hurst exponents of the original traces and those found by the method. This can be seen from figure 7.8 where the slope of the best-fit lines is approximately 1.2. The authors proceeded to test the stability of the method against noise, drift and distortion before using it to interrogate two real data

sets-fracture profiles and stock prices. Figure 7.9(b) shows the A we analysis of 211 granite profiles, one example of which is plotted in figure 7.9(a). The Fourier power spectral analysis (figure 7.9(c» and the AWe analysis are in good agreement. Figure 7.10 contains the stock market index for shares taken from the Milan stock exchange over a two and a half year period. Both the A we and Fourier analysis of this single time series are plotted below the index. The large spread in the Fourier spectrum is evident in the plot. The authors concluded that for large numbers of samples the wavelet method and Fourier (power spectrum) method perform equally well. However, when only a small number of examples are available the wavelet method outperforms the Fourier method. This was also concluded by Dougan el a/ (2000). Copyright © 2002 lOP Publishing Ltd.

10'

24 (a)

(b)

22

-..""• "

~ 20

.~

18 16 14

10' 10° 10- 2 10-4

0

20

40

10- 3

80

60

x (mm)

10-2

to-I

10°

a

10' (e)

10'

..

SlO4

10' 10° 10-2

10- 4

10- 3

10-2

10- 1

f Figure 7.9. Wavelet and Fourier analysis of a granite fracture profile. (a) One single representative profile from the granite fracture. The number of points in the profile is 2050. (b) Average wavelet coefficient (A We) analysis of the entire sel of(2050 x 211) data points. The solid line is the regression fil to the scaling region. The corresponding Hurst exponent is H = 0.81 ± 0.02. (c) Fourier power spectral (FPS) analysis of the data. Here the solid line corresponds to a Hurst exponent of H = 0.79 ± 0.03. After Simonsen er at (1998). Reproduced with the kind permission of the American Physical Society, the authors and J Schmittbuhl.

Another related method of computing H directly from the wavelet coefficients using the scaling of the variance of discrete wavelet coefficients at scale index 111 which we saw in chapter 4, section 4.2.1, is given by 2M "'-I

I:

')

( T,71,1I

(T"",,)' = ---'=2"'''1----:''::-'-11=0

//I

(7.6)

From chapter 4, equation (4.1 Ob), we know that coefficient variance is simply related to the power spectrum as (77) Combining this expression with the power law relationship for fBm of index H given above in expression (7.3) and the fact that frequency fis inversely proportional to the wavelet scale a (=2"'), we obtain the scaling relationship 2

(Til/,ll) III ex: Copyright © 2002 lOP Publishing Ltd.

(2H+I) (f/ll

(78a)

-:E

-

14

12

N

0

~

10

."8

8

~

6

0.

.c

•

4

200

0

600

400 days

10 14

10' (b)

10'

10 12

~104

10 10

...

S

~

2< 10' la'

(c)

"- 10'

10'

•

10' 10-'

10-2

10-1

10"

10' 10-'

10-3

a

10-2

10'

f

Figure 7.10. Wavelet and Fourier analysis of share prices. (a) Fiat share prices taken from the Milan

Stock Exchange for the period from September 1988 (day I) to May 1991, with three observations per day. (b) The result of the wavelet analysis for the data in (a). The estimated Hurst exponent, corresponding 10 the solid line, is J-I = 0.65 ± 0.03. (c) The result of the Fourier analysis for the data in (a). The Hurst exponent in this case is H = 0.62 ± 0.06. Note the mOfe well behaved scaling

region for the wavelet method compared with the Fourier method. After Simonsen el al (1998). Reproduced with the kind permission of the American Physical Society, the authors and B Vidakovic.

In the literature, ~, is often used as a compact notation for the variance of the discrete wavelet coefficients at index m, hence the relationship is written

(7.8b)

If we take the square root of both sides of expression (7.8b) to get am ex: ([/IIH+!•

(7.9)

we can see that the scaling of the coefficient standard deviations is consistent with that given by expression (7.4). We expect this, as both the mean absolute value of the coefficients and the standard deviation of the coefficients are first-order measures of spread. Furthermore, for an orthonornmal multiresolution expansion using a dyadic grid, the scale a is proportional to 2'''. We can, therefore, take base 2 logarithms of both sides of expression (7.8b) to get the often-seen expression

,

log2(T~,) =

(2H

+ 1)111 + constant

(7.10)

where the constant depends both on the wavelet used and the Hurst exponent. See, for example, Flandrin (1992) who defines the constant and gives it explicitly for the Haar Copyright © 2002 lOP Publishing Ltd.

wavelet. Figure 7.II(a) shows plots of log, (IT;',) against m for two fBms, one of index H = 0.6 and the other H = 0.8. These were computed using Haar wavelets. According to expression (7.12), the plotted points should fall on lines of slopes 2.2 and 2.6 respectively. These gradients are superimposed on the plots. Good agreement is found between the experimental plots and the theoretical line. Remember that this is for a single realization of the fBm trace, and ensemble averaging over many

traces will produce a much more accurate result. Figure 7.II(b) contains a plot of the coefficients in sequential format for the fBm of index H = 0.6 in figure 7.II(a). The increase in the variance with scale is evident from the plot. Figure 7.11 (c) contains example signal reconstructions using only the coefficients at the levels specified. Li el al (1996) have used this approach in a geophysical context to examine the fractal structure of velocity logs. Note that, from the above arguments, we can also see that the wavelet scale-

dependent energy 2M "'-I

£11I

=

L

(Tm,II)2

(7. I I)

+ constant

(7.12)

11=0

has the fractal scaling law 10g,(E",) = 2Hm

that is

(7.13) This is the same scaling as that given by equation (7.1). This makes sense, as E", is a measure of the scale dependent variance of the signal. Figure 7.II(d) shows the 10g,(E",) against m plot for the traces shown in figure 7.II(a). These plots have slopes of around 1.2 and 1.6 respectively as we would expect. We have concentrated on discrete transform coefficients Tm ,11 as they are prevalent in the fractal-wavelet literature. However, continuous transforms, T(a ,b), obviously exhibit the same

scaling law as (7.13), i.e. E(a) ex a'H

(7.14)

(refer back to chapter 2, equation (2.22)) but now the a scale parameter is continuous (or at least for practical purposes a discretized approximation based on a non-dyadic grid) and the slope of the plot of 10g(E(a)) against log(a) is used to find H. There are now large numbers of scientific papers concerning the identification

and modelling of fractional Brownian objects and processes using wavelets. Meharabi el al (1997) compared seven different methods for analysing fBm, including one based on eq uation (7.10). They found that the wavelet decomposition method offers a highly accurate and efficient tool for characterizing long-range correlations in complex

distributions and profiles. Kaplan and Kuo (1993, 1995) have presented a method based on the Haar wavelet to estimate the fractal dimension of fBm from fractal signals using the scaling relationship between the wavelet coefficients of the discrete fGn increments of the fBm signal. The fractal feature extraction technique they developed was used for texture segmentation of synthetic test images and aerial Copyright © 2002 lOP Publishing Ltd.

I

1. -

I

..E:

N

$N

30 25 20 15

30 25 ..N E: 20 $ N 15 0010 .Q 5

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00 1 2 3 4 5 6 7 89101112

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3 4 5 6 7 89101112

m

m

m=6,7,8,9,10 m=5

100

/

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m=3

50 T· l

0

m=2

'llllJII ".l.' II" 1'1 '1

m=1

-

I.

