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ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 118

EDITOR-IN-CHIEF

PETER W. HAWKES CEMESÑCentr e National de la Recherche ScientiÞque Toulouse, France

ASSOCIATE EDITORS

BENJAMIN KAZAN Xerox Corporation Palo Alto Research Center Palo Alto, California

TOM MULVEY Department of Electronic Engineering and Applied Physics Aston University Birmingham, United Kingdom

Advances in

Imaging and Electron Physics EDITED BY

PETER W. HAWKES CEMESÑCentr e National de la Recherche ScientiÞque Toulouse, France

VOLUME 118

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∞ This book is printed on acid-free paper.  C 2001 by ACADEMIC PRESS Copyright 

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CONTENTS

CONTRIBUTORS . . . . . . . . . . . . . . . . . . . . . . . . . . PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FUTURE CONTRIBUTIONS . . . . . . . . . . . . . . . . . . . . . .

vii ix xi

Magnetic Resonance Imaging and Magnetization Transfer JOSEPH C. McGOWAN

I. II. III. IV. V. VI.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . Development of Magnetization Transfer Theory . . . . . . . . . . Magnetization Transfer Imaging . . . . . . . . . . . . . . . . Application in Human Studies . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . Appendix I: Solution of the Complete Coupled Bloch Equations for Two-Site Chemical Exchange . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

2 12 21 54 65 77 78 80

Noninterferometric Phase Determination DAVID PAGANIN AND KEITH A. NUGENT

I. II. III. IV. V. VI.

Introduction and Overview . . . . Methods of Phase Imaging . . . . A New Approach to Phase . . . . Propagation-Based Phase Recovery Experimental Demonstrations . . Conclusion . . . . . . . . . . References . . . . . . . . . .

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86 87 93 99 108 122 123

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129 130 151 191

Recent Developments of Probes for Scanning Probe Microscopy EGBERT OESTERSCHULZE

I. Introduction . . . . . . . II. Atomic Force Microscopy . III. Near-Field Optics . . . . References . . . . . . .

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vi

CONTENTS

Morphological Image Enhancement and Segmentation IVAN R. TEROL-VILLALOBOS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . II. Some Basic Tools in Mathematical Morphology . . . . . . . . . . III. Morphological Nonincreasing Filters Using Gradient Criteria (Morphological Slope Filters) . . . . . . . . . . . . . . . . . IV. A Sequential Family of MSFs . . . . . . . . . . . . . . . . . V. Image Segmentation using MSFs . . . . . . . . . . . . . . . . VI. Nonlinear Multiscale Approach Using a Sequential Family of MSFs . VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

208 210

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

275

214 229 235 248 271 272

CONTRIBUTORS

Numbers in parentheses indicate the pages on which the authorsÕcontributions begin.

JOSEPH C. MCGOWAN (1), United States Naval Academy, Annapolis, Maryland 21402 KEITH A. NUGENT (85), School of Physics, The University of Melbourne, Victoria 3010, Australia EGBERT OESTERSCHULZE (129), Institute of Technical Physics, University of Kassel, 34132 Kassel, Germany DAVID PAGANIN (85), School of Physics, The University of Melbourne, Victoria 3010, Australia IVAN R. TEROL-VILLALOBOS (207), Centro de Investigacion y Desarrollo Technol« ogico en Electroqu«õmica, Parque Technol« ogico Queretaro S/N. Sanfandila-Pedro Escobedo.CP, 76700-APDO 064 Queretaro, Mexico

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PREFACE

The four chapters that make up this volume are all in the general area of imaging and image processing. We begin with an account of magnetic resonance imaging, which older readers will think of as nuclear magnetic resonance imaging, and of the related technique of magnetization transfer imaging. In this, J. C. McGowan Þrst describes in detail the physics of the magnetic resonance imaging process and then goes on to discuss the recently developed technique of magnetization transfer imaging. The purpose of this is to obtain information about the interactions between water protons, which are visible using magnetic resonance imaging, and the protons of larger molecules that are of physiological interest. Among the applications areas are multiple sclerosis and other diffuse brain disorders. The chapter can be read at several levels since it contains all the technical details that will interest the specialist and, in parallel, a very readable commentary that can be appreciated by those from other Þelds. Next comes a very welcome account of the highly original work of D. Paganin and K. A. Nugent on phase determination by noninterferometric methods. My attention was caught by their paper in Phys. Rev. Letters in 1998, in which their ideas on phase determination were Þrst sketched and I am delighted that they have agreed to write this full account, which puts the problem in context, explains clearly the basis of their approach, and contains much new material. A particularly interesting feature of this work is the role occupied by generalized radiance and the associated problems of radiometry for partially coherent radiation (cf. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge 1995). Both theory and practice are examined here and this account of these subtle ideas should render them much more accessible. Scanning probe microscopy is still a young subject and is in rapid growth, with continuing new developments in instrumentation and experimental techniques. The chapter by E. Oesterschulze Þrst discusses developments in atomic force microscopy and then turns to near-Þeld optics. The Þrst part covers essentially the technological aspects of these microscopes; the section on near-Þeld microscopy opens with a succinct but very clear account of far-Þeld optics, so that we can appreciate the difference between this and the newer near-Þeld instruments. Passive probes are then examined after which E. Oesterschulze introduces us to light-emitting and light-detecting active probes. Altogether a very full account of present preoccupations in this area.

ix

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PREFACE

The Þnal chapter, by I. R. Terol-Villalobos, is a new addition to the numerous articles published here on aspects of mathematical morphology. Here, the theme is selective enhancement and segmentation based on a type of gradient Þlters, in which the gradient is the difference between the image and the same image after erosion or dilation. After a brief introduction, in which the toggle mappings are examined, the morphological slope Þlters are deÞned and analyzed at length. This long contribution forms a short monograph on this branch of mathematical morphology. I am most grateful to all the contributors to this volume for the care that they have brought to their manuscripts and conclude with a list of surveys planned for the next few volumes. Peter Hawkes

FUTURE CONTRIBUTIONS

T. Aach Lapped transforms G. Abbate New developments in liquid-crystal-based photonic devices S. Ando Gradient operators and edge and corner detection A. ArnŽodo,N. Decoster, P. Kestener, and S. Roux A wavelet-based method for multifractal image analysis D. Antzoulatos Use of the hypermatrix M. Barnabei and L. Montefusco Algebraic aspects of signal and image processing L. Bedini, E. Salerno, and A. Tonazzini (vol. 120) Discontinuities and image restoration C. Beeli Structure and microscopy of quasicrystals I. Bloch Fuzzy distance measures in image processing R. D. Bonetto (vol. 120) Characterization of texture in scanning electron microscope images G. Borgefors Distance transforms A. Carini, G.L. Sicuranza, and E. Mumolo V-vector algebra and Volterra Þlters Y. Cho Scanning nonlinear dielectric microscopy E. R. Davies Mean, median, and mode Þlters H. Delingette Surface reconstruction based on simplex meshes xi

xii

FUTURE CONTRIBUTIONS

A. Diaspro Two-photon excitation in microscopy R. G. Forbes Liquid metal ion sources E. Fšrster and F. N. Chukhovsky X-ray optics A. Fox The critical-voltage effect L. Frank and I. MŸllerov« a Scanning low-energy electron microscopy A. Garcia A brief walk through sampling theory L. Godo & V. Torra Aggregation operators P. Hartel, D. Preikszas, R. Spehr, H. Mueller, and H. Rose (vol. 120) Design of a mirror corrector for low-voltage electron microscopes P. W. Hawkes Electron optics and electron microscopy: conference proceedings and abstracts as source material M. I. Herrera The development of electron microscopy in Spain J. S. Hesthaven Higher-order accuracy computational methods for time-domain electromagnetics K. Ishizuka Contrast transfer and crystal images I. P. Jones ALCHEMI W. S. Kerwin and J. Prince The kriging update model B. Kessler Orthogonal multiwavelets G. Kšgel Positron microscopy

FUTURE CONTRIBUTIONS

W. Krakow Sideband imaging N. Krueger The application of statistical and deterministic regularities in biological and artiÞcial vision systems B. Lahme KarhunenÐLoeve decomposition J. Marti (vol. 120) Image segmentation C. L. Matson Back-propagation through turbid media S. Mikoshiba and F. L. Curzon Plasma displays M. A. OÕKeefe Electron image simulation N. Papamarkos and A. Kesidis The inverse Hough transform M. G. A. Paris and G. dÕAriano Quantum tomography C. Passow Geometric methods of treating energy transport phenomena F. A. Ponce Nitride semiconductors for high-brightness blue and green light emission T.-C. Poon Scanning optical holography H. de Raedt, K. F. L. Michielsen, and J. Th. M. Hosson Aspects of mathematical morphology H. Rauch The wave-particle dualism D. Saad, R. Vicente, and A. Kabashima Error-correcting codes O. Scherzer Regularization techniques

xiii

xiv

FUTURE CONTRIBUTIONS

G. Schmahl X-ray microscopy S. Shirai CRT gun design methods T. Soma Focus-deßection systems and their applications I. Talmon Study of complex ßuids by transmission electron microscopy M. Tonouchi Terahertz radiation imaging N. M. Towghi Ip norm optimal Þlters T. Tsutsui and Z. Dechun Organic electroluminescence, materials and devices Y. Uchikawa Electron gun optics D. van Dyck Very high resolution electron microscopy J. S. Walker Tree-adapted wavelet shrinkage C. D. Wright and E. W. Hill Magnetic force microscopy F. Yang and M. Paindavoine Pre-Þltering for pattern recognition using wavelet transforms and neural networks M. Yeadon Instrumentation for surface studies S. Zaefferer Computer-aided crystallographic analysis in TEM

ADVANCES IN IMAGING AND ELECTRON PHYSICS VOLUME 118

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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 118

Magnetic Resonance Imaging and Magnetization Transfer JOSEPH C. McGOWAN United States Naval Academy, Annapolis, Maryland 21402

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Fundamentals of Magnetic Resonance Imaging . . . . . . . . . . . B. Spin Flips and Relaxation . . . . . . . . . . . . . . . . . . . . C. Three Fundamental Signals in Magnetic Resonance . . . . . . . . . . II. Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . A. Field Gradients and Slice Selection . . . . . . . . . . . . . . . . B. Imaging with a Spin Echo Technique . . . . . . . . . . . . . . . . C. Contrast in the MR Image . . . . . . . . . . . . . . . . . . . . D. Gradient Echoes and Rapid Imaging Techniques . . . . . . . . . . . III. Development of Magnetization Transfer Theory. . . . . . . . . . . . . A. The Bloch Equations . . . . . . . . . . . . . . . . . . . . . . B. The Chemical Exchange Model . . . . . . . . . . . . . . . . . . C. Investigation of Magnetic Exchange with Double Resonance . . . . . . D. Magnetization Transfer between Unresolvable Spins . . . . . . . . . E. Analytical Models for Magnetization Transfer . . . . . . . . . . . . F. Analytic Solutions of Coupled Bloch Equations . . . . . . . . . . . G. Analytic Solution of SimpliÞed Bloch Equation Sets . . . . . . . . . H. Comparison of Predicted Z-Spectra from the Complete and SimpliÞed Solutions . . . . . . . . . . . . . . . . . . . . . . . I. Implication of the Equivalence of the Predicted Z-Spectra . . . . . . . J. Three-Site Models of Biological Tissue . . . . . . . . . . . . . . . K. Solutions of the Three-Site Models. . . . . . . . . . . . . . . . . L. Three-Site Cyclic Exchange . . . . . . . . . . . . . . . . . . . M. General Three-Site Detailed Balance . . . . . . . . . . . . . . . . N. Three-Site Exchange through an Intermediate Site . . . . . . . . . . O. Relaxation in an Exchanging System . . . . . . . . . . . . . . . . P. Transient Solution for Longitudinal Magnetization (Exact Solution for T1 ) Q. Approximate Solution for T1 . . . . . . . . . . . . . . . . . . . R. Exact Solution for T2 . . . . . . . . . . . . . . . . . . . . . . S. Approximate Solution for T2 . . . . . . . . . . . . . . . . . . . T. Effect of Exchange on Observed T1 . . . . . . . . . . . . . . . . U. Effect of Exchange on Observed T2 . . . . . . . . . . . . . . . . V. Selective Saturation . . . . . . . . . . . . . . . . . . . . . . . W. Saturation Dependence on External B1 Field . . . . . . . . . . . . . X. Saturation in the Two-Spin System . . . . . . . . . . . . . . . . . Y. Saturation in a Two-Spin Exchanging System . . . . . . . . . . . . IV. Magnetization Transfer Imaging . . . . . . . . . . . . . . . . . . . A. Pulsed Off-Resonance Magnetization Transfer Techniques . . . . . . . B. On-Resonance Pulsed MT . . . . . . . . . . . . . . . . . . . .