-50 -100

-

o

200

400

600

800

1000

1200

i (=2 M - m +n) (b)

Figure 7.11. The relationship between the wavelet coefficient variance and Hurst exponent for fractional Brownian motion. (a) fBm traces (H = 0.6 leftjH = 0.8 right) with their corresponding log2((}~) against m plots given below. The slopes of 2H + I = 2.2 (left) and 2.6 (right) are shown on the plots above the data points. (Arbitrary axis units.) (b) The sequentially indexed wavelet coefficients for the H = 0.6 fBm trace in (a). Note that the coefficient values have been cut off at the largest scales in the plot as the coefficients at these scales dominate.

images of natural scenes. The paper by Malamud and Turcotte (1999) contains results from the variance analysis of both fractional Brownian motion and fractional Gaussian noises. They compared the wavelet technique with three others for determining the scaling relationship and noted that wavelet variance analyses lack many of the inherent problems associated with Fourier power analysis. The identification of fBm within white noise is considered by Hwang (1999) and a method for the detection of transients within l/! Gaussian noises using undecimated discrete wavelet transforms is described by Liu and Fraser-Smith (2000). Chen et al (1997a) used wavelets to define the fractal dimensions of satellite images. They then employed neural networks to classify the images, e.g. urban, bare soil, ocean, forest, etc. The energy content and self-similarity of the detail signals of fractional Brownian images is Copyright © 2002 lOP Publishing Ltd.

I

50

50

-

0

0

I

I

500

1000

-50

100

I

-

0

\

-100

0

0

m=lO

500

1000

-50

0

500

m=9

1000

m=7

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o

500

1000

o

m=5

500

1000

m=2

(c) 30

30

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25

25

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-. 20

~

~

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15

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..9 10

r--1----r----.--.---.---...--.---r-T----r--.---,

15

OIl

.-~,~

..9 10 5

5

o L.......l----'-----'---'---'--...L..-.L......JL........L----,----,--....u

o1

2 3 4 5 6 7 8 9 10 1112 m

o o1

L.......l----'-----'---'---'---'--J.....-L...-L----'----'----"

2 3 4 5 6 7 8 9 10 1112 m

(d)

Figure 7.11 (continued). (c) Reconstruction using only those coefficients at the indicated scale. Note the reduction in the vertical axis scale with decreasing coefficient scales. (d) log2(Em ) against m plots corresponding to the fBm traces given in (a) (H = 0.6 left/ H = 0.8 right). The slopes of 2H + 1 = 1.2 (left) and 1.6 (right) are shown on the plots above the data points.

discussed briefly by Mallat (1989). Turner et al (1995) have employed wavelet-based techniques to determine fractal characteristic measures of precision-machined surfaces and Ikeda et al (1999) used two-dimensional wavelets to estimate the dimension of fractal surfaces. Frantziskonis et al (2000) have determined the Hurst exponent of corrosion pits using Daubechies wavelets. Arneodo et al (1998c) found fBm scaling in DNA sequences using Mexican hat wavelets. Li and Ulrych (1999) have used fBms synthesized using wavelets to model geological processes. Jones et al (1996) have suggested a method for the computation of Hurst exponents using wavelet packet analysis. They tested their method on synthetic fBm traces before using it to characterize the spatial distribution of local enzyme concentration in fungal colonies. Zeldin and Spanos (1996) have outlined a wavelet method for the synthesis of random fields including fBm, Elliot et al (1997a) have detailed a Fourierwavelet (Meyer) method for synthesizing fractal random fields and Pesquet-Popescu (1999) has detailed a wavelet packet-based method for analysing two-dimensional fBm fields. Masry (1998) has described the spectral properties of two-dimensional self-affine fBm fields (and other types) in terms of the discrete wavelet transform. Copyright © 2002 lOP Publishing Ltd.

Vergassola and Frisch (1991) and Vergassola et al (1993) have discussed the application of wavelet transforms to self-similar processes, considering both turbulent flow and fBm signals. Elliot et al (1997b) considered a wavelet-based technique to study the relative spreading of pairs of particles for a family of anisotropic velocity fields. Thresholding methods for use with fractional Gaussian noise (the derivative of fBm) are discussed by Wang (1996). A number of methods, including waveletbased methods, for the determination of H from fGn signals and its relationship to fBm is discussed in the context of heart rate variability by Fischer and Akay (Fischer and Akay, 1996, 1998; Akay and Fischer, 1997). Liu et al (2000b) have developed a two-dimensional fractal parameter estimation method for natural scenes and textures. They demonstrated their technique by performing coastline detection and texture segmentation of both synthetic and natural images. There are many other scientific papers which detail in greater depth the analysis and synthesis of fractional Brownian motion using wavelet transforms. See for example Masry (1993), Hirchoren and Attellis (1997, 1998), Tewfik and Kim (1992), Dijkerman and Mazumdar (1994), Ramanathan and Zeitouni (1991), Kato and Masry (1999) and Veitch and Abry (1999). 7.2.3

Multifractals

Multifractal theory concerns itself with fractal objects which cannot be completely described using a single fractal dimension (monofractals). They have in effect an infinite number of dimension measures associated with them. This section presents a short summary of wavelet-based multifractal characterization. The multifractal scaling of an object is characterized by NE: ex c-!(a)

(7.15)

where NE: is the number of boxes of length c required to cover the object andf(a) is the dimension spectrum, which can be interpreted as the fractal dimension of the set of points with scaling index a (Hillborn, 1994). We can find the multifractal spectrum ofa signal by partitioning it into Nboxes oflength c. A probability density, P(c, i), of the signal in each box, labelled i, is calculated where P( c, i) is the fraction of the total mass of the object in each box. The qth-order moments M(c, q) are then calculated as follows: N(E:)

M(c, q)

L

=

P(c, i)q

(7.16)

i= 1

For a multifractal object this moment function scales as (7.17) From this scaling both a and thef(a) spectrum can be calculated from

a(q)

Copyright © 2002 lOP Publishing Ltd.

=

dr(q) dq

(7.18)

and

I(a)

=

qa(q) - T(q)

(7.19)

In the wavelet-based method for calculating the I(a) spectrum, the function of

equation (7.16) is replaced by the wavelet-based moment function

M(a,q) =

L

IT(a,bi)I"

(7.20)

where IT(a,bi)1 is the ith wavelet transform modulus maxima found at scale a. By summing only over the modulus maxima, this incorporates the multiplicative structure of the singularity distribution into the ealeulation of the partition function (Muzy el ai, 1991). For a multifractal object this wavelet-based moment function scales as

M(a,q) rxar(q)

(7.21)

This relationship is then used to ealeulate a and the f(a) spectrum and hence characterize the mullifractal object or process under investigation. Note that for negative q values equation (7.20) becomes unstable in the neighbourhood of points on maxima lines where the wavelet transform is close to zero. However, this can be remedied by replacing the value of the wavelet transform modulus at each maximum by the supremum value along the corresponding maxima line at all scales smaller than a (M uzy el ai, 1993). Haase and Lehle (1998) have described a modulus maxima method based on Gaussian derivative wavelets for determining multifractal spectra. They have

illustrated their method on time series from a circle map and a turbulent velocity signal. Figure 7.12 contains the turbulent velocity signal, together with its multifractal spectrum. The signal is shown in figure 7.12(a), acquired from within an axisymmetric jet. Figure 7.12(b) shows the energy spectrum of the signal showing the classic -5/3 power law of the inertial range of turbulence. Figure 7.12(c) contains the wavelet transform plot with its modulus maxima shown directly below in figure 7.12(d). A Mexican hat wavelet was used in the decomposition. Finally, the f(a) spectrum, computed from the modulus maxima lines, is shown in figure 7.12(e). Note that if the velocity signal were a monorractal, such as fBm, the multifractal spectrum would collapse to a single point. Full coverage of multifractal theory is outside the scope of this text. For more information, including the use of T(q) to determine the generalized fractal dimension D", see for example Degaudenzi and Arizmedi (1999) who have studied the multifractal character of an airborne pollen time series using a wavelet transform

modulus maxima method incorporating the third derivative of a Gaussian as the analysing wavelet. Arraull el al (1997) have used a wavelet-based mullifractal analysis technique to probe high resolution satellite data of marine stratocumulus, comparing their results with the analysis of monofractal Brownian surfaces. Riedi el al (1999) have developed a wavelet-based multi fractal model for use in computer traffic network modelling (see also Gilbert and Willinger, 1999). They suggested the use of their model in other disciplines such as finance and geophysics. M uzy el al (1991) considered wavelet transforms and multifractal signals, comparing a Copyright © 2002 lOP Publishing Ltd.