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1 Volume 118 ISBN 0-12-014760-2

C 2001 by Academic Press ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright  All rights of reproduction in any form reserved. ISSN 1076-5670/01 $35.00

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JOSEPH C. McGOWAN C. D. E. F.

A Relationship between Magnetization Transfer Contrast and T2 . . . Correlation in Images of Biological Tissue . . . . . . . . . . . . Correlation in Images of Agarose Gel Phantoms . . . . . . . . . . Solving the Inverse Problem: Elucidation of Fundamental Model Parameters from the Z-Spectrum . . . . . . . . . . . . . . . . V. Application in Human Studies . . . . . . . . . . . . . . . . . . . A. Quantitative MTI . . . . . . . . . . . . . . . . . . . . . . . B. Example: Applications of Magnetization Transfer to Multiple Sclerosis and Diffuse Brain Disorders . . . . . . . . . . . . . . . . . . VI. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix I: Solution of the Complete Coupled Bloch Equations for Two-Site Chemical Exchange . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction Nearly 30 years ago, Paul LauterburÕs suggestion that nuclear magnetic resonance might be used for creating images in humans (Lauterbur, 1973) set in motion rapid change in diagnostic radiology. Development of magnetic resonance imaging (MRI) technology proceeded at a phenomenal rate, and today MRI has supplanted ionizing radiographic techniques in many diagnostic applications. For detailed noninvasive examination of soft tissues there is very little alternative to MRI. There also exist new applications, including cerebral functional MRI and imaging of water diffusion, that could not have been imagined in the context of plain radiographs and ultrasound examinations. Magnetic resonance spectroscopy (MRS) is also Þnding use in the clinic. Long established as a technique for chemical analysis, it provides a number of advantages in probing the biochemical basis for physiological processes. Combined examinations including MRI and MRS can offer synergistic advantages in certain disease evaluations. Current research in MRI includes an emphasis on reÞning and enhancing techniques that are still relatively young by comparison with other radiological modalities. Additionally, there is a great deal of interest in establishing novel forms of MRI contrast, reßecting characteristics of biological tissue that are not probed by standard diagnostic MRI. There is also increased emphasis on the use of MRI-obtained information in a quantitative vice qualitative sense. It must be understood that this could be seen to conßict with the traditional interpretation of radiological data as practiced by radiologists. The training of a radiologist is founded upon developing the ability to form an ÒimpressionÓof a study and to evaluate that impression in light of experience with previously reviewed cases, patient outcomes, and ancillary information provided by other tests and consulting physicians. The impression primarily takes into account the apparent contrast developed between different tissues by

MRI AND MAGNETIZATION TRANSFER

3

the imaging modality, but must also include factors such as image quality and the presence of artifact and confounding appearance. Expert practitioners of radiology synthesize this information instantly and can gain valuable insight from a seemingly ambiguous study with a skill that must be classiÞed as both art and science. Although there is ongoing research aimed at modeling these thought processes and creating artiÞcial intelligence algorithms, replacement of radiologists with computers is not contemplated in the near term. On the contrary, the objective of quantitative image analysis, and particularly quantitative MRI analysis, is to provide the radiologist with additional information that cannot be obtained via evaluation of apparent contrast. This additional information might include an intrinsic comparison with a norm as well as the ampliÞcation of Þne differences that may not be visually detected on the image. Diagnostic MRI is primarily based upon the magnetic resonance properties of ubiquitous (in living tissue) water protons, including analysis of the empirical time constants T1 and T2 that were Þrst proposed by Bloch (1946), in parallel with the work of Purcell et al. (1946), later recognized with the shared Nobel prize. Typically in clinical application images are obtained that reßect (but do not measure) the time constants. More recent quantitative techniques are used to explore mechanisms affecting the magnetic resonance properties of a tissue that may represent an underlying phenomenon relevant to a physiological question. An example of such a technique is the subject of the present work. In this paper we will undertake a brief review of the fundamentals of magnetic resonance and magnetic resonance imaging in order to motivate a discussion of a technique aimed at probing interactions between MRI-visible water protons and protons of larger molecules of physiologic interest. The underlying assumption for this technique is that a magnetization state may be transferred between such protons, and thus it is referred to as magnetization transfer (MT). Contrast obtained via this mechanism is called magnetization transfer contrast (MTC) and imaging that reßects MTC is known as magnetization transfer imaging (MTI). Applications of MTI will be discussed in brain, together with analysis techniques being developed to exploit the MT phenomenon. We note that although the fundamental phenomena of magnetic resonance are described by quantum mechanics, the observations that are essential to the arguments presented herein can be explained with classical arguments, and for the purpose of application to medical imaging and diagnosis this is nearly always the case. The theory of magnetic resonance has been developed in both disciplines, beginning with the work of Bloch (principally a quantum physicist who wrote the classical description of magnetic resonance) and Purcell (principally a classical physicist who wrote the quantum description of magnetic resonance).

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JOSEPH C. McGOWAN

A. Fundamentals of Magnetic Resonance Imaging Nuclear magnetic resonance (NMR) refers to the enhanced absorption of energy that occurs when certain nuclei are exposed to radiofrequency energy at a characteristic frequency. The effect was Þrst described and observed in particle beams by Rabi (1937) and this work led to the 1946 advances of Bloch and Purcell (Bloch, 1946). Quantum mechanics deÞnes a property of some nuclei known as Òspin.ÓThe spin quantum represents an inherent angular momentum and associated with each spin state there is a speciÞc energy level. The smallest number of spin states possible is two. In that case, the states are referred to as ±1/2 or as ÒupÓ and Òdown.ÓThe energy levels associated with the spin states vary according to the strength of the external magnetic Þeld that is present. An individual nucleus can experience a transition between spin states by emitting or absorbing energy, but these transitions occur only when the amount of energy involved is exactly the correct amount, given by PlanckÕs Law: E = hν

(1)

where E is the energy of transition, ν is frequency, and h is PlanckÕs constant. From this relationship it is seen that the energy associated with the transition must be associated with a characteristic frequency. This remains true, and the energy involved is the same, whether the transition corresponds to absorption or emission of energy. It can be shown that when the spin, or, more correctly, a group of spins acting together, emits energy it takes the form of a magnetic Þeld rotating at the characteristic frequency. Thus it is reasonable that to cause absorption of energy one should apply a rotating magnetic Þeld as well. Interestingly, a rotating magnetic Þeld can be obtained from a linearly oscillating magnetic Þeld as follows. Consider the Þeld Bx described by the relationship Bx = 2B1 cos(ωt + ϕ)ex

(2)

with ex the unit vector in the x direction. This can be decomposed into two counterrotating magnetic Þelds. Br = B1 cos(ωt + ϕ)ex + B1 sin(ωt + ϕ)e y Bl = B1 cos(ωt + ϕ)ex − B1 sin(ωt + ϕ)e y

(3)

For the purpose of MR only the Þeld rotating in the direction of spin is relevant, with the other effectively insigniÞcant as a result of being far off resonance. The mechanism of MR is illustrated by considering a sample containing a population of identical nuclei placed in a static magnetic Þeld. By convention,

MRI AND MAGNETIZATION TRANSFER

5

Figure 1. Relationship between angular momentum (J) and magnetic moment (μ) in a rotating charged particle.

the nuclear spins at equilibrium are characterized by the lower energy Òspin downÓcondition. Initially, RF energy at the resonance frequency is applied, causing the nuclei to absorb energy and to undergo a transition to the ÒspinupÓ state. The magnetic Þeld during that part of the experiment is described as the sum of the large static and small rotating Þelds. The RF energy is then turned off. Subsequently, we observe that the nuclei emit RF energy at precisely the resonance frequency as they undergo a transition back to the lower energy ÒspindownÓstate. There is no precise analogy in classical physics for nuclear spin. However, the time-dependent behavior of ensembles of nuclear spins is accurately represented by theories of classical mechanics. These ideas aid in providing a classical description of the motion of spins in magnetic resonance. As discussed above, each nucleus that possesses spin has associated with it a magnetic moment. A spinning charged particle, such as a proton, constitutes a current loop described by the motion of the charge. Referring to Figure 1, this current I is given by I =q·

v 2πr

(4)

where q is the charge, v is the velocity, and r is the radius. As a result of the current there is a magnetic dipole moment which is the product of the area of the loop and the current. The magnitude of this moment is given as qvr/2, and its direction is oriented along the axis of rotation of

6

JOSEPH C. McGOWAN

the particle, as depicted in Fig. 1. The magnetic moment is thus parallel to the angular momentum of the particle, and is related to angular momentum by: μ=

q ∗J 2m

(5)

where μ represents the magnetic moment, q and m are the charge and mass associated with the particle, and J is the angular momentum. By summing the magnetic moments associated with many nuclei comprising a sample, we arrive at the net magnetic moment, or magnetization, M. This vector quantity can assume a continuum of magnitudes and can point in any direction. In the presence of an external magnetic Þeld (B0), the spins tend to align themselves either parallel or antiparallel to the Þeld. There exists a differential in the number of spins aligned one way or the other, related to the energy state of the system. For protons, this difference is only one part in 108 but the difference is still signiÞcant, because of the large numbers of protons involved. Thus, a net spin vector equal to the sum of all of the individual spin vectors is oriented along the direction of the Þeld. The net spin vector is referred to as spin magnetization and this is the fundamental quantity that is manipulated in NMR. Associated with a nuclear particle in a magnetic Þeld is a characteristic, or resonance, frequency, as noted above. The Larmor relationship establishes that this frequency is linearly related to the magnetic Þeld strength. Thus: f = γ B0

(6)

with f the resonance frequency and B0 the applied Þeld. The constant of proportionality (γ ) is known as the gyromagnetic (or magnetogyric) ratio and is characteristic of the individual nuclei under study. Some nuclei of interest that exhibit this phenomenon include protons of water (1H), phosphorus (31P), ßuorine (19F), and sodium (23Na). The gyromagnetic ratio for a single proton is approximately 4258 Hz/gauss (G). For diagnostic use of MR one might employ a magnetic Þeld strength of 1.5 tesla (i.e., 15,000 G), with Eq. (4) giving a value of 63.87 MHz for the resonance frequency, corresponding to channel 3 in the television range of radiofrequency (RF) emissions. The equilibrium state for the direction of spin magnetization is in alignment with the external Þeld. This orientation is referred to as the longitudinal direction, or along the ÒzÓaxis, and it is associated with a low-energy state. If energy is added to the system in some way, the spin magnetization may no longer be aligned with the external magnetic Þeld, and it will tend toward regaining that alignment. However, because of the spinning motion of the magnetization vector, the path taken by the magnetization toward the equilibrium state is not

MRI AND MAGNETIZATION TRANSFER

7

Figure 2. Precession of a gyroscope under the inßuence of gravity. The axis of rotation is indicated as is the path of precession. Spin magnetization behaves in an analogous manner under the inßuence of an external magnetic Þeld.

direct. Rather, it is inßuenced by a torque given as the cross product of the magnetic moment and the vector corresponding to the external Þeld. τ =μ×B

(7)

This torque causes the motion of the spin vector to be a precession, analogous to that of a gyroscope under the inßuence of gravity (Fig. 2). In fact, hydrogen nuclei associated with water in our bodies precess innocuously, at a rate of approximately 2 kHz, about the earthÕs magnetic Þeld (5 × 10−5 tesla). The spin magnetization vector can point along the direction of the external Þeld, or at some angle to it. It is useful to decompose this vector into the component of the magnetization that is aligned with the external Þeld (z magnetization or Mz) and the component that is perpendicular to the external Þeld (transverse magnetization or Mxy). Transverse magnetization can be detected by a properly positioned radiofrequency (RF) coil (receive coil), taking advantage of Faraday induction. Referring to Figure 3, relative motion between the spin magnetization and the receive coil exists whenever there is a transverse component to the spin magnetization. It is apparent that purely longitudinal magnetization gives no signal, while purely transverse magnetization maximizes the signal. We observed that an energy state higher than equilibrium is associated with spins out of alignment with the external Þeld. With the exception of a special case (inversion or 180◦ ßip of spins and subsequent recovery) it will always be true that energy above the equilibrium state is associated with spin magnetization out of alignment with the z axis. The resonance phenomenon is exploited

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JOSEPH C. McGOWAN

Figure 3. The transverse component of precessing spin magnetization induces voltage in a properly positioned surface coil by FaradayÕs law (a). The longitudinal component of the magnetization (b) does not induce a voltage as the ßux lines of the magnetization remain parallel to the wires of the receive coil.

to add energy to the spin magnetization in order to establish transverse magnetization and develop an MR signal. This overcomes the presence of the strong external Þeld, which tends to keep the spins aligned in the longitudinal direction. An analogy is that of pushing a child on a swing under the inßuence of gravity, where one observes that pushing at the ÒcorrectÓfrequency allows one very easily to increase the height to which the child is propelled (stored energy in the system) while pushing at the wrong frequency is not effective. This will be explored in the following section.

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9

B. Spin Flips and Relaxation Consider a collection of spins under the inßuence of a large stationary external Þeld B0. It is desirable to add energy to the system in order to perform a magnetic resonance experiment. As noted, this can be done through the use of a second, and much weaker, magnetic Þeld rotating at the resonance frequency. The effects of this Þeld can be most easily seen with the use of a rotating reference frame. First, in a reference frame rotating at the characteristic frequency of the spin system, the effect of the stationary magnetic Þeld (B0) will disappear. This is clear from the observation that the spinning motion is entirely due to the inßuence of the external (stationary) Þeld, as is apparent from the Larmor relationship. Thus, in such a reference frame it is possible to explore the inßuence of a second magnetic Þeld (B1) that is applied in such a way that it rotates at the resonance frequency. In the rotating reference frame the B1 Þeld is stationary. Since, as noted, the inßuence of the B0 Þeld disappears in the rotating frame, the spin magnetization in the rotating frame is inßuenced only by the B1 Þeld, and obeying the relationships outlined earlier it tends to precess around it. If we allow this precession to proceed for a quarter of a cycle (90◦ ) we are said to have applied a pulse with a 90◦ ßip angle. Turning off the B1 Þeld at that point restores the effective Þeld to B0 and allows precession (in the laboratory frame) around the axis of B0. As noted above the precessing magnetization with a transverse component will result in detectable signal in a properly positioned receiver coil. In this case magnetization is said to have been ÒßippedÓ into the transverse plane. A 90◦ ßip of the spin magnetization results in the maximal signal possible, as all of the z magnetization is transformed into xÐy, or transverse, magnetization. From this point the signal decreases as a result of dephasing of individual spins or groups of spins. The spins get out of phase with one another because of small variations in the Larmor frequency arising from chemical differences and also because of physical inhomogeneity of the applied magnetic Þeld, with the latter typically the larger effect. Dephasing results in signal decrease as the spins no longer add constructively and tend toward canceling each other out. The rate of dephasing, and thus signal decrease, due to chemical effects is known as T2 and is one of the fundamental constants described by Bloch in his original characterization of the magnetic resonance phenomenon (Bloch, 1946). This signal loss is irreversible as it results from random processes, and T2 is the time constant of the exponential function describing that decay. The process of T2 relaxation is also known as spinÐspin relaxation. In contrast, the signal loss due to magnetic inhomogeneity is reversible to the degree that the inhomogeneity is constant. This effect is exploited in

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Òspin-echoÓbased techniques (Hahn, 1950), which are brießy outlined next. The overall signal loss including both random and nonrandom processes is described by the constant T2∗ . The remaining fundamental constant is T1, the time constant describing the exponential recovery of longitudinal magnetization. T1 differs from T2 in that the T1 process describes a transfer of energy to or from the system and is related to the aggregate spin magnetization as opposed to individual spins. This process is also known as spinÐlatticerelaxation. The two relaxation mechanisms can be viewed as independent, although there may exist some correlation between them. It is apparent that T1 must always be as long or longer than T2, since the longitudinal magnetization cannot be fully restored until all of the transverse magnetization has disappeared. Thus, T1 can be considered an upper limit for T2, a limit which is approached in some aqueous solutions. In biological tissue, however, T2 is typically observed to be shorter than T1 by an order of magnitude. The T1 and T2 relaxation times appear as constants in the Bloch equations, a set of three differential equations which describe the time-dependent behavior of the magnetization. Although BlochÕs use of these relaxation terms was entirely empirical, they have been spectacularly successful in characterizing differences in tissue properties that are correlated with information of physiological and/or biochemical signiÞcance. C. Three Fundamental Signals in Magnetic Resonance The application of RF energy at the resonance frequency (taking the form of a rotating magnetic Þeld) is typically described as a Òpulse,Ówith which can be associated a Òßipangle.ÓSeveral pulses may be given in succession, and the results of these trains of pulses can be described as one of three types of signals. The Þrst, free induction decay or FID, follows each pulse and is seen as a decaying sinusoid with a time constant of T2∗ , deÞned by the following relationship. 1 1 + γ π B0 = ∗ T2 T2

(8)

with B0 representing the gradients of magnetic Þeld strength due to inhomogenieties (of the magnetic Þeld) within any given region. The FID begins at its maximum value as depicted in Figure 4. The FID is a time-domain signal which can be Fourier transformed to the frequency domain for analysis. If the material under study is homogeneous (deÞned by all spins being characterized by the same resonance frequency) and the stationary magnetic Þeld is likewise (and thus B0 = 0), then the frequency content of the Fourier-transformed FID

Figure 4. Free induction decay (FID) following an excitation pulse at time zero. The signal is characterized by an exponential envelope (with characteristic decay constant T2∗ ) around a decaying sinusoid with frequency reßecting the difference between the spin magnetization characteristic frequency and the receiver setting.

will be limited to a single frequency. In this case the frequency spectrum of the FID is a Lorentzian line. If other resonance frequencies are present in the spin population, due either to chemical variance within the spins or to magnetic Þeld inhomogeneity, the frequency-domain representation of the signal will be more complex and will reßect the additional frequency content through the appearance of distinct spectral lines and/or through broadening of the characteristic line. Two successive pulses will produce a spin echo (Hahn, 1950). This exceptionally useful phenomenon results from the rephasing of spins that were dephased by magnetic Þeld inhomogeneities (i.e., reversible dephasing). The result is a signal building to a maximum value, which is reached at a time TE (echo time), equal to twice the time between the two pulses. The echo thus formed gives a signal reßecting T2 instead of T2∗ , and also has the advantage (over an FID) of allowing acquisition of both the buildup and the decay of the signal. As Þrst observed by Hahn (1950), and subsequently reÞned by Carr and Purcell (1954), the spin echo reverses the static dephasing process upon application of a second RF pulse following the initial excitation pulse. The spin echo appears in time as two FIDs placed back to back, building to a maximum intensity and then decaying with characteristic time T2∗ . The spin echo is depicted in Figure 5. Assuming a spin system at equilibrium (also referred

12

JOSEPH C. McGOWAN

Figure 5. Spin echo created by a pair of RF pulses. The total time to the echo is TE (echo time). The FID following the Þrst pulse is shown. Note that there would also be an FID associated with the second pulse (omitted for clarity).

to as fully relaxed) the largest possible spin echo results when the two pulses given are a 90◦ pulse followed by a 180◦ pulse. The rephasing of reversible dephasing is illustrated in Figure 6. The remaining possible NMR signal is known as a stimulated echo and results from a succession of three pulses. Here the maximum signal is obtained when all of the pulses are 90◦ . The stimulated echo is distinguished by Tm, the mixing time, in which transverse magnetization created by the initial 90◦ pulse is converted to longitudinal magnetization and thus decays with time constant T1 instead of T2, preserving the magnetization for subsequent detection following the third pulse. Figure 7 diagrams the stimulated echo and relevant times.

II. Magnetic Resonance Imaging A magnetic resonance image can be obtained via acquisition of FIDs corresponding to each volume element (voxel) in a slab under study. This can be accomplished by manipulating the stationary magnetic Þeld with the application of smaller Þelds (gradient Þelds) that add a linear distortion to the B0

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Figure 6. Demonstration of dephasing and rephasing leading to a spin echo. Nine spin isocromats are simulated to have slightly different resonance frequencies. Four of the individual sinusoids corresponding to these signals are shown offset from the summary curve at −7 units on the y axis. At time zero, the spins are all aligned with B0 and are ßipped into the transverse plane, yielding individual signals which are summed to produce the simulated MR signal (upper curve). As time progresses, the difference in resonance frequencies causes spins with different frequencies to acquire phase differences with one another. The simulated signal decreases as dephasing progresses. At time 1000 (arbitrary units), an inversion pulse is simulated, ßipping all spins 180◦ about one of the transverse axes (that is, not the longitudinal axis). This has the effect of reversing the positions of relatively fast and slow spins. As time continues, the spins continue to acquire phase in the same direction as previously, but now faster and slower spins are approaching the baseline position at the same rate that they were previously diverging from it. At a time equal to twice the pulse interval, the phase dispersion will be completely undone and a peak in signal strength (spin echo) will be observed. The height of the peak will be less than the original FID by virtue of pure T2 dephasing processes that are irreversible and cannot be recovered by the spin echo method.

Þeld. If gradient Þelds are applied in three dimensions it is possible to assign a resonance frequency to a single voxel, which can then be excited and allowed to generate a FID. Application of the Fourier transform to each FID yields a single Lorentzian line whose area corresponds to the density of spins in that location. The image is formed by constructing a two-dimensional map of the slab such that the gray-scale values assigned to each point on the map correspond to the peak area for its spatial equivalent volume in the slab. This technique, though cumbersome, was the Þrst to be advanced (Hinshaw, 1974)

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JOSEPH C. McGOWAN

Figure 7. Stimulated echo created by a train of three pulses. Pulse timing is indicated on the diagram. Each pulse will be associated with an FID, and a spin-echo is created by each combination of two pulses. With the exception of the initial FID, these signals have been omitted for clarity.

and is referred to as the Sensitive Point Method. Fortunately, the use of multidimensional Fourier transform techniques makes the modern acquisition of an MR image much faster and more straightforward.

A. Field Gradients and Slice Selection It can be seen by Eq. (6) that resonance frequency varies directly with effective magnetic Þeld strength. Thus it is possible to associate spatial position with the frequency detected in the MR experiment through the use of Þeld gradients that vary the static magnetic Þeld strength along a particular axis. Figure 8 depicts an MR scanner with static B0 Þeld oriented in the z direction. A linear Þeld gradient is established by positioning two electromagnets along the axis of B0 with the center of the stationary magnetic Þeld midway between the electromagnets. Thus, two additional magnetic Þelds are generated along the B0 axis, equal in magnitude but opposite in direction. The net B0 at any point along the axis is the sum of both of these Þelds and the static Þeld, establishing a gradient of Þeld strength, and thus a gradient of resonance frequency in the longitudinal direction. Gradient coils are typically installed

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Figure 8. An MR scanner with static B0 Þeld oriented in the z direction. A linear Þeld gradient is established by positioning two electromagnets along the axis of B0 with the center of the B0 Þeld midway between the electromagnets. The electromagnets serve to generate additional magnetic Þelds which are added to B0 to arrive at net total Þeld. The direct proportionality of magnetic Þeld strength and frequency may be used to encode spatial position into the frequency of the signal.

in the three orthogonal directions x, y, and z, allowing the manipulation of resonance frequencies in three dimensions. Referring to Figure 9, a Þeld gradient may be energized for the duration of an excitation pulse. This will have the effect of establishing spatial boundaries where the spins within the boundaries will possess resonance frequencies in the range of the excitation pulse, and those outside will not. This process is known as slice selection and is the Þrst step in what is known as 2D (two dimensional) MRI, or spin-warp imaging. Alternatively, it would be possible to do three iterations of slice selection, ending up with a single region (the intersection of slices) that would produce a signal. This technique can be related to the Òsensitive pointÓmethod referred to above, which is the basis for all spatial localization schemes (Hinshaw, 1974). It has found more recent use in MR spectroscopy as the foundation of single-voxel localization methods (Bottomley, 1987; Frahm et al., 1989).