(a)

~1010 kl

I'" ~~

m' 105

'1Ir~

I

I

I

IO'

100

I

,

10"

k (b)

(e)

1.0 2.e -•

-

0.8

~ 0.6

"1

.5

0.4 0.2

4 .e -2

f

1.0

1.4

1.2

1.6

a (d)

(e)

Figure 7.12. Multifractal spectrum of a turbulent velocity signal. (a) Turbulent velocity signal. (b) The energy spectrum of the signal in (a) showing the -5/3 law. (e) Wavelet transform of (a) using a Mexican hat wavelet. Cd) Skeleton of maxima lines from (c). (e) The multifractal spectrum f(o:)

resulting from the wavelet modulus maxima method. Figures kindly provided by Dr Maria Haase, Institut nir Compuleranwendungen, Universitat Stuttgart.

multifractal lurbulent signal with a monofractal fErn signal both with the same spectral power law. A number ofpraclical examples of the connection belween wavelets and (multi)fractals can be found in the book on wavelets, fraclals and Fourier transforms edited by Farge et al (1993).

7.3

Finance

This is a very interesting application of wavelet analysis which concerns a pertinent

problem (which there is often a large financial incentive to solve!). The data sets (financial indices, census data, spatially distributed economelric measures, etc.) are typically highly nonstationary, exhibit high complexity and involve both (pseudo-) random processes and intermittent deterministic processes. A good example would be financial indices generated from, among other effects, a large number of small scale (pseudo-random) share dealings combined with large scale (deterministic) Copyright © 2002 lOP Publishing Ltd.

Coffee sample period: 1960.1 until 1995.12

300

250 200

ISO 100

5~ l=::::::~,~~~r==~'~:'~-~~-T,--"--,,r----r, ~~~__, o

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

trend

Figure 7.13. Wavelet analysis of coffee price index. Top, dyadic wavelet scalogram. Bottom, original time series with trendline obtained using the wavelet-transformed data at the base level of resolution. After Davidson el a/1998. Reproduced with kind permission of Academic Press Ltd and the authors.

interest rate adjustments. The application of wavelets to finance is still in its infancy when compared with other subject areas. However, wavelet theory is beginning to make a number of inroads into the area. A good overview of the application of wavelets in economics and finance is given by Ramsey (1999). The paper provides selective details of recent advances in the analysis of economic data using wavelet transforms. More specific examples of applications are detailed below. We have already come across the work of Simonsen ef at (1998) in the previous section, who showed fractal scaling of stock market indices lIsing a method based on wavelet coefficient scaling (figure 7.10). Davidson ef at (1998) have used the orthogonal dyadic Haar transform to perform semi-nonparametric regression analysis of commodity price behaviour. In their study, they present dyadic scalograms and take account of edge effects in the data. An example of one of their commodity price scalograms is given in figure 7.13. See also Kim (1999) who presented brief results for the Korean stock price index. Ramsey ef at (1995) searched for evidence of self-similarity in the US stock market price index. They investigated the power law scaling relationship between the wavelet coefficients and scale and found some evidence of quasi-periodicity in the occurrence of some large amplitude shocks to the system. In addition, they concluded that there may be a modest amount of predictability in the data, and that it may be more than just a simple Brownian motion. In Ramsey and Lampart (1998), the authors highlighted the importance of Copyright © 2002 lOP Publishing Ltd.

timescale decomposition in analysing economic relationships. Using an SI2 Symmlet they analysed the different relationships that can occur between two economic variables at different levels of decomposition. Wavelet-based methods to remove hidden cycles from within financial time series have been developed by Arino er al (1995). Their methods first decompose the signal into its wavelet coefficients then compute the energy associated with each scale. The dominant scales are defined as those with the highest energies. New coefficient sets are then produced related to each of the dominant scales by either one of two methods developed by the authors. These reapportion the coefficient values according to the scales of the dominant energies. Hence, for the signal containing two dominant scales, two new complete sets of wavelet coefficients are computed. These are then used to reconstruct two separate signals, corresponding to each dominant scale. Arifio and Vidakovich

illustrated their method using a time series of Spanish cement production. This is shown in figure 7.14, where the original time series (shown dashed in the figure) is partitioned into a business cycle component and a seasonal component. A Daubechies D4 wavelet was used in the decomposition of the original time series. Evidence for a cascade mechanism in market dynamics has been found by Arneodo er al (1998d) using wavclet transforms. They suggested that this may be attributed to a variety of mechanisms, including the heterogeneity of traders and their different time horizons causing an 'information' cascade from long to short

timescales, the lag between stock market fluctuations and long-run movements in dividends, and the effect of the release (monthly, quarterly) of major economic indicators which cascades to fine timescales. In a related paper, Muzy er al (2000) have proposed a multi fractal 'stochastic volatility' model that captures the features of financial fluctuations where wavelet transforms are employed to determine the multifractal nature of the time series. A multifractal random walk is then employed as the 'stochastic volatility' model, whose parameters were estimated for real financial data. Aussem er al (1998) have used wavelet-transformed financial data as the input to a neural network which was trained to provide five-days-ahead forecasts for the S&P500 closing prices. They performed the analysis on the wavelet coefficients over the four smallest scales using a B-spline wavelet. In addition, they examined each wavelet series individually to provide separate forecasts for each timescale and recombined these forecasts to form an overall forecast. Morehart er al (1999) have employed wavelets as a spatial analysis tool for economic and financial measures used in agriculture. They illustrated their method using decompositions of wheat dependency data and debt utilization ratio data displayed on two-dimensional maps of the United States. Figure 7.15 shows one of the plots from the study containing the spatial distribution of the decomposition of the debt capacity utilization ratio. Morehart and co-workers recommended the method for the graphical presentation of information, density estimation and wavelet-based non-parametric regression. Matching pursuits have been used by Ramsey and Zhang (1996) to decompose the S&P500 index. The data are characterized by periods of quiet interspersed with intense activity over short periods of time. The authors found that fewer coefficients are required to specify the data than for a purely random signal signifying some form of deterministic structure to the signal. Ramsey and Zhang have also applied matching pursuits to foreign exchange rate data sets, specifically the Deutschmark~US dollar, yen-US Copyright © 2002 lOP Publishing Ltd.

3000

2000

1000

o

1960

1970

1980

1990

Figure 7.14. Spanish cement production. Top, total series (dashed) and business cycle series. Bottom, seasonal component. After Arino et at (1995). Reproduced with kind permission of the authors.

dollar and yen-Deutschmark (Ramsey and Zhang, 1997). Their analysis revealed underlying traits of the signal. However, they went on to state that although most of the energy of the system occurs in localized bursts of activity, there seems to be no way of predicting the occurrence of these random events and hence little opportunity to improve forecasting. Shin and Han (2000) have also investigated exchange rate forecasting using a method which combines wavelet transforms, genetic algorithms and artificial neural networks. They used their method to forecast the daily Korean won/US dollar returns one day ahead of time and found that the genetic algorithm-based wavelet thresholder in their method performed better than three other wavelet thresholding algorithms: cross-validation, best level and best basis. Finally, a method for the filtering out of intraday periodicities in exchange rate time series has been proposed by Gencay el al (2001) using the maximal overlap discrete wavelet transform.

Copyright © 2002 lOP Publishing Ltd.