B. Imaging with a Spin Echo Technique Combinations of RF pulses and the application of gradients are known as pulse sequences, and specialized sequences have been developed for a wide variety of applications. Certainly the most important class of these techniques

16

JOSEPH C. McGOWAN

Figure 9. Slice selection in an MR scanner. An excitation pulse may be given in conjunction with the application of a gradient in the longitudinal direction. The bandwidth of the excitation pulse is directly related by the Larmor relationship to the Þeld strength associated with spins that will be excited by the pulse. The gradient application extends this association to the spatial dimension, in this case ÒselectingÓthe region of tissue with Larmor frequency corresponding to the applied RF. (Gradient coils omitted for clarity).

for diagnostic imaging is based upon the spin echo. The fundamental principle at work in these techniques is that, in practice, the observed dephasing of the spins (and the decay of the signal) is primarily a function of T2∗ as opposed to T2. Recall that the T2∗ processes may be reversible while the T2 processes are random, and that neither represent an energy transfer process. The source of the T2 dephasing is small gradients of the static magnetic Þeld that result from imperfect construction of the main magnetic Þeld. The spin echo serves to reverse the dephasing processes described by T2∗ . This concept is illustrated in Figures 5, 6, and 10, which also serve to describe two experimental parameters that determine the nature of the contrast in MR images derived with spin echoes: echo time (TE) and repetition time (TR). There are two pulses in this sequence, and the acquisition of the echo occurs after the second pulse such that the second pulse is exactly midway between the initial excitation and the echo. The echo time (TE) is deÞned as time between excitation and acquisition, and the repetition time (TR) is measured between successive excitation pulses. The spin-echo sequence differs from a FID acquisition in an important way. As the excitation and acquisition are separated in time, it is possible to perform spatial encoding. Additionally, there is an advantage in being able to acquire

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Figure 10. Pulse timing diagram for a spin-echo sequence, demonstrating the relationship between experimental timing parameters. Depicted are the RF pulses (excitation and inversion), three gradient proÞles corresponding to the three orthogonal directions, and the detected signal.

not only the complete decay of signal from maximum to zero, but also the buildup of signal as the spins rephase. In some ways this is like doing a double experiment, and it effectively increases the signal-to-noise ratio of the resulting measure by a factor of the square root of 2. The excitation pulse initially rotates some or all of the spin magnetization into the transverse plane. Typically, the excitation pulse is given as a 90◦ pulse, and the second pulse as a 180◦ or ÒinversionÓpulse. Although any two pulses will produce a spin echo, it is this combination that produces the strongest spin echo. After the initial excitation, T1 and T2 (and T2∗ ) relaxation processes commence, with the T2∗ relaxation being described by the loss of signal due to the dephasing of individual spins. As demonstrated in Figure 10, this dephasing results from the small differences in static Þeld experienced by the groups of spins (isochromats) in particular locations, making the individual spin vectors appear to rotate at different speeds. After a time TE/2, the second RF pulse is applied. The effect of the second pulse on the spin vectors is to reorient them so that continuation of the precession as determined by the Þeld inhomogeneity tends to make the spins regain their original phase at time TE. As the spins rephase, the signal strength builds to a maximum value which is the spin echo, and then the spins once again dephase. This sequence of Òpulse,invert, and detect (acquire)Óthe echo is played out in conjunction with the application

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of gradients that provide spatial encoding. Finally, any remaining TR time allows for partial restoration of equilibrium magnetization in preparation for the next excitation. In order to obtain an MR image from the data obtained using spin echoes, the two-dimensional Fourier transform method is most commonly employed (Edelstein et al., 1980). In this way the time-domain data are acquired and stored in a matrix known as k-space (k derives from the German word for inverse). The slice select gradient has at this point already limited the signal to that originating from the desired slice. Gradients in the two other dimensions provide encoding of spatial information into the frequency and phase of the signal. The phase encoding gradient is applied during the time period when the RF is off in order to establish a phase difference among spins along the phase encoding axis. The sequence is repeated with different amplitudes of phase encode gradient, and the number of such acquisitions determines the spatial resolution of the image in the phase encode direction. The frequency encoding gradient is turned on during acquisition of the echo to establish a frequency difference among spins along that axis. In the frequency encode direction, the limiting spatial resolution is determined by characteristics of the RF receiver. The two-dimensional Fourier transform operates on the time-domain data to obtain a frequency (and phase) domain representation. Since the gradients have encoded spatial information into the frequency and phase of the signal, the result of the Fourier transform is a mapping of intensity on spatial position.

C. Contrast in the MR Image The key advantage of MRI when compared to X-ray based modalities is the ability to obtain excellent contrast between diverse soft tissues which are similar in terms of water content (Koenig and Brown, 1993). Additionally, MRI offers the ability to manipulate experimental parameters to alter the appearance of the image. A potential pitfall of this ßexibility is that it is possible to make diverse tissues appear isointense on an MR image via improper adjustment of the spin-echo imaging parameters TR and TE. Consider an image acquired using a value of TR that is long compared to T1, and a value of TE that is short compared to T2. This is referred to as proton density weighting, and the objective of the timing is to establish essentially complete restoration of equilibrium magnetization between pulses. In this type of image the contrast will result primarily from differences in the number of proton spins contributing to the signal reßected in each pixel intensity. Relaxation effects have little inßuence, as the short TE minimizes T2 effects while the long TR minimizes T1 effects. In order to obtain contrast reßecting primarily T1 values (T1-weighting), TR is adjusted to be shorter than T1, while maintaing TE short. In this way tissue regions with long T1 values will not fully recover to the equilibrium

MRI AND MAGNETIZATION TRANSFER

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magnetization state before the excitation pulse for the subsequent acquisition is given. Since the maximum transverse magnetization (that is, the maximum MR signal) immediately following the acquisition pulse is equal to the longitudinal magnetization immediately preceding the pulse, a smaller signal will be detected in regions of longer T1. Thus, in this image regions of relatively long T1 will be dark, while regions of relatively short T1 will be bright. In order to obtain T2-weighting, perhaps the most useful kind of image for diagnostic purposes, one uses a TE on the order of or longer than T2, allowing signiÞcant T2 relaxation to occur before acquisition of the signal. The effects of T1 are minimized as in proton density imaging by using a long TR. In T2-weighted images dark areas indicate short T2 values, while regions of long T2 experience less loss of signal during the TE period and will be relatively bright. The three types of images that have been described make up the bulk of clinical MRI examinations. Images with T1 weighting are most useful for determination of anatomical structure and provide excellent delineation of fat, ßuids, soft tissue structures, and bone. Images with T2 weighting have been found to be useful for identifying a great many disorders in soft tissue. For example, malignant tumors are often bright on T2-weighted images when compared with surrounding normal tissue. Proton density weighted images are somewhat less used, but are still diagnostic in some cases. Historically, these images were obtained primarily because the technique for acquiring T2weighted images (with long TR) included a substantial amount of ÒdeadÓtime spent waiting for relaxation recovery. The proton-density images were acquired by causing a spin echo to occur during the dead time and thus did not extend the examination. Presently, most T2-weighted imaging is performed using a much more efÞcient imaging technique known as Òfast spin echo,Ówhich takes advantage of the creation of multiple spin echoes with successive inversion pulses that repeatedly rephase the magnetization until pure T2 dephasing is such that usable signal cannot be detected. In this technique, essentially all of the imaging time is devoted to acquiring multiple echoes that contribute to the T2-weighted image. Because of this, proton-density weighted images represent a real cost in terms of imaging time and are today only obtained when needed to make a particular diagnosis.

D. Gradient Echoes and Rapid Imaging Techniques A number of experimental parameters can be manipulated in MRI, and it has already been seen that they affect imaging time as well as the contrast obtained. It comes as no surprise that there are also image quality trade-offs and that, in general, time and quality are inversely related. SpeciÞcally, it is often possible to improve quality by repeating acquisitions and averaging the results (that is, by trading time for quality). There are also situations where a more rapid scan

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is advantageous and worth a penalty in signal strength. For example, some examinations are limited by patient motion, either voluntary or involuntary, that may tend to reduce the time available for examination and/or the maximum practical TR for scans that comprise the examination. Rapid imaging techniques have been developed to decrease the time required to perform an MR examination, sometimes by trading off signal strength or quality. Recall that the spin echo technique is used to refocus the individual spin magnetizations in order to counteract the effects of small Þeld inhomogeneities. Additionally, it was noted that this method maximizes signal strength while also allowing time within the pulse sequence for spatial encoding and slice selection. Modern superconducting magnets achieve Þeld homogeneity to such an extent that it may not be necessary to employ a spin echo technique to accomplish the former objective. However, it may still be necessary to use an ÒechoÓtype technique in order to manipulate the spin magnetizations to encode speciÞc information. This can be accomplished with gradient echoes, which are achieved through the use of dephasing and rephasing gradients in a manner somewhat analogous to the spin echo technique (Haase et al., 1986; Frahm et al., 1986). The gradient echo technique offers the opportunity to exchange signal strength for speed and may be advantageous when the strength of the available signal is not limiting. In gradient echo techniques the second RF pulse of the spin echo is eliminated, thus eliminating effects due to inaccuracies of that pulse. More importantly, the excitation pulse is in general much smaller than a 90◦ pulse. This allows rapid recovery of longitudinal magnetization, allowing the pulse sequence to be repeated without saturation effects becoming pronounced. Thus, relatively short TR periods may be used without suffering unacceptable signal loss. The following equation can be used to predict the effect of reduced ßip angles on the steady state magnetization: M x y = M0

1 − E1 sin β 1 − E 1 cos β

with

E 1 = e−TR/T1

(9)

where Mxy is the steady-state maximum value of transverse magnetization, M0 is the equilibrium longitudinal magnetization and is proportional to proton density, and β is the ßip angle (Ernst et al., 1987). Through the use of Eq. (9) one can predict the optimal ßip angle that maximizes signal strength for given values of TR and T1. This is called the Ernst angle and is equal to arccos (E1) (Ernst et al., 1987). There are a number of other methods that exist to reduce imaging time while still obtaining relaxation time weighted contrast. Examples include methods which collect fewer data than the standard examination, such as the use of a smaller number of phase encode steps with the same Þeld of view, resulting in coarser spatial resolution in the phase encode direction. On the other hand, one can save time and phase encode steps by using the same spatial resolution in the phase encode direction but a smaller Þeld of view. A trade-off of time

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for quality may be achieved by reducing the number of acquisitions that are averaged, and the inherent redundancy of the information content of the two halves of a spin echo can be eliminated to achieve a so-called half-Fourier image. The fast spin-echo techniques referred to earlier employ a succession of spin echoes with individual phase encoding for each echo (Hennig et al., 1986). The most extreme extension of the fast spin-echo technique collects a succession of echoes sufÞcient to acquire the whole of the k-space matrix with only one excitation. This is known as single-shot fast spin echo and has found clinical application, especially in patients for whom holding still is not possible. Single-shot gradient echo methods preceded the spin-echo methods and are called echo planar imaging, by which it is possible to acquire an entire image in less than 50 ms (MansÞeld et al., 1976). Hybrid combinations of the two multiple echo techniques have also been implemented.

III. Development of Magnetization Transfer Theory Conventional magnetic resonance techniques can be employed for characterization of ensembles of spins that differ in resonance frequencies and in the observed dynamic behavior of relaxation to an equilibrium state (Ernst et al., 1987). Among samples that are observed through the NMR characteristics of a single nucleus, variations in resonance frequency are termed chemical shifts and occur as a result of the chemical structure of the molecule in which the nucleus is found (Proctor and Yu, 1950). Samples of nuclei with similar or identical chemical shifts are distinguished primarily through variations in nuclear spin relaxation between different magnetic environments, which give rise to unique spinÐlattice(longitudinal) and spinÐspin(transverse) relaxation times. In biological tissue, the NMR-visible nucleus that is most commonly studied is the hydrogen nucleus, or single proton. This resonance is overwhelmingly the strongest NMR signal in biological tissue, because of the natural abundance of water. The observable signal is not, however, restricted to water, arising additionally from other hydrogen-containing molecules. While the chemical shift of the proton resonance varies with different magnetic environments, the magnitude of the shift is typically small in relation to the linewidth of the resonance, and consequently may be difÞcult to observe. In contrast, observed relaxation times are highly variable in tissue, and differences in these times are correlated with anatomical structure as well as pathology. This observation has formed the basis for the widespread success of MRI in diagnostic imaging (Bottomley et al., 1984, 1987; Heard et al., 1992; Martin and Edelman, 1990). The Bloch equations (Bloch, 1946) predict that in a homogeneous sample the relaxation of the observed magnetization is described by two monoexponentially decaying functions corresponding to the longitudinal and transverse

22

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relaxation times. Magnetic resonance can therefore be exploited to produce images with contrast between homogeneous regions based upon spin density, longitudinal relaxation time, and transverse relaxation time. These variables essentially comprise the parameter space of the clinical magnetic resonance examination, and images that exhibit contrast primarily reßecting one or another of the parameters are said to be weighted with respect to that parameter. The measurement or comparison of relaxation times in tissues makes the implicit assumption that the relaxation behavior can be described similarly as monoexponential decay. This assumption is valid to the extent that the differentiation of tissues with respect to observed relaxation times has been successful in clinical magnetic resonance imaging. However, biological tissues are complex structures incorporating large macromolecules. The nuclear magnetic environment for the macromolecular protons is solid-like, characterized by long correlation times and correspondingly short transverse relaxation times. Macromolecular protons are unlikely to make a signiÞcant direct contribution to the observed magnetization because of the extremely rapid decay of transverse magnetization, but through spin exchange or cross relaxation may contribute signiÞcantly to the observed dynamic behavior of the magnetization. This hypothesis forms the basis of the Þeld of study known as magnetization transfer. That is, in the presence of actual or effective spin exchange, the observed relaxation behavior of the water proton resonance, which is the only NMR visible resonance under consideration, is not expected to be monoexponential. Instead, it may be described by a number of characteristic times. This presents the possibility of a more accurate characterization of tissue with magnetic resonance in terms of a multicompartment model, in effect an expansion of the space of parameters that comprise the characterization. A. The Bloch Equations The Bloch equations provide a classical description of the relaxation of a single spin, or an ensemble of spins in a homogeneous sample. These equations are (Bloch, 1946):   M z − M0 d Mz =− (10) − ω1 M y dt T1 Mx d Mx = ω0 M y − dt T2

(11)

My d My = −ω0 Mx + ω1 Mz − dt T2

(12)

MRI AND MAGNETIZATION TRANSFER

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Here M represents the magnitude of the magnetization vector in the direction of the unit vector corresponding to the subscript. The resonance frequency is ω0, and M0 is the net longitudinal z magnetization in the steady state with the external radiofrequency (B1 Þeld) equal to zero. The magnitude of B1 expressed in frequency units is ω1 ≡ γ B1 where γ is the gyromagnetic ratio. In this formulation the B1 Þeld is applied along x, and T1 and T2 are the spinÐlattice and spinÐspinrelaxation times, respectively. In a homogeneous, or single spin, environment, these equations predict an exponential approach to equilibrium values of longitudinal and transverse magnetization. The time constants that describe this behavior are the spinÐlatticerelaxation time (T1) and the spinÐspin relaxation time (T2). B. The Chemical Exchange Model Biological tissues consist of macromolecules in an aqueous gel and may be represented by a two-compartment model by assuming that only two magnetic environments exist for protons, one applicable for protons attached to water and the other for protons attached to macromolecules. The typical description of this two-site exchange network involves distinct magnetic environments ÒaÓ and ÒbÓ,corresponding to water protons (free spins, ÒaÓ)and macromolecular protons (bound spins, ÒbÓ).This model is shown in Figure 11. A variety of environments is possible for bound spins, corresponding to every chemically distinct appearance of a nonwater proton in the tissue structure. Additionally, the relaxation of spins in diverse locations may be modulated by different correlation times and motions. The model therefore incorporates the assumption that these spins are combined in a bound proton pool with a single set of representative relaxation times. As proposed by McConnell (1958), the free and bound proton pools interact through chemical exchange of protons, with transfer of spins between sites occurring at a rate that is rapid compared to the Larmor frequency. Thus, the relaxation of an individual spin is dependent upon its local environment at any given time. In McConnellÕs formulation the behavior of the magnetization is modeled by two sets of coupled Bloch equation modiÞed to include chemical exchange (McConnell, 1958). These equations incorporate a rate constant kxy, which represents transfer of spins from pool x to pool y. The three equations that describe the ÒaÓsite are:   d Mza Mza − M0a =− − ω1 M ya − Mza kab + Mzb kba (13) dt T1 Mxa d Mxa − Mxa kab + Mxb kba = ω0a M ya − dt T2

(14)

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JOSEPH C. McGOWAN

Figure 11. Two-site model for magnetization transfer. The sample contains both water molecules and large macromolecules designated by R (top panel), each with protons that may exchange. The individual spin environments are depicted below (lower two panels) and annotated with characteristic relaxation times. Exchange between compartments is described by two variables that, with the four characteristic relaxation times, completely characterize the model.

M ya d M ya − Mxa Uab + Mxb Ub = − ωoa Mxa + ω1 Mza − dt T2

(15)

and there are three analogous equations for the ÒBÓspins. Although these equations were written to describe speciÞcally chemical exchange, it is essential to note that they are not limited in applicability to chemical exchange. For example the idea of spin diffusion, whereby the magnetization state of the nuclear spins moves from one site to another, and which may be invoked to explain relaxation, is equivalently well represented by McConnellÕs formulation. In fact, all mechanisms by which groups of spins undergo relaxation governed by more than one environment can be phenomenologically equivalent in this formulation given proper deÞnition of the constant terms (Hoffman and Forsen, 1966).

C. Investigation of Magnetic Exchange with Double Resonance Forsen and Hoffman applied McConnellÕs equations to the development of the Ònuclearmagnetic double resonance technique,Ówhich they used to investigate the exchange of spins between two magnetic sites with different chemical shifts

MRI AND MAGNETIZATION TRANSFER

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(Forsen and Hoffman, 1963a, 1963b, 1964). Recognizing that the modiÞed coupled Bloch equations are greatly simpliÞed if the magnetization of the ÒbÓpool is held at zero, they used the application of a strong RF Þeld at the resonant frequency of the ÒbÓspins in order to ÒsaturateÓthem, that is, to reduce their magnetization state to zero magnitude. A discussion of saturation follows below. Concurrently, the ÒaÓspins were assumed to be unaffected directly by the applied RF, as it was sufÞciently separated in frequency from the ÒaÓresonance. The time dependence of the ÒaÓspins under this condition is governed by the differential equation M0a Mza d Mza = − dt T1a τ1a

(16)

1 1 = kab + τ1a T1a

(17)

where τ1a is given by

The solution of Eq. (16) is given by   τ1a τ1a −t/τ1a e + Mza = M0a τa T1a

(18)

with τa ≡ 1/kab , and it follows from Eq. (18) that the new equilibrium value of Mza is   τ1a (19) Mza (t → ∞) = M0a T1a

The foregoing development is not strictly correct, in line with arguments of Boulat and Bodenhausen (1992) regarding the interpretation of the Solomon equations (Solomon, 1955) from which Forsen and HoffmanÕs relationships are derived. SpeciÞcally, the substitution of the boundary condition Mzb = 0 into Eq. (13), from which Eq. (16) is derived, should also be carried out in the ÒbÓspin version of Eq. (13). Even in the simple case where M0a = M0b , this results in a paradox, imposing the requirement that τa = T1a − T1b

(20)

which is not in general true. The paradox is resolved by the necessary inclusion of at least one more of the coupled equations, allowing nonzero transverse magnetization in the ÒbÓpool. Incorporating this equation and considering the physically realistic situation where the amplitude of the RF Þeld is much greater than the relaxation and exchange rates (ω1 ≫ k and ω1 ≫ τ1 ), the system of equations does indeed reduce to Forsen and HoffmanÕs result. Returning to the double resonance technique, the apparent longitudinal relaxation time under the experimental conditions is simply τ1a , which can be

26

JOSEPH C. McGOWAN

measured. This parameter, together with the observed magnetization at equilibrium and saturation conditions, allows calculation of the exchange constant τa . In order to calculate the remaining exchange constant τb , the experiment is reversed with the ÒaÓpool saturated and the ÒbÓpool measured. Invoking detailed balance, the ratio of the number of spins in the two pools is also determined with the use of the following relationship. τa M0a = M0b τb

(21)

The double resonance technique was demonstrated by observations of the exchange of the hydroxyl proton in a mixture of salicyl aldehyde and 2-hydroxyacetophenone, as well as in other systems (Forsen and Hoffman, 1963a, 1963b). In later work, Forsen and Hoffman extended this theory and demonstrated experimental characterization of exchange in three site systems (Forsen and Hoffman, 1964). D. Magnetization Transfer between Unresolvable Spins Complicating the application of the double resonance technique in tissue is the observation that, for the proton resonance, the chemical shift of the free water protons is similar or identical to the chemical shift of the bound protons. This can be seen in the symmetry of the magnetization transfer effect with respect to off-resonance irradiation at frequencies on opposite sides of the water proton peak (Wolff and Balaban, 1989). The similarity of the chemical shift complicates the magnetization transfer experiment in two ways. First, the signal observed at the proton resonance arises almost completely from free water protons, rendering the bound pool invisible. Therefore it may be possible to perform only one of the two experiments necessary to characterize the exchange network, that is, to saturate the bound spins and observe the free. Second, application of RF energy at an appropriate frequency for saturation of the bound pool also tends to saturate the free pool, whereas for Eq. (16) to be valid one pool must be selectively saturated. Edzes and Samulski (1977) proposed a selective hydration inversion technique to address this problem, which represented an experimental attempt to selectively invert the free water proton pools. This method was designed to exploit the relatively short T2 of the bound proton species by applying a relatively long inversion (π) pulse to the sample. The pulse length chosen was much greater than the transverse relaxation time of the bound spins, but was much shorter than the T2 of the free water spins. Therefore the free spins were completely inverted, while the bound spins were essentially

MRI AND MAGNETIZATION TRANSFER

27

saturated, since they experienced signiÞcant transverse relaxation (i.e., dephasing) during the pulse. Observation of the recovery of the signal from inversion revealed the effect of spin exchange. Analysis of the results of these experiments was complicated by the difÞculty of resolving the double exponential character of the inversion recovery, although the investigators were able to report relaxation times and exchange rates in collagen samples (Edzes and Samulski, 1978).

E. Analytical Models for Magnetization Transfer A truncated set of coupled Bloch equations has been used to investigate magnetization transfer assuming that the two proton pools are coupled through the exchange of longitudinal magnetization only. For example, the following equations, equivalent to those presented by Grad and Bryant (1990; Grad et al., 1990) are applicable in a reference frame rotating at the Larmor frequency, with the offset from the Larmor frequency given by ω. 1 d Mza = − (Mza − Mz0 ) + f kab (Mzb − M0b ) + ω1 M ya dt τ1a

(22)

d Mzb 1 = − (Mzb − M0b ) + kb (Mza − M0a ) + ω1 M yb dt τ1b

(23)

d Mxa,b Mxa,b =− − ωa,b M ya,b dt T2a,b

(24)

M ya,b d M ya,b =− − ωa,b Mxa,b − ω1 Mza,b dt T2a,b

(25)

A simpliÞcation of this equation set is obtained with the assumption that the transverse magnetization of the ÒaÓspins is unaffected by partial saturation of the ÒbÓspins, which may be valid at relatively low saturation powers and large values of ω. If these conditions hold, the steady-state solution for the longitudinal magnetizations may be derived by solving Eqs. (22) and (23) as well as the equations for the transverse magnetization of the ÒbÓspins (Eq. (24) as applied to the ÒbÓpool). This approach is equivalent to that suggested byBoulat and Bodenhausen (1992), discussed earlier. The solution for the ÒaÓ magnetization, in terms of the reduced magnetization Mza =

M0a − Mza 2M0a

(26)

28

JOSEPH C. McGOWAN

is given by α β + (ω)2 γ

(27)

kb T2b ω12 T1a T1b 2f

(28)

Mza = where α=

1 (kb T1a + 1)(T2b ω12 T1b + 1) f   1 γ = T2b2 kb T1b + (kb T1a + 1) , f

β = kb T1b +

(29) (30)

and f is deÞned as the ratio of ÒaÓsites to ÒbÓsites. In the limit where kb T1a ≫ 1, an expression for the reduced magnetization is similar to the steady state solution of the Bloch equations for a single spin. This enables plotting of the Z-spectrum, describing the behavior of the spin system in the offset frequency space with constant ω1:   1 ω12 T1b T2b Mza = (31)   2 (1 + (ω)2 T2b2 1 + f TT1b 1a + ω12 T1b T2b