·

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7.4

Geophysics

We have already covered the application of wavelet analysis to geophysical flows in chapter 4, including canopy flows, cloud formation processes, wind-generated ocean surface waves, and large scale oceanic and atmospheric flow phenomena. In Copyright © 2002 lOP Publishing Ltd.

this section we cover some of the other areas of geophysics that have employed wavelet transform methods including seismology, well logging, topographic feature analysis and the analysis of climatic data. A good place to begin a literature search is with the comprehensive summary paper by Kumar and Foufoula-Georgiou (1997). A more detailed overview can be found in the earlier collection of papers edited by the same authors (Foufoula-Georgiou and Kumar, 1994).

7.4.1

Properties of subsurface media

Il is generally accepted that modern wavelet transform analysis began in the early 1980s with the Morlet wavelet, developed to aid in the interrogation of seismic signals (e.g. Goupillaud el ai, 1984). Seismology can, therefore, be thought of as the birthplace of the modern wavelet transform. Since then, many other wavelets have been developed and used to analyse these and many other signals in geophysics. There are, in fact, large numbers of techniques available to measure the properties of subsurface media including geophysical techniques (seismology, gravimetry, magnetic methods, electrical methods, radioactive methods), ground penetrating radar (GPR) and well logging. A few examples of the wavelet analysis of these signals are covered in this section. The role of wavelets is discussed briefly in the review of recent developments in the processing of seismic data by Talwani and Zeit (1998). The use of orthonormal discrete wavelet transforms as a tool for characterizing seismic time series has been examined by Grubb and Walden (1997). They decomposed seismic signals using both Daubechies wavelets (D4 to D20) and Symmlets (S8 to S20) providing simple illustrative examples of the wavelet decompositions in their explanatory paper. Deighan and Watts (1997) have employed Battle-Lemarie wavelets to filter seismic signals in order to suppress unwanted surface waves (ground roll) present within the signals. Fedorenko and H usebye (1999) have developed a method, which includes the Daubechies D20 wavelet transform, for the automatic detection ofP and S arrival times in seismic records. Li el al (1996) have used a wavelet-based synthesis technique to model fractal-like velocity logs of a zero offset vertical seismic profile. They compared their results with a classical damped least squares method for modelling this type of data and found that the wavelet-based technique reduces significantly the artefacts associated with the classical technique. (Note that in this paper the authors used a 'scale index' which is actually a 'level index' as we have defined it in chapter 3, section 3.3.4.) In a later paper, Li and Ulrych (1999) used wavelet transform-based models of fractal fBm processes (refer back to section 7.2.2) for the modelling and analysis of processes with features of geological and physical interest. Kaleic ef al (1999) have used Coiflets to perform wavelet analysis of acoustic imagery from shallow seismic data taken at the sea floor. By examining the wavelet transform coefficients of the signals at different scales they could highlight different features below the surface such as steel cables and methane gas pockets. Hachiya and Amao (1996) have presented a wavelet-processing method to isolate pertinent features from acoustic images of subsurface media containing buried artefacts. They proposed the use of their technique for separating the desired signal from archaeological acoustic survey data at shallow depth. Bergeron el al (1999) have Copyright © 2002 lOP Publishing Ltd.

frequency (Hz) 0

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10001.02.03.0

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0.50 B

--•• .g

0.75

LOO C 1.25

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(b)

(c)

(d)

Figure 7.16. Decomposition of a seismogram using STFT, CWT and MP. (a) A synthetic seismogram used in the comparison study. (b) A time-frequency decomposition produced by an STFT. (e) A scale-index translation plot produced by the CWT. (d) A time-frequency energy distribution produced by the MP. After Chakraborty and Okaya (1995). Reproduced with the kind permission of the Society of Exploration Geophysicists and the authors.

recently applied three-dimensional Mexican hat wavelet transforms to a threedimensional seismic tomographical modeL They used the maximum wavelet energies and associated wavenumbers as proxy quantities for viewing seismic velocity anomalies (SYAs). They found that the distribution of these wavelet quantities reveals information which is not obvious from direct visual examination of SYAs, e.g. depth extent of tectonic boundaries and inference of plumelike objects. Nagano and Niitsuma (2000) have reported on a Morlct wavelet-based study to measure the dispersion of crack waves in subsurface media. These waves are trapped seismic waves which propagate along fluid-filled crack interfaces and their waveforms (e.g. velocity-frequency dispersion and amplitude-space distribution) are strongly dependent on both the geometry and physical properties of the crack. Chakraborty and Okaya (1995) have compared short term Fourier transforms, continuous (Morlel) wavelet transforms and matching pursuit (MP) decomposition in a study of seismic data. Figure 7.16 shows a synthetic seismogram together with three decompositions using short time Fourier transforms, wavelet transforms and matching pursuits. Comparing the three signals we can see that the MP decomposition localizes the signal events best in the time-frequency plane. Figure 7.17 shows Copyright © 2002 lOP Publishing Ltd.

18

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Figure 7.17. Shot Gather 19, Siljan, Sweden. Reflections A, B, C caused by dolerite sills of 60m, layered 30m and 20m, respectively. After Chakraborty and Okaya (1995). Reproduced with the kind permission of the Society of Exploration Geophysicists and the authors.

a reflection seismogram from Slijan, Sweden. Several major reflections can be identi-

fied in the figure. These reflections, labelled A, Band C in the figure, were identified as being caused by dolerite sills: one of 60 m thickness, a layered 30 m sill and a 20 m sill. Figure 7.18 contains the matching pursuit decomposition for two of the traces (18 and 56) of the seismogram. Their locations are identified in figure 7.17. Four different feature shapes were identified in the matching pursuit time-freq uency plane by the authors. The first type, elliptical shapes elongated in the frequency direction, represent events that are localized in time but contain many frequencies. These are associated with reflections. A second type, elongated in time but narrow in

frequency, is identified with low-frequency surface waves. The third type is circular, representing events which have only one or two frequencies present and exist for

only a short period of time. The fourth type is a long streak in the time direction. This represents a single frequency such as 60 Hz noise which occurs over a long time duration. Figure 7.18(a) contains the MP decomposition of signal 18. Reflected Copyright © 2002 lOP Publishing Ltd.

c;;;;;:J 0.0

o

frequency (Hl) 2S 50 7S 100 125

0.0 S

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LO B

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Figure 7.18. Time-frequency energy distribution of the Siljan data using MPD. (a) Trace 18. (b) Trace 56. (The boxes on top of the time-frequency plots represent the one-dimensional Fourier spectra of the respective traces.) Arter Chakraborty and Okaya (1995). Reproduced with the kind permission of the Society of Exploration Geophysicists and the authors.

events A, Band C can all be identified in the time-frequency plane by elliptical atoms with shorter time axes and longer frequency axes. The circular atoms are caused by random noise present in the data and the vertical peaks are due to shot-generated noise. Events Band C can also be located in figure 7.18(b) for signal 56. Verhelst (1998) has also interrogated seismic data using the matching pursuits method with Gabor atoms and found phase attributes that could be related to facies types in a delta system. The properties of subsurface media, especially the permeabilities of soils and rock, have in many cases been found to be multiseale in nature and hence particularly amenable to both fractal and wavelet analysis. Prokoph and Barthclmes (1996) have used Morlet wavelet transforms both to detect and localize abrupt changes and to differentiate bctween periodic and chaotic cycling sequences in marine sedimentary Copyright © 2002 lOP Publishing Ltd.

TaakenOUersberg 1

.~ =s

trans-/regressions stages

(tr./re.)

polyplocum reo

mucrolUlta tr. pilula tr.

M.Santonian tr. koeneni reo

(llsede tectoev.) Hyphantoceras tr. plenus tr.

80

27.5

6.9 4.8

period (m)

Early Cenomanian tr.

Figure 7.19. Wavelet scalogram (left) of the SP logging data (centre) of borehole Taaken-Ottersberg I near Bremen. After Niebuhr and Prokoph (1997). Reproduced with kind permission of Academic Press Ltd.

successions. They tested their analysis on sample data sets containing periodic and chaotic sequences, before examining gamma-ray well logs from marls of the Cretaceous North German basin. Again using Morlet-based transforms, Niebuhr and Prokoph (1997) investigated self-potential logging data taken from sites in North Germany. Figure 7.19 shows an example of the borehole logging data together with its corresponding scalogram. Features at 80 m, 6.9 m and 4.8 m dominate the scalogram plot. In addition, an abrupt cut-off in the 27.5 m feature in the scalogram can be seen to occur at the Turonian/Conician (Tu/Co) boundary. Using this type of analysis, the authors identified regions of cyclicity and chaos in the borehole data. Prokoph and Agterberg (1999) have further examined the use of wavelet transforms in the detection of cycles, trends and discontinuities in sedimentary successions. Moreau el 01 (1996) proposed a filtering method for non-stationary geophysical data using orthogonal 020 wavelets and a thresholding technique based on chisquared statistics. The authors compared their method with two other thresholding criteria and applied it to both synthetic data and field data composed of thermistance measurements made in a limestone underground quarry. Li (1996) has presented preliminary results concerning the use of continuous complex wavelets in the analysis of synthetic sonic log data. In a later paper, Li (1998d) presented the method in more detail, applying the wavelet transform to both synthetic and real data. He performed Copyright © 2002 lOP Publishing Ltd.

Figure 7.20. Characteristic multiscales of spatial homogeneity of permeability determined by wavelet analysis. The data are from the north-east wall of area 5, pit 3 in the alluvial fan deposit, and were gathered from four sets of vertical transect data: group A at I-12m, group B at 12-24m, group C at 24-36 m and group 0 at 36-52 m. Curves indicate the magnitude of the variability as a function of scale, arrows indicate scale of local maximum variance. Taken from Li B-L and Loehle C 1995 'Wavelet analysis of multiscale permeabilities in the subsurface' Geophysical Research Leffel's 22(23) 3123-3126. Copyright 1995 American Geophysical Union. Reproduced by permission of the American Geophysical Union.

wavelet power spectral analysis of the data sets to reveal the structural properties of the underground heterogeneities. Saito and Coifman (1997) have analysed acoustic well-logging signals using a wavelet dictionary employing two different methods to select the best basis. Their methods allow them to both differentiate between sand and shale and estimate the volume fractions of minerals at each depth. Li and Loehle (1995) have investigated the spatial series of permeabilities taken during a geological cross-section of alluvial fan deposits. They used the wavelet variance of (Mexican hat) transformed data to characterize the multiseale heterogeneity of the subsurface permeabilities. A plot of the wavelet variance for four subsets of the vertical transect permeability data is shown in figure 7.20. The authors argued that the plot shows the heterogeneity of the spatial structure in permeability across scales and between transects. Panda ef af (1996) have applied wavelet transforms and wavelet packets to both one- and two-dimensional permeability data to determine both the location of layer boundaries and other discontinuities and reduce the amount of data required to represent the signal. Figure 7.21 shows their original permeability signal together with a denoised version. The denoising was performed using a wavelet threshold Copyright © 2002 lOP Publishing Ltd.

denoiscd data

25

i l

10

~r~\1,

. . r'~ "\(';.

0.1 original data

25,-----'--=-'-'-'-'------,

llO g &. 0.1 ' - - - - - - - - - " - - - ' - ' - - - - ' - - ' 10 o 30 40 Figure 7.21. Denoising Page sandstone permeability using wal'elet threshold method. After Panda ef al (1996). Copyright 1996 The Society of Petroleum Engineers. Reproduced with kind permission of the Society of Petroleum Engineers.

melhod and allowed the authors to identify the Irue discontinuities in the data. Figure 7.22 shows the original dala set (bottom) containing 4096 data points, together with two reduced permeability data sets. The top plot was reduced to 32 points using a wavelet packet scheme and the middle plot was reduced, again to 32 data points, using a traditional pressure-solver scaleup tool. Comparing the two figures we can see that the wavelet method preserves better the most important features of the permeability field, especially the low and high permeability streaks. Chu (1996) has outlined a technique based on multiresolution analysis for upscaling one- and two-dimensional petroleum rock reservoir parameters for single and multi phase flows. Both Mehrabi and Sahimi (1997) and GUnal and Visscher (1997) have detailed wavelet-based numerical methods for modelling transport in heterogeneous disordered media, and Sahimi (2000) has developed a fractal-wavelet neural-network method that can characterize and model fractured subsurface reservoirs. Fenton (1999) has shown how wavelet coefficient variances may be used to estimate the characteristic parameters in spatially stochastic soil variation. In another study of the spatial variability of soil characteristics, Lark and Webster (1999) used wavelet-based statistics, based on the Oaubechies 04 wavelet, to characterize two soil transects taken from contrasting landscapes. Vermeer and Alkemade (1992) have suggested the use of Mexican hat wavelets for the analysis of well logs. They used the zero crossings in the scalogram to detect edges in the data which they interpreted as boundaries between different geological units. In this way they were able to perform a multiscale segmentation of gamma-ray well log data. 7.4.2

Surface feature analysis

We have already considered the use of wavelets in the analysis of engineering surfaces in chapter 5 and fractal surfaces earlier in this chapter. The analysis of geophysical Copyright © 2002 lOP Publishing Ltd.

r>

wavelet solved results, number of points 32 ,,------=-:.::.:.:::.==::.:..:=====-:::...!:.::.:.:=--=-=---:=l =

20

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raw data, number of points 4098

n

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20 1

o

10

20 depth (m)

30

40

Figure 7.22. Comparing wa\'elet-based local variability preserving scaleup results with pressure solver scaleup data. Fine grid =4096, coarse grid = 32. After Panda el 01 (1996). Copyright 1996 The Society of Petroleum Engineers. Reproduced with kind permission of the Society of Petroleum Engineers.

surface feature distributions is carried out for a number of reasons including the monitoring of pollution, vegetation growth, the form and spread of towns and cities or the analysis of natural topographic features, The topography of the surface also affects the fluid dynamic properties of the fluid-surface boundary layer (oceanic and atmospheric). The interannual variability of normalized difference vegetation index (NOVI) data has been interrogated by Li and Kafatos (2000) using an S8 Symmlet wavelet decomposition, They found a possible relationship between the £1 Nino/Southern Oscillation Index and an NOVT variation signal from an II-year data set. They proposed the variability of the NOVT as a good proxy for climate variations, Bradshaw and Spies (1992) have considered both Haar and Mexican hat wavelets in a study to characterize the nature of canopy gaps within forests, They used wavelet variance to identify the dominant scales in forest canopy structure. In another study of the response of vegetation to landscape structure, Brosofske et al (1999) used Mexican hat wavelets to interrogate two diversity indices, Plotting the wavelet variance resulting from the transformed data, they found that quite different scales were dominant in the two indices, A fractal-wavelet approach has been employed by Solka et al (1998) for the identification of man-made regions in unmanned aerial vehicle imagery, Adolphs (1999) has employed a number of techniques including fractal and wavelet methods to analyse the roughness variability of sea ice and snow cover thickness profiles, The study found that wavelets proved to be superior Copyright © 2002 lOP Publishing Ltd.