Equation (31) differs from the steady-state solution of the Bloch equations for a single spin (Bloch, 1946) only in the term f T1b /T1a , which appears in the denominator of the expression. This term gains signiÞcance as f ≫ 1, as long as the T1 of both spin pools is approximately equivalent. If f T1b ≪ T1a , the expression reduces to the steady-state solution of the Bloch equations describing the ÒbÓspins. Thus, measurement of the ÒaÓspins in a partially saturated system yields the spectrum of the ÒbÓspins (Grad and Bryant, 1990). As noted previously, this formulation is valid only under conditions that do not affect the transverse magnetization of the ÒaÓspins, so that the change in the longitudinal magnetization of ÒaÓis wholly due to exchange. This state is also referred to in the literature as zero direct saturation. Qualitatively, in the offset frequency space of the Z-spectrum, this is where the ÒdoubleexponentialÓbehavior of the Z-spectrum is not apparent, that is, where the Z-spectrum may be sensitive to characteristics of the ÒbÓpool, but not to the interaction of the ÒaÓand ÒbÓspins. At very small and very large offset frequencies the Z-spectrum of the tissue sample resembles the lineshape of a single spin. It may be only in the transition area between these two extremes that the information regarding exchange is contained. The truncated model may seriously

MRI AND MAGNETIZATION TRANSFER

29

overestimate the ÒaÓmagnetization in this region, and in general can not be Þt to the entire Z-spectrum (Grad et al., 1990). Wu (1991) as well as Caines et al. (1991) expanded the applicability of this formulation by restoring the two equations describing the ÒaÓtransverse magnetization. Equivalently, WuÕs formulation differs from McConnellÕs complete set of coupled Bloch equations only in the neglect of the transfer of transverse magnetization. The solution of this set of six coupled equations is given by (Wu, 1991) M za = with

Aω14 + Bω12 ω2 + Cω12 2[Aω14 + Dω12 ω2 + Eω4 + Gω12 + H ω

(32)

A = R2a F T1a T1b R2b B = R2a T1a (F + ka T1b + Fka T1a T1b R2b ) C = R2a R2b T1a [R2a FkaT1b + R2b (F + ka T1b )] D = R2a T1a (F + ka T1b ) + F T1b R2b (ka T1a + 1) E = Fka T1a + F + ka T1b G = R2a R2b [(R2a F T1b (kaT1a + 1) + T1a R2b (F + ka T1b )]   2 2 + R2b (Fka T1a + F + ka T1b ) H = R2a 2 2 I = R2a R2b (Fka T 1a + F + ka T1b )

Here, F is deÞned as the ratio of the number of bound sites to free sites. The reduced magnetization is deÞned in accordance with Eq. (26). This formulation includes the direct saturation of the ÒaÓspins, and therefore is applicable at relatively small offset frequencies, including the frequencies that have been utilized for in vivo magnetization transfer experiments (Wu, 1991; Bryant and Lester, 1993).

F. Analytic Solutions of Coupled Bloch Equations In the previous section three possible analytical models were proposed to describe the two-site magnetic exchange network. These models, consisting of the complete coupled Bloch equations and two simpliÞed forms, predict the shape of the Z-spectrum given the intrinsic relaxation and exchange parameters

30

JOSEPH C. McGOWAN

that characterize the system. The two simpliÞed models have been used in the analysis of experimentally derived Z-spectra (Grad and Bryant, 1990; Grad et al., 1990; Wu, 1991). We obtained solutions to these models in terms of the absolute, rather than reduced magnetizations, resulting in greatly simpliÞed forms (McGowan, 1993). In addition, the general analytical solution to the coupled Bloch equations was obtained by us and others and the solution is given in Appendix I (McGowan, 1993; Roell et al., 1998). For this derivation relaxation rates were used instead of relaxation times and were deÞned as Rx = 1/Tx . The variable f was deÞned as M0a /M0b , and detailed balance was assumed. Although the expression is lengthy, it is easily incorporated into computer algorithms which generate predicted Z-spectra quite rapidly.

G. Analytic Solution of SimpliÞed Bloch Equation Sets The solution of a truncated set of four Bloch equations describing the longitudinal magnetization of both pools and the transverse magnetization of the bound pool, with the transverse magnetization assumed constant at zero, can be expressed as kab R1b − R1a Mza β = f k2 M0a −R1a − kab − βab

(33)

with β = −R1b − f kab +

ω12 −

ω2 R2b

− R2b

differing from the previously published equation (32) in that absolute rather than reduced magnetization is used. Restoring the two equations for transverse magnetization of the free spins to the model and deÞning the term α in a manner analogous to β, we obtain the following solution: Mza = M0a

kab R1b β

− R1a

(34)

f k2

α − βab

with α = −R1a − kab + δ, δ =

ω12 −ω2 R2a

− R2a

MRI AND MAGNETIZATION TRANSFER

31

Viewed another way, Eq. (34) is obtained by adding one term (δ) to the denominator of the right side of Eq. (33). Since the two models represented by the equations that lead to Eqs. (33) and (34) have been demonstrated to differ only in near resonance behavior (that is, where signiÞcant direct saturation is present) (Wu, 1991), the term δ can be seen as related to direct saturation of free spins.

H. Comparison of Predicted Z-Spectra from the Complete and SimpliÞed Solutions As noted earlier, the Z-spectra predicted by Eqs. (33) and (34) have been previously compared in the literature. We add the comparison of the complete solution to Eq. (34). The Z-spectrum predicted by the complete solution was computed for systems characterized by parameters that have been reported for biological tissue (Eng et al., 1991; Morris and Freemont, 1992; Wolff and Balaban, 1989) and was compared to the spectrum that arises from Eq. (34). As might be expected given the short transverse relaxation time that is associated with the bound spins, the complete solution yields results that do not differ appreciably from those obtained with the approximation of Eq. (34). This is particularly the case under experimental conditions that might be reasonably applied in vivo, that is, in accordance with the guidelines of the United States Food and Drug Administration (FDA, 1982). Figure 12 compares the two solutions under example sets of intrinsic system parameters. The experimental condition is that of constant irradiation at 156 Hz. In these cases the MT effect predicted by the two models would be indistinguishable. The complete solution does diverge from the approximation under some conditions, speciÞcally at small saturation offsets with high power irradiation. A comparison of the two solutions under an example of these conditions is shown in Figure 13.

I. Implication of the Equivalence of the Predicted Z-Spectra Simulations demonstrate that the Z-spectra predicted by the ÒcompleteÓ solution are essentially equivalent to predictions of Eq. (34), which suggests that the simpliÞed form is adequate for analysis of measured longitudinal magnetization in the two-site system, without regard to whether transverse magnetization exchange should be included in the model. Although this is true, we observe that the two models, similar in prediction of the Z-spectrum, differ dramatically in prediction of observed transverse relaxation time. With that in mind, the analysis of coupled Bloch equations

32

JOSEPH C. McGOWAN

Figure 12. Comparison of Z-spectra predicted by the complete solution of the two-site coupled Bloch equation model (Appendix I) and the approximate solution obtained by excluding exchange of transverse magnetization. Over the range of parameters considered, the complete solution (solid lines) yields results that are nearly identical to the results of the approximate solution (symbols). The curve plotted with asterisks (∗ ) was obtained with the following parameters: T1a = 2.0 s, T2a = 0.05 s, T1b = 2.0 s, T2b = 40 μs, kab = 1.0, f = 2.0, ω1 = 156 Hz. The other curves were obtained by varying one parameter while holding the rest constant. These variations (plot symbols) are kab = 4.0 (diamonds), f = 5.0 (triangles), T1b = 0.2 s (boxes).

that neglect transverse magnetic exchange may be appropriate to give insight into the behavior of longitudinal magnetization in multisite exchanging systems. For example, three-site systems provide additional degrees of freedom when considering biological tissue, allowing the inclusion of intermediate Òhydration layerÓ sites or alternately, two classes of bound spins. These three-site systems have been solved in two ways. An analytic solution for steady-state longitudinal magnetization was obtained with the assumption that the transfer of transverse magnetization can be neglected. These solutions, presented later, may be used to generate and analyze Zspectra. To predict relaxation rates, the entire equation set including transverse magnetization exchange can be solved numerically as a matrix as

MRI AND MAGNETIZATION TRANSFER

33

Figure 13. Comparison of Z-spectra predicted by the two-site coupled Bloch equation models for magnetization exchange, illustrating inclusion of the exchange of transverse magnetization. The solid line represents the Z-spectrum corresponding to intrinsic parameters T1a = 4.8 s, T2a = 1.0 s, T1b = 4.8 s, T2b = 70 μs, kab = 1.0, f = 40.0, ω1 = 500 Hz. The dashed line represents the Z-spectrum under identical conditions that is predicted when transverse exchange is neglected.

discussed in the following. The eigenvalues of this matrix correspond to the time-dependent relaxation behavior.

J. Three-Site Models of Biological Tissue There are three independent exchange schemes that are appropriate for a three site system. The Þrst, called cyclic exchange, is shown in Figure 14 and is an example of a system without detailed balance. The second is the general detailed balance form (Fig. 15), which might be appropriate when considering two classes of bound spins that exchange with free water but also with

34

JOSEPH C. McGOWAN

Figure 14. Three-site cyclic exchange is a simple example of an exchanging system that violates detailed balance. A physical example could be envisioned as a tumbling molecule where nuclei pass through three distinct environments (G. Radda, private communication).

each other. Finally, Figures 16a and 16b describe a limited form of detailed balance which could be envisioned two ways. The Þrst of these is again the system with two classes of bound spins, but in this case the bound spins cannot exchange with each other (Fig. 16a). Alternatively, Figure 16b describes free spins and bound spins connected exclusively through an intermediate hydration layer with a separate characteristic magnetic environment. Such a model was previously proposed as potentially valid for tissue (Zhong et al., 1989).

Figure 15. The general detailed balance condition for a three-site network. This exchange network can also be referred to as a maximally connected three-site network.

Figure 16. (a) A special case of detailed balance in a system that is not maximally connected. This network corresponds to a system with two independent bound sites, both exchanging with the free water but not with each other. (b) A network that is the mathematical equivalent of (a). The free spins are labeled A and the exchange occurs between free spins A and bound spins C via intermediate spins B.

36

JOSEPH C. McGOWAN

K. Solutions of the Three-Site Models The proposed three-site models have been solved using coupled Bloch equations with transfer of longitudinal magnetization and no transfer of transverse magnetization. These solutions correspond to the exchange schemes described previously.

L. Three-Site Cyclic Exchange With the notation introduced for the complete coupled Bloch equation solution, and f1 deÞned as the ratio of the number of ÒaÓsites to ÒbÓsites, f2 deÞned similarly as the ratio of the number of ÒaÓsites to ÒcÓsites, the expression for observed (ÒaÓsite) longitudinal magnetization for the cyclic exchange case is Mza =

−kca kbc R1b cb f 1

kca R1c + c f2 kca kbc kab cb

+

a+

R1a

(35)

with ω12

a = −R1a − kab +

−ω2

b = −R1b − kbc +

−ω2

c = −R1c − kca +

R2a

− R2a

ω12 R2b

− R2b

ω12 −ω2 R2c

− R2c

M. General Three-Site Detailed Balance With a slight modiÞcation to the terms a, b, and c deÞned in Eq. (35), it is possible to write the relationship for this more general condition in a fairly compact form: Mza =

kab kbc R1c bc

+

a−

kab R1b + f2 kcac2 kbbcf1R1b b f 1 kab 2 + 2 kab kbcbckac f2 b

− −

f 2 kac kbc R1b + kaccR1c − cb f 1 2 k 2 ac k 2 bc f 2 2 − f2 kcac c2 b f 1

R1a

(36)

MRI AND MAGNETIZATION TRANSFER

37

with a = −R1a − kab − kac + b = −R1b − kbc − kba + c = −R1c − kca − kcb +

ω1 2 −ω2 R2a

− R2a

ω12 −ω2 R2a

− R2a

−

2 kbc f2 f1c

ω12 −ω2 R2c

− R2c

N. Three-Site Exchange through an Intermediate Site As noted, this formulation applies equivalently to exchange with two different bound sites that do not exchange with one another, or to exchange through an intermediate site. Again, appropriate deÞnition of terms a, b, and c simpliÞes the form. Mza =

−kab kbc R1c bc

a = −R1a − kab + c = −R1c − kcb +

+

a−

kab R1b b 2 f 1 kab b

− R1a

(37)

ω12 −ω2 R2a

− R2a

ω12 −ω2 R2c

b = −R1b − kbc − kba +

− R2c ω12 −ω2 R2a

− R2a

−

2 f2 kbc f1c

Note that in these equations there are only three independent exchange rates, as detailed balance allows the reverse exchange rate between two sites to be written in terms of the forward exchange rate between those sites and the ratio of spin populations. Extension to numbers of sites greater than three is straightforward.

38

JOSEPH C. McGOWAN

O. Relaxation in an Exchanging System Relaxation refers to the restoration of a state of equilibrium, which in the current context could represent the equilibrium magnetization in the presence or absence of an external perturbing Þeld. The relaxation times that appear in the Bloch equations are intrinsic relaxation times, that is, they describe the behavior that would be observed in a homogeneous sample in the absence of exchange. These intrinsic times are distinguished from relaxation times that are observed in the laboratory and are in general not equal to observed times if exchange is occurring. To explore the effect of exchange on relaxation times it is necessary to consider the transient solution of the coupled modiÞed Bloch equations. The complete modiÞed Bloch equations form a system of linear homogeneous differential equations, which can be represented as the following 6 × 6 characteristic matrix: −

1 − ka T1a 0

0 −

1 − ka T2a −

ω1

kb

0

0

−ωa

0

kb

0

1 − ka T2a

0

0

kb

0

ω1

−ω1

ωa

ka

0

0

0

ka

0

0

0

0

ka

−ω1

−

1 − kb T1b −

1 − kb T2b ωb

−ωb −

1 − kb T2b (38)

Here, ωa and ωb refer to the frequency offset of the irradiating RF from Larmor frequencies of the spins. The RF irradiation is assumed applied along the x axis with amplitude ω1. The time dependence of the solutions is related to the eigenvalues of this matrix, analysis of which is simpliÞed greatly with the observation that relaxation times may be measured in the absence of an RF Þeld. In this case the two equations describing the longitudinal magnetization [Eqs. (13a), (13b)], corresponding to rows 1 and 4 of the matrix, decouple from the rest of the system and can be solved for the longitudinal relaxation times. Similarly, the four remaining equations may be solved for the transverse relaxation times. Note that this analysis predicts the variation of observed transverse relaxation, as well as longitudinal relaxation, with exchange. It is in this aspect that the complete Bloch equation model differs most signiÞcantly from the simpliÞed models.

MRI AND MAGNETIZATION TRANSFER

39

P. Transient Solution for Longitudinal Magnetization (Exact Solution for T1) In the absence of an external B1 Þeld, the two equations that describe longitudinal magnetization were given by Leigh (1971): Mza Mzb M0a d Mza =− + + dt τ1a τb T1a

(39)

Mza Mzb M0b d Mzb = − + dt τa τ1b T1b

(40)

Their general solutions are +

−

−t/T1+

−t/T1−

Mza = C1 e−t/T1 + C2 e−t/T1 + M0a Mzb = D1 e

+ D2 e

+ M0b

(41) (42)

where −

1 T1obs

   1/2 1 1 1 2 = − ± = −A1 ± A1 − − τ1a τ1b τa τb T1

(43)

with A1 =

1 1 + 2τ1a 2τ1b

(44)

Note that the observed T1 values are not inßuenced by the difference in resonance frequencies between the ÒaÓand ÒbÓspins. In practice and in many experimental settings only the longer of the two T1 values is observed. Even if two exponential decays are similar, they are often difÞcult to resolve. This accounts for the fact that T1 relaxation in biological tissue is typically assumed to be monoexponential. Assuming initial conditions of rotation of Mza and Mzb through ßip angles θ a and θ b, the initial conditions of Mza = M0a cos θa and Mzb = M0b cos θb , Schotland and Leigh (1983) derived the following values for the constants in Eqs. (41) and (42):      1 1 1 −1 1 1 1 cos θb + cos θa − − − − − − M0a C1 = T1a τ1a τa T1+ T1 T1− T1 (45)   1 1+ − C1 (46) D1 = τb τ1a T1

40

JOSEPH C. McGOWAN

 1− 1 − (47) C2 τ1a τ1      1 1 −1 1 cos θb 1 1 1 − − M0a − + − cos θa − + + C1 = T1a τ1a τa T1+ T1 T1 T1 (48) D2 = τb



These equations complete an exact description of longitudinal relaxation in the absence of external irradiation, and are equivalent to forms derived by other investigators (Morris and Freemont, 1992). Q. Approximate Solution for T1 Two approximate solutions for T1 arise from assumptions of fast and slow exchange with regard to T1 and were derived by McLaughlin and Leigh (1973). For fast exchange,



1 1

− ka f ≫

T1a T1b

and the longer of the two relaxation times is given by 1 T1obs

=

f 1− f + T1a T1b

(50)

For slow exchange, with



1 1



− ka f ≫

T1a T1b

(51)

the approximate relaxation times are 1 T1obs

=

1 1 1 , = τ1a T1obs τ1b

(52)

R. Exact Solution for T2 The characteristic matrix for T2 is of fourth order, but can be transformed into two matrices of second order that are complex conjugates of one another. It is sufÞcient to solve one of these matrices, as their roots will be identical except in phase. This is equivalent to stating that the transverse

41

MRI AND MAGNETIZATION TRANSFER

relaxation times along x and y are identical. The real part of the solution to this system is (Leigh, 1971) 1/2  G + (G 2 + H 2 )1/2 1 (53) = A2 ± T2obs 2 with G=

1 4



1 1 − τ2a τ2b

2

+

1 1 − (ωa − ωb )2 τa τb 4

(54)

and H=

1 2



1 1 − τ2a τ2b



(ωa − ωb )

(55)

As with T1, this solution predicts two observed relaxation times. In practice it may be difÞcult to resolve the two times, and the observed time will typically be close to the longer of the two, particularly if they are greatly different.

S. Approximate Solution for T2 For the special case where ωa = ωb , that is the two resonances occur at the same chemical shift, Eq. (53) reduces to the following (for the longer T2 value) (McLaughlin and Leigh, 1973): 1/2      1 1 1 1 1 2 1 1 1 (56) − = + + + T2 2 τa τb 4 τ2a τ2b τa τb which can be written to Þrst order as     1 1 1 1 1 1 1 = + − + + T2obs 2 τ2a τ2b 2 τ2a τ2b

ka kb − τ12b

1 τ2a

(57)

which is equal to 1 1 = + T2obs τ2a

ka kb − τ12b

1 τ2a

(58)

This gives rise to two approximate solutions that depend on the concentration of bound spins as well as the exchange and relaxation parameters. In the fast

42

JOSEPH C. McGOWAN

exchange case, the assumption is 1 T2b

(59)

1 1 + T2a f T2b

(60)

1 T2b

(61)

1 + ka + ka2 T2b f T2a

(62)

1 τ2a

(63)

ka f ≫ and the resultant expression for T2 is 1 T2obs

=

For slow exchange, ka f ≪ T2 is given by 1 T2obs

=

or simply 1 T2obs

=

T. Effect of Exchange on Observed T1 The solutions just given demonstrate that in a multisite system experiencing exchange of magnetization, there may be a difference between intrinsic and observed relaxation times. The variation of observed T1 with exchange is predicted by Eq. (39) and is seen to include contributions from intrinsic T1 as well as exchange. This effect is generally acknowledged and exploited in the double resonance technique and variations thereof. These methods typically involve measuring apparent T1 (through inversion recovery or another technique) in the presence of an RF irradiation that is assumed to completely saturate the magnetization of the bound spins. The observed longitudinal relaxation time is often referred to as T1sat and is equivalent in the case of full saturation to τ 1a (Carr and Purcell, 1954). The elucidation of this parameter, along with the ratio of Mza to M0a under saturating conditions, allows the intrinsic T1a , and consequently the ka, to be obtained. This calculation forms the basis of some quantitative magnetization transfer imaging (MTI) applications that have been proposed and demonstrated (Wolff and Balaban, 1989). However, the accuracy of exchange rate estimates using this method is highly dependent upon the validity of the assumption of complete selective saturation

MRI AND MAGNETIZATION TRANSFER

43

of the bound spins with no effects of off-resonance irradiation on the free spins (Yeung, 1993). In the slow exchange case discussed earlier, the observed T1a is in general approximately equal to τ1a and therefore the T1sat measurement may not be signiÞcantly different from T1 without saturation. Conversely, in fast exchange, the observed T1 is essentially independent of ka and may be insensitive to variations in exchange rate. This suggests that care must be taken in analysis of double resonance technique results in cases where both resonances cannot be measured, for example, in biological tissue.

U. Effect of Exchange on Observed T2 In contrast to predicted T1 effects, the effect of exchange on the apparent transverse relaxation times is not generally acknowledged. This arises from the utilization of models that include the exchange of longitudinal magnetization only (Grad and Bryant, 1990; Grad et al., 1990; Wu, 1991; Yeung and Swanson, 1992). JustiÞcation for use of these models is attribution of the magnetization transfer effect to dipolar cross relaxation as opposed to chemical exchange. However, there is nothing intrinsic to the chemical exchange model that requires actual chemical exchange (Hoffman and Forsen, 1966). For example, dipolar cross relaxation may be modeled by the exchange of magnetization. A possible advantage of the inclusion of transverse exchange in the model is the suggestion of a mechanism for enhanced transverse relaxation that appears to be characteristic of some heterogeneous systems. This is illustrated by examination of measured relaxation times in biological tissue and comparison with theoretical predictions. In a homogeneous water sample, the intrinsic relaxation times are given by (Dwek, 1973) 3 γ 4hø2 1 f 1 (τc ) = T1 10 r 6

(64)

with f 1 (τc ) =

τc 4τc2 + 1 + τc ω02 1 + 4τc2 ω02

(65)

and 1 3 γ 4hø2 f 2 (τc ) = T2 20 r 6

(66)

44

JOSEPH C. McGOWAN

Figure 17. Intrinsic relaxation times plotted against correlation times. The solid line represents longitudinal (spinÐlattice)relaxation time (T1 ) and the dashed line represents transverse (spinÐspin)relaxation time (T2 ).

with f 2 (τc ) = 3τc +

5τc 2τc + 1 + ω02 τc2 1 + 4ω02 τc2

(67)

where ω0 is the Larmor frequency, γ is the gyromagnetic ratio, hø is PlankÕs constant, r is the proton intermolecular distance, and τc is the rotational correlation time. Based on a value of T1 = 2.3 s and τc = 0.35 × 10−11 , which corresponds to a Larmor frequency of 63.87 MHz, the value of the constant term γ 4hø2 /r 6 is calculated to be 8.28 × 1010 , and the logÐlogplot of T1 and T2 can be generated. This is shown in Figure 17. Referring to Figure 17, a homogeneous sample of water with rotational correlation time less than about 10−10 s is characterized by identical T1 and T2 values. As the correlation time increases, the T1 and T2 curves diverge, giving rise to an increasing T1 to T2 ratio. Assuming a T1 value on the order of 1 s (which is reasonable for biological tissue), it is apparent that the measured values of transverse relaxation times in tissue (on the order of 70 ms) are much shorter than would be expected. Qualitatively, an explanation for this

MRI AND MAGNETIZATION TRANSFER

45

observation may be that spin exchange has shortened the observed transverse relaxation time. The observed effect on T1 resulting from this process could be minimal, as can also be seen from the Þgure. For example, a bound spin ensemble with characteristic correlation time of approximately 5 × 10−6 will have a T1 on the order of the free water T1 . As such the two magnetic environments will be similar with regard to T1 , so exchange between them will have little effect on the magnetization. Conversely, the same bound spin ensemble will have a T2 10,000 times shorter, causing a dramatic effect on observed T2 by way of exchange. This might suggest that the exchange of transverse magnetization could explain part of the observed behavior of transverse relaxation in tissue. A mechanism for this exchange has been postulated by Koenig and Brown (1993). The shape of the Z-spectrum predicted by the complete solution of the Bloch equations is approximated by the simpliÞed model that assumes no transfer of transverse magnetization. In some cases the simpler model will be adequate and will give results equivalent to the complete solution in terms of predicting or analyzing the Z-spectrum. In contrast, models that neglect the exchange of transverse magnetization may not be appropriate for interpretation of transverse relaxation rates in the presence of exchange, while the complete model provides a mechanism for exchange modulated transverse relaxation. The observation that MT contrast is qualitatively similar to T2 weighted contrast (Wolff and Balaban, 1989) is consistent with the inclusion of transverse magnetic exchange.