to visual inspection in objectively finding hidden segments and substructures within the profiles. The scale analysis of the surface properties (surface temperature and albedo) of sea ice has been carried out by Lindsay et al (1996) using the discrete wavelet transform. The techniques developed in this earlier investigation were later used in a study of the temporal variability of the energy balance of thick Arctic pack ice by Lindsay (1998). Using the Morlet wavelet, Little et al (1993) have detected a 200 km long anomalous topographic zone in a 1600 km bathymetric profile, which they suggest is the site of a short-lived, abandoned spreading centre associated with a thermal topographic swell. Also using a Morlet wavelet, Cazenave et al (1995) have analysed medium-wavelength ('"'-'IOOOkm) geoid anomalies over the central Pacific. In an early paper on the application of wavelet transforms, Teti and Kritikos (1992) analysed azimuth cuts taken from two different SAR ocean images using Morlet wavelets in both a running discretized form and a tight frame form. More recently, Fukuda and Hirosawa (1999) investigated the smoothing effect of wavelet-based speckly filtering of SAR images, and Ferretti et al (1999) have exploited multibaseline SAR interferometry using a wavelet base weighted average method for the reconstruction of high-quality digital evaluation models. The method leaves important peaks in the signal much more intact than traditional highpass filtering. Simhadri et al (1998) have developed a wavelet-based feature extraction computational scheme to extract fine edges from oceanographic images. Wang et al (1999) have developed a wavelet-based technique (using a redundant tight wavelet frame) for the removal of clouds and their shadows from Landsat TM images. Ranchin and Wald (1993) have reviewed the application of wavelet transforms to remotely sensed images (specifically SPOT HRV panchromatic images). Later, Wald and Baleynaud (1999) employed wavelet-based techniques in a study of air quality in cities using Landsat thermal infrared data. Zhou et al (1998) used a multiresolution wavelet-based method to merge two types of remote sensing image (SPOT panchromatic images and Landsat TM images). They did this to produce a hybrid image containing both good terrain detail and useful spectral information. There are many other papers in the literature concerning the fusion of mulitsensor geophysical data images using wavelet methods. See for example Nunez et al (1999), Zhukov et al (1999), Ranchin and Wald (1997) and the references contained therein. 7.4.3

Climate, clouds, rainfall and river levels

An introductory tutorial on the use of wavelet transforms in climatic time series analysis is given by Lau and Weng (1995). In their paper, an analogy between the wavelet transform representation and music is used to illustrate the difference between the local and global information in climatic time series before applying the methods to Northern hemisphere surface temperature measurements (over a period of 140 years) and a deep sea sediment record (over 2.5 million years). Baliunus et al (1997) have employed both Mexican hat and Morlet wavelets in a study of temperature data taken over three centuries in central England. The better temporal resolution of the Mexican hat is put to use in filtering trends in the time domain, whereas the superior frequency resolution of the Morlet wavelet is used for spectral Copyright © 2002 lOP Publishing Ltd.

studies of the data. In a paper covering the monitoring and analysis of multiple microclimate variables for characterizing the ecology of the physical environment, Chen et al (1999b) used Mexican hat wavelets to look at the scaling in temperature and overstorey cover at ground surface along a transect within a pine forest. Hu and Nitta (1996) used Mexican hat wavelets to detect dominant timescales in the rainfall records over North China and India over the period 1891-1992. They found that the dominant timescales in the rainfall variations are located in two timescale bands: shorter than 10 years and 14-28 years. Fraedrich et al (1997) have compared a number of analysis techniques to identify abrupt climate changes from historic time series of the flood levels of the river Nile. They found that a multiscale moving t-test has advantages over the wavelet transform (using the first derivative of Gaussian wavelet) in detecting and specifying the degree of significance of the abrupt climate change. Labat et al (2000) have provided a well illustrated, comprehensive account of the use of a variety of wavelet methods to analyse rainfall-runoff relationships from karstic springs. They employed both discrete and continuous wavelet tools in their investigation including wavelet spectra and cross-spectral analysis, wavelet coherence, hard thresholding and multiresolution covariances. They studied periodic pumping and intermittent runoff processes in two separate karstic spring outflow data sets. The wavelet methods allowed them to separate different subprocesses connected with the data. They then analysed rainfall and springflow rates in three karstic basins and found that the wavelet cross-analyses gave meaningful information on the temporal variability of the rainfall-runoff relationship. Jay and Flinchem (1997) used continuous wavelet analysis to interrogate the modulation of the external tide in a river by variations in streamflow. They found the wavelet transform method superior to Fourier and harmonic techniques for studies of nonstationary river tides because of its time-frequency resolution properties. Nakken (1999) used Morlet wavelet transforms to investigate the rainfall and runoff records of a river catchment in order to differentiate components due to climate change from those due to man-made effects (e.g. land use changes). The method is proposed for the detection of streamflow response to climate change, especially large-scale circulation phenomena. Serio and Tramutoli (1995) found evidence for the existence of scaling laws in a cloud system generated and advected by strong baroclinic instability. Using twodimensional Mexican hats to decompose infrared cloud images collected by satellite, they were able to discern two scaling behaviours in this type of frontal instability. Using purpose-built wavelets whose shape was inspired by the general shape of the signatures of the aircraft echoes within the time series, Boisse et al (1999) have developed an automated wavelet-based method for removing aeroplane and other transitory echoes in strato-tropospheric (ST) radar measurements. Grecu and Krajewski (1999) have used wavelet and fractal preprocessing techniques within a neural network-based methodology for the detection of anomalous propagation echoes in weather radar data. Low total ozone events over northern Sweden have been analysed using Mexican hat wavelet transforms by Weinberg et al (1996). See also the interesting discussion on the merits of the wavelet-based analysis of the data ensuing from this paper involving Flynn and Celarier (1997) and Weinberg et al (1997). Kumar and Foufoula-Georgiou (1993) have decomposed two-dimensional rainfall Copyright © 2002 lOP Publishing Ltd.

fields using two-dimensional orthogonal Haar wavelets in a multiresolution representation. They proposed the use of wavelets as a consistent method for the decomposition of inhomogeneous and anisotropic rainfields. In a later paper (Kumar and Foufoula-Georgiou, 1996a), the authors established empirical connections between wavelet-based statistical characteristics of the rainfall data and physical storm characteristics. The results from this work were then used to develop a model for the disaggregation of spatial rainfall based on a coupling between its meteorological and scaling descriptions (Kumar and Foufoula-Georgiou, 1996b). Other studies in this area include those by Gollmer et al (1995) who used Daubechies D8 wavelets to analyse liquid water path (LWP) data in a study of marine stratocumulus cloud inhomogeniety, Tian et al (1999) who employed wavelet packets in a neural network study of cloud classification, and Chapa et al (1998) who used Morlet wavelet transforms to interrogate cold cloud index data in order to detect periodicities in convective activity in South America.

7.5

Other areas

7.5.1

Astronomy

There has been a recent surge in the use of wavelet transforms for the analysis of astronomical data, both time series and spatial data. Aschwanden et al (1998) have performed a multiresolution analysis of hard x-ray time series of solar flares. Their analysis allowed them to determine the shortest timescales associated with strong, smoothly varying, and weak flares. Furthermore, the fastest significant time structures (in strong flares) were found to be related to physical parameters of propagation and collision processes. Morlet wavelets were used by Townsend (1999) to analyse the line-profile variations of rapidly rotating stars exhibiting non-radial pulsation. Hughes et al (1998) used Morlet wavelets to analyse over 20 years of radio flux data from the astronomical object BL Lac object OJ 287. They found modulation of the total flux over the whole time period. However, this modulation changed its period midway through the time series. This behaviour is explained through a 'shock in jet' model by the authors. Morlet wavelets have also been used by Lucek and Balogh (1998) in a study of Alfvenic fluctuations in solar wind data. Bedding et al (1998) have reported on a wavelet-based investigation of the switching between two pulsation modes of a star. The switching of the pulsation modes between 332 and 175 days is clearly seen in their transform plots associated with the light curve time series. Bijaoui et al (1996) used discrete wavelet transforms to provide a new statistical indicator of cosmological distributions according to scale. The spatial structure in cosmic microwave background (CMB) maps has been assessed by Tenorio et al (1999) using spherical Haar wavelets. They went on to use the soft-thresholding of planar Daubechies wavelets coefficients to denoise the map: both to remove local point sources and to suppress (nonlocal) instrument noise. Sanz et al (1999) used a variety of two-dimensional transforms together with wavelet thresholding to denoise CMB anisotropy maps with added noise at different signal to noise ratios and then compared their results to other methods of filtering. Pagliaro et al (1999) have Copyright © 2002 lOP Publishing Ltd.

analysed the substructure of clusters of galaxies using three-dimensional cubic B-spline wavelets. Fang et al (1998) considered the problem of galaxy clustering. They used Daubechies D4 wavelets to investigate the bias associated with scale dependence of the clustering associated with galaxies. Pando and Fang (1998) have developed a discrete wavelet spectrum estimator which they have used to analyse the power spectrum of the spatial distribution of Lya clouds. Meiksin (2000) employed Daubechies D20 wavelets to provide a statistical characterization of the absorption properties of the Lya forest. The method was applied to the Keck HIRES spectrum of Q1937-1009 and proposed as an easily automated procedure for basing a comparison between measured and predicted properties of the Lya forest. Rabadi and Myler (1998) used an image of Saturn to illustrate their fast wavelet-based algorithm for the image reconstruction from Fourier domain information. Finally, Aussem and Murtagh (1997) have employed neural networks to predict sunspot time series using wavelet decomposition. 7.5.2

Chemistry and chemical engineering

A good place to begin the search for information concerning the use of wavelet analysis in chemistry is with the review paper by Leung et al (1998). This comprehensive review covers the application of wavelet transform techniques in chemical analysis from 1989 to 1997 and contains 130 references to other articles in the field. The authors cover a variety of topics in chemistry including flow injection analysis, chromatography, infrared spectroscopy, mass spectrometry, nuclear magnetic resonance and ultraviolet-visible spectroscopy, voltammetry, quantum chemistry and chemical physics. The paper by Alsberg et al (1997) provides a comprehensive overview of the wavelet transform targeted at chemometricians. It gives a brief history of the wavelet transform before tackling both the continuous and discrete wavelet transform, including wavelet packets, and their applications. Depczynski et al (1997) provide a brief but clear introduction to the discrete wavelet transform (and associated multiresolution analysis) and its use in the analysis of chemical signals. See also the paper by Shao and Cai (1998). In a more general review of chemometrics, Wold and Sjostrom (1998) have highlighted the wavelet transform as an emerging tool for dealing with large sets of very similar variables (from spectra to chromatograms). More specific studies concerning wavelet transforms in chemistry and chemical engineering include: the use of spline wavelets by Zheng et al (1999b) in the study of electrochemical signals; the analysis of flow regime in gas-liquid bubble columns by Bakshi et al (1995) using wavelet packets; wavelet-based methods for process monitoring developed by Shao et al (1999) and Tsuge et al (2000); an integrated wavelet and neural network-based framework for process monitoring and diagnosis by Zhao et al (1998), Chen et al (1998, 1999a), Wang et al (1999) and Yang et al (2000); a brief mention of the role of wavelet transforms in the treatment of noisy electrochemical data by Cottis et al (1998); the analysis of magnetic resonance spectroscopy data using continuous wavelets by Ding and McDowwell (2000) and Serrai et al (2000); the application of wavelet-based neural networks to hybrid modelling and optimization of a chemical process by Safavi et al (1999); the wavelet-based error control and adaptation strategy within the simulation of Copyright © 2002 lOP Publishing Ltd.

multicomponent mixture processes by Briesen and Marquardt (2000); the use of discrete wavelets by Stephanopoulos et al (1997) in the mining of fermentation databases; the scale and magnitude thresholding (which they call smoothing and denoising respectively) of experimental spectra using wavelet and other transforms by Barclay et al (1997); and the numerical procedure based on wavelet collocation suggested by Liu et al (2000a) for the solution of models for packed-bed chemical reactors and chromatograph columns. 7.5.3

Plasmas

There are many papers concerning the use of wavelet transforms in this area. Some recent ones include: the studies by Dose et al (1997) on fusion plasma transients; Santoso et al (1997) on nonstationary plasma fluctuations; van Milligan (1997) on the nature of plasma edge turbulence; Dong et al (1998) on coherent structures within tokamak plasma turbulence; Bruskin et al (1998) on the reconstruction of the plasma density profile using wavelet analysis of microwave reflectometer signals; Bruskin et al (1999) on plasma fluctuations using wavelet spectra and cross spectra; Heller et al (1999) on scrape-off layer intermittency in the Castor tokamak using wavelet bicoherence; and Jakubowski et al (1999) on the analysis of plasma fluctuation measurements using beam emission spectroscopy. It is interesting to note that most of these studies employed the Morlet wavelet. 7.5.4

Electrical systems

The use of wavelets in the detection of transmission line transients in power cables has been investigated by a number of researchers. A brief introductory tutorial intended for power system engineers is provided by Kim and Aggarwal (2000). Goswami (1998) has used semi-orthogonal wavelets to evaluate the reflection coefficient for open/short transmission lines. A wavelet transform approach to the solution of multiconductor transmission line equations has been developed by Raugi (1999). Huang et al (1999b) employed a Morlet wavelet transform approach to detect various simulated power system disturbances, including voltage sag, voltage swell, momentary interruption and oscillatory transients. Zhao et al (2000) have also proposed a wavelet-based approach to fault detection and classification in power cable systems. Their system, however, uses a multiresolution algorithm based on discrete orthonormal wavelets such as Daubechies, Coiflets and Symmlets. Both wavelet packets and continuous Morlet wavelet transforms have been used by Pham and Wong (1999) to perform harmonic analysis of power system waveforms. They validated their approach on both synthesized waveforms and power system waveforms measured in the Western Australia system. A discrete wavelet transform-based feature extraction technique for the discrimination between internal faults and inrush currents in power transformers has been developed by Mao and Aggarwal (2000). A method for the reliable analysis of ultrawide-band detected partial discharge (PD) currents has been developed by Angrisani et al (2000) based on the Daubechies Dl6 wavelet. They illustrated their method on experimental PD signals. Morency and Lemay (1999) have described a method based on the Daubechies D4 Copyright © 2002 lOP Publishing Ltd.

wavelet for the detection of erroneous data points (spikes) present in a signal sampled from an analogue-to-digital convertor. 7.5.5

Sound and speech

Early papers by Grossman et al (1987) and Kronland-Martinet (1988) discuss the use of the Morlet wavelet in the analysis of speech and music sounds. Guillemain and Kronland-Martinet (1996) have presented techniques for the estimation of relevant parameters from sound signals such as frequency and amplitude modulation corresponding to each spectral component of the sound. Pielemeier et al (1996) provide a review of time-frequency analysis of musical signals which includes wavelet transforms among a variety of other techniques and, more recently, Fujinami (1998) has described a wavelet analysis of a sound image localization transfer function in headphone listening which reveals important features for spatial hearing. The compression of high quality audio signals using wavelet packets is described by Sablatash and Cooklev (1996) within a more general treatise on the subject. A Daubechies wavelet-based method for detecting transient underwater signals from within a changing background sound environment has been developed by Bailey et al (1998). They applied their method to the detection of dolphin sounds-clicks and whistles-from within noisy sound data segments. Quyen et al (1998) have developed a method based on the Daubechies D4 wavelet for the classification of underwater mammals (whales and porpoises). Speech recognition is tackled with Haar wavelet transforms by Uchaipichat and Parnichkun (2000) in work which combines wavelet decomposition and neural network classification. Weiss and Dixon (1997) have outlined a wavelet-based denoising method for the removal of unwanted backscatter from high-frequency underwater acoustic signals. Unoki and Akagi (1999) have addressed the problem of extracting a desired acoustic signal from a noisy signal using wavelet analysis. Kermit and Eide (2000) have given details of an audio signal identification method in which Haar wavelets are used to preprocess audio signals (speech and music) prior to identification using a neural network. Obaidat et al (1999) have described a wavelet-based method for the detection of pitch period, an important parameter in the design of automatic speaker recognition systems (see also Obaidat et aI, 1998). A text-to-speech synthesis system is described by Kobayashi et al (1998) which incorporates discrete wavelet analysis in the determination of pitch period. Sarikaya and Hansen (2000) have investigated the impact of stress on monophone speech recognition accuracy and proposed a new set of acoustic parameters based on high resolution wavelet analysis. A method for the encoding and enhancement of wideband speech signals using wavelet packets has been described by Carnero and Drygajlo (1999). Mikhael and Ramaswamy (1995) used speech signals to illustrate a method of signal representation using mixed transforms and Singh et al (1997) have selectively enhanced acoustic events in speech signals using wavelet subtransform domains. Pinter (1996) has given a summary account of many of the types of wavelets that have been employed in the representation of speech signals including the Morlet, Haar, Le Marie-Meyer and Malvar wavelets, together with wavelets derived from biophysical models of the inner ear and wavelets with psychoacoustical features. Copyright © 2002 lOP Publishing Ltd.