V. Selective Saturation Magnetic resonance techniques that probe magnetization transfer between spins in different magnetic environments are typically designed to saturate one of the several spin pools that make up the system, allowing observation of the effects of magnetic exchange or cross-relaxation. In biological tissue, the relevant spin pools have chemical shift values that are thought to be essentially identical, and thus they are distinguished only by relaxation parameters. The concept of saturation, and particularly selective saturation, is critical to the understanding of these experiments. This section explores the meaning of saturation in single- and two-spin environments, with attention to the case of two spin pools with identical chemical shifts (McGowan and Leigh, 1994). Relationships derived from the Bloch equations demonstrate that selective saturation of a spin pool (while leaving the other unperturbed) is possible in a nonexchanging system to a degree determined by the intrinsic relaxation parameters of the two pools, that is, by the relaxation times which would be observed in the absence of exchange. Optimal saturation conditions are described

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in terms of the intrinsic relaxation parameters as well as the experimental choices of saturation offset frequency and amplitude. In the two-spin exchanging system, which is often used as a model for biological tissue, theory predicts that the degree of saturation depends additionally on the exchange characteristics of the system.

W. Saturation Dependence on External B1 Field In a single-spin environment, saturation refers to the maintenance of the spin longitudinal magnetization at some level less than the equilibrium magnetization through the application of RF irradiation. This is predicted by the steady-state solution of the Bloch equations (Bloch, 1946) in the presence of continuous RF irradiation at a frequency offset ω from the resonance frequency. 1 + (ωT2 )2 Mz = M0 1 + (ωT2 )2 + S

(68)

where Mz is the spin magnetization in the longitudinal direction, T2 is the spinÐ spin relaxation time, and S is the saturation factor (Bloch, 1946), deÞned as follows: S = ω12 T1 T 2

(69)

Here T1 is the spinÐlatticerelaxation time, ω1 = γ B1 , γ is the gyromagnetic ratio, and B1 is the Þeld resulting from applied RF energy. This relationship shows that for a given spin environment and saturation frequency offset, the saturation factor increases as the square of ω1, and the fraction of remaining longitudinal magnetization is Lorentzian in ω1. Complete saturation is achieved as S → ∞ and corresponds to the steady-state condition of zero longitudinal magnetization. This theory predicts the behavior of the saturation effect at constant ω1 and variable ω, as in the Z-spectrum of an exchanging system (Purcell et al., 1946). Assuming that (ωT2 )2 ≫ 1

(70)

which is reasonable for many systems of interest, Eq. (68) may be simpliÞed and rearranged to ω12 TT21 Mz 1− = 2 T1 M0 ω1 T2 + ω2

(71)

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47

√ which is Lorentzian in ω. The width of this line at half height is ω1 (T1 /T2 ), emphasizing that the degree of saturation of a single line varies in proportion to the square root of the ratio of spinÐlatticeto spinÐspinrelaxation times.

X. Saturation in the Two-Spin System Forsen and Hoffman (1963a) pointed out that complete saturation of one spin pool effectively decouples the Bloch equations that describe the twospin system, greatly simplifying their solution. As noted, measurement of relaxation times in the presence and absence of complete saturation enables calculation of the Þrst order rate constant that describes the exchange between pools (Forsen and Hoffman, 1963a, 1963b, 1964). However, the cited experiments were conducted in systems that comprised spin pools with different chemical shifts, which allows saturation of one pool with minimal effect on the other (Forsen and Hoffman, 1963a; Mann, 1977). This is not the case in the proton magnetization transfer experiment as applied in biological tissue. Here, the two spin pools are resonant at the same or similar chemical shift, differing only in relaxation parameters. It is clear that increasing the saturation factor by raising saturation power increases the degree of saturation of the target pool. However, the saturation power increase will have a similar effect on the other pool, in which the objective is that it be unaffected. Since it is not possible to achieve complete selective saturation in this manner, it is desirable to have an optimal saturation condition that maximizes the effect on the target pool while minimizing the effect on the other. A complication to the determination of an optimal condition is that the system exchange and relaxation parameters must be known in order to determine the effect of magnetic exchange on degree of saturation. Since these parameters are in general unknown, their determination being typically the object of the experiment, an estimate of saturation must be made. An analysis of the nonexchanging system provides a mechanism for such an estimate. Consider a nonexchanging two-spin system composed of ÒaÓand ÒbÓpools, where the pools have identical chemical shifts and differ only in the ratio of T1 to T2. Assuming that the ÒbÓpool has the larger T1 /T2 ratio, it is possible to examine two cases, the Þrst of which corresponds to the objective of reducing the magnetization of the ÒbÓspins to zero while leaving the ÒaÓspins unperturbed. For simplicity, we assume without loss of generality that M0a = M0b = 1, which removes these parameters from the relationships of Eqs. (68) and (71). Thus Mza or Mzb describe the fractions of maximum longitudinal magnetization in the two pools, and we wish to maximize Mza while minimizing Mzb.

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We assume the condition (70) on spin pool ÒaÓand apply Eq. (71) to each spin pool. In this case the equations describing the steady-state magnetization are 1 (72) Mza = ω2 1 + ω1 2 TT2a1a and Mzb =

1 + (ωT2b )2 1 + (ωT2b )2 + ω12 T1b T2b

(73)

In general, an increase in saturation amplitude (ω1) or a decrease in saturation offset frequency (ω) will decrease the magnetization of both pools, but neither experimental parameter will independently deÞne conditions to saturate the ÒbÓ pool. We might therefore deÞne as optimal conditions on both ω and ω1 that will minimize the departure from the desired state of both pools, giving equal weighting to each pool. This is an arbitrary deÞnition, but provides the general form for resolving any sets of conditions that might be proposed. We deÞne xz as the fractional departure from the desired condition of the ÒzÓspin pool, and thus the optimal saturation condition as that which achieves the smallest value of x in both pools. Hence, xa = 1 − Mza =

1

ω12 T1a ω2 T2a ω2 + ω1 2 TT2a1a

(74)

and xb = Mzb

(75)

Equating xa and xb gives the relationship between ω and ω1 which corresponds to optimal saturation, or the maximal degree of selective saturation for a given value of ω:

 1/4 ω1 = 1 + T2b2 (ω)2



T2a T1a T1b T2b

1/4

(ω)1/2

(76)

Figures 18a and 18b demonstrate the behavior of the error parameter x in the ÒaÓand ÒbÓpools as offset frequency varies from 1 to 4 kHz. The intersections between respective xa and xb curves correspond to optimal conditions, and represent the minimal values of x that are achievable for the speciÞed ω. The value of x decreases with increasing offset frequency, and approaches a

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49

limiting value as   1/4 1/4 1 + ωT2b2 → ωT2b2

(77)

which is similar to Eq. (68) although less stringent, as a consequence of the fourth root. Clearly, if Eq. (70) is satisÞed in the ÒbÓpool, Eq. (77) must also be. Under conditions that satisfy Eq. (77), Eq. (76) reduces to a condition on

Figure 18. (a) Selective saturation of a spin pool with larger T1 /T2 ratio. The error parameters xa (= 1 − Mza , solid lines) and xb (= Mzb , dashed lines), corresponding to fractional departure from the desired saturation condition, are plotted as a function of saturation amplitude (ω1 ) for various offset frequencies (ω) (from left, within each set of curves, offset frequency = 1, 2, 3, 4 kHz). Spin pool ÒaÓhas T1 = 4.2 s and T2 = 0.07 s. Spin pool ÒbÓhas T1 = 1.0 s and T2 = 10−4 s. The intersections of xa and xb at each offset frequency (circles) correspond to optimal saturation conditions, as deÞned in the text. (b) Selective saturation of a spin pool with larger T1 /T2 ratio. The error parameters xa (= 1 − Mza , solid lines) and xb (= Mzb , dashed lines), corresponding to fractional departure from the desired saturation condition, are plotted as a function of saturation offset frequency (ω) for various amplitudes (ω1 )(from left, within each set of curves, amplitude = 50, 100, 150 Hz). Spin pool ÒaÓhas T1 = 4.2 s and T2 = 0.07 s. Spin pool ÒbÓhas T1 = 1.0 s and T2 = 10−4 s. The intersections of xa and xb at each offset frequency (circles) correspond to optimal saturation conditions, as deÞned in the text.

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JOSEPH C. McGOWAN

Figure 18. (Continued)

the ratio of ω1 to ω: ω1 = ω



T2a T2b T1a T1b

1/4

and the departure from selective saturation at the optimal condition is

(T1 /T2 )a x= (T1 /T2 )b

(78)

(79)

We refer to the quantity ω1/ω as the offset ratio. The saturation behavior of the two spin pools, assuming condition (77), is illustrated in Figure 19, which shows the variation of xa and xb with offset ratio, and the intersection of the two curves corresponds to the optimal saturation condition, which is achieved for any combination of ω1 and ω that have the correct ratio. Thus optimal conditions for the assumed constraints are obtained by setting ω large enough to satisfy condition (77), and then applying Eq. (78) to determine the appropriate saturation amplitude. The departure from selective saturation is then given by Eq. (79).

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Figure 19. Selective saturation of a spin pool with larger T1 /T2 ratio, approximate solution with (ωT2 )2 -1 in both pools. The error parameters xa (= 1 − Mza , solid line) and xb (= Mzb , dashed line), corresponding to fractional departure from the desired saturation condition, are plotted as a function of offset ratio (ω1 /ω). (T1 /T2 )a = 60.0 and (T1 /T2 )b = 104 . The intersection of these curves corresponds to the experimental conditions which maintain both pools equally close to the desired saturation condition (optimal saturation).

A similar argument has been pursued to examine the case where the pool to be saturated has the smaller T1 /T2 ratio (McGowan and Leigh, 1994). These arguments are made with the assumption that the chemical shifts of both pools are identical. In either case, if the chemical shifts are not identical, a greater degree of selective saturation is possible by irradiating at the frequency of the pool to be saturated. The maximal degree of saturation determined as outlined above can then be taken as a lower bound.

Y. Saturation in a Two-Spin Exchanging System In a two-spin system undergoing magnetic exchange, the expression for Mz is signiÞcantly more complicated, tending to preclude the simple analysis outlined above. However, it is clear that the longitudinal magnetization of the

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JOSEPH C. McGOWAN

ÒsaturatedÓpool tends to increase through exchange as well as relaxation. Further, the steady-state value of longitudinal magnetization with exchange in the saturated pool must be greater than the theoretical value that would be achieved in the absence of exchange. This is equivalent to stating that the steady-state longitudinal magnetization of the ÒnonsaturatedÓspins is decreased, which is the object of the experiment. However, assuming that the T2 value of the saturated pool is very small compared to the exchange rate, the degree of saturation of that pool may not vary signiÞcantly with exchange. This situation may exist in biological systems, where the T2 of the bound spins has been estimated to be on the order of 12Ð60μs (Henkelman et al., 1993; Morris and Freemont, 1992; Wolff and Balaban, 1989) and the Þrst-order exchange rate to be on the order of 0.3Ð5.0s−1 (Henkelman et al., 1993; Morris and Freemont, 1992; Wolff and Balaban, 1989). In the cases of systems where these estimates are reasonable the steady-state value of the longitudinal magnetization for the saturated pool is essentially the value in the nonexchanging case. This is illustrated by a comparison of the theoretical values of Mza and Mzb, calculated with an analytical solution of the coupled Bloch equations (Appendix I), as the pseudo Þrst-order exchange rate kab increases from zero. As before, the objective of the simulated experiment is to selectively saturate the ÒbÓpool. The no-exchange case corresponds to kab = 0, and complete saturation is deÞned as Mz = 0. Figure 20a shows the behavior of the ÒaÓspin longitudinal magnetization. For exchange rates close to zero there is a strong dependence on kab, which varies with changes in the intrinsic T1 of the ÒaÓspins. As the exchange rate increases, it does so with diminishing effect on Mz. This suggests that the potential accuracy of the estimation of kab from the Mza depends on the value of kab. Figure 20b shows the effect of selective saturation of the ÒbÓspins with exchange. In this case variations