Pinter then described the development of perceptual wavelets and their application to speech enhancement. 7.5.6

Miscellaneous

In an introductory review of patterns that occur in extended dissipative systems,

including sand waves in the desert, the stripe patterns of fish, Rayleigh-Benard convection in fluids and magnetic domains, Bowman and Newell (1998) have suggested the continuous wavelet transform as a useful tool for the extraction of order parameters from these patterns. Li et al (1996) have provided a review of the application of wavelet transforms to optics and Gharbi and Barchiesi (2000) have used Daubechies wavelets in a method to characterize the local spatial frequency separation in near-field microscopy. Wen et al (1996) and Deng et al (1999) have detailed wavelet-based approaches to the verification of handwritten signatures. Other applications of wavelets include the digital watermarking for tamper-proofing of images (Kundur and Hatzinakos, 1999; Tsai et ai, 2000); the use of discrete wavelet analysis for pattern generation in architectural design (Sariyildiz et ai, 1998); the automatic detection of incidents on motorways (Cohen and ling, 1995); the study of intermittency in discrete dynamical systems, namely the logistic map (Figliola and Schuschny, 1995); the analysis of the three-dimensional topography of a clamshell surface (Toubin et ai, 1999; Diou et ai, 1999); the tracking of moving targets using multiresolution analysis (Hong, 1999) and matching pursuits (Chen and Ling, 1999); the construction of a Lorentz-covariant superposition oflight waves (Han et ai, 1995); lip reading using both Coiflet wavelets and Fourier methods (Yu et ai, 1999) and automatic human face recognition and expression analysis using both wavelets (Wiskott, 1999; Kondo and Yan, 1999; Lien et ai, 2000) and wavelet packets (Garcia et ai, 2000).

Copyright © 2002 lOP Publishing Ltd.

Appendix

Useful books, papers and websites

This appendix aims to provide the reader with a shortlist of useful books, papers and websites. They have been selected by the author for their extensive content and/or clarity of presentation.

1 Useful books and papers There are a large number of wavelet books in the literature, from those aimed at a mathematical audience to those which deal with a specific scientific discipline. The mathematical and statistical literature is greatly weighted towards the discrete orthonormal wavelet transform and associated transforms, e.g. the nondecimated discrete wavelet transform, due in part to its nice mathematical properties. However, the applied scientific literature is more balanced between the continuous and discrete wavelet transforms. There are many good texts available covering the background theory or specific applications. However, this section is restricted to three texts. •

• •

The World According to Wavelets. In her book Hubbard (1996) provides an excellent account of the history and use of the wavelet transform. As it subtitle says, the book is 'the story of a mathematical technique in the making'. Much of the text is written using no mathematics at all, and where mathematical explanations of the concepts are employed to convey some of the concepts, the treatment is minimal. Ten Lectures on Wavelets. By Ingrid Daubechies (1992), this is one of the first wavelet texts and has become a standard in the field. A Wavelet Tour of Signal Processing. A more recent text by Mallat (1998), this book provides much of the mathematical detail underlying useful wavelet transform techniques used in practice.

There are a number of papers in the literature which transcend their target audience to provide a lucid explanation of some aspect or aspects of wavelet transform theory which can be accessed by a wider audience. Some of these papers together with a brief description of their contents are listed below: Copyright © 2002 lOP Publishing Ltd.

•

•

Kumar and Foufoula-Georgiou (1994): This is the introductory paper in the book edited by Kumar and Foufoula-Georgiou. It provides a concise account of both the continuous and discrete wavelet transforms. Alsberg el al (1997): Targeted at chemometricians, this comprehensive paper

contains some historical background, some basic theory of the continuous and discrete wavelet transforms and wavelet packets before ending with applications including denoising, baseline removal and compression. The appendix contains

•

some useful information too, including details of the HYBRID and VISU thresholding methods. Labat el al (2000): This is a well illustrated, comprehensive account of the use ofa variety of continuous and discrete wavelet methods (e.g. wavelet spectra and cross-spectral analysis, wavelet coherence, hard thresholding, multiresolution

•

covariances) in the analysis of rainfall-runoff relationships. Wong and Chen (2001): This paper concerns the analysis of multidegree-of-freedom systems exhibiting nonlinear and chaotic behaviour using the Morlet wavelet transform. The paper uses a large number of clearly presented examples of the modulus and phase plots for the Morlet wavelet transform of both test signals and the nonlinear signals under consideration.

•

Jawreth and Sweldens (1994): This overview paper is packed full of information concerning the continuous and discrete wavelet transform.

•

Williams and Armatunga (1994): A good place to find out more practical information on the implementation of the multiresolution algorithm. The paper makes explicit the mathematics involved with signal decomposition and reconstruction.

• •

•

Mallat (1989): The original multiresolution paper. Very insightful. Meneveau (199Ia): Although this paper is aimed at the fluid mechanic community, it contains a lot of useful information concerning the discrete orthonormal wavelet transform. Abramovich el al (2000): This is a good place to begin a literature search on the statistical application of wavelet transforms. Among other things, it contains a

concise overview of the various methods of wavelet thresholding.

2

Useful websites

Typing 'wavelet' into a search engine should produce a very large number of sites containing wavelet material. A brief list of useful websites is given below with a short note on the contents of each site. This is not a comprehensive list, but rather has been compiled to give the reader some good places to begin a search. Most of them contain a large number of hyperlinks to other useful sites. These sites were all active at the time of writing.

• http://www.mathsoft.com/wavelets.html This site contains (or rather is) a very large list of other websites which contain papers for download covering introductory material, applications and theory. The list is fairly comprehensive and it is a very good site from which to begin a search for information. Copyright © 2002 lOP Publishing Ltd.

•

http://www.wavelet.org/wavelet/index.html

This is the website for Wavelet Digest, the e-mail periodical which contains news and

views from the wavelet community including details of new books and papers, recent theses, software, courses and conferences, jobs and not least a section devoted to questions and answers from the subscribers. It contains a lot of useful information

for both novice and expert. You can sign up to Wavelet Digest from this site. The newsgroup is open to all and the questions posed range from the very basic to the thought provoking. • http://paos.colorado.edu/research/wavelets/ Christopher Compo and Gilbert Torrence's site. This is a very nicely presented site. It has an interactive bit where you can submit your own data for wavelet analysis. The answers to frequently asked questions and software are also available. • http://www.mame.syr.edu/faculty/lewalle/wavelets.htmI Jacques Lewalle's site-the continuous wavelet transform is explained through many illustrations. http://www.public.iastate.edu/~rpolikarfWAVELETS/waveletindex.htm I • Robi Polkar's site-lots of well illustrated introductory information.

•

http://www.amara.com/curren t/wa velet.html

Another good place to begin. • http://www.stat.,ta nford.ed u/ ~wa velab/ Site containing WA VELAB software-a long list of routines for MATLAB@

Copyright © 2002 lOP Publishing Ltd.

References

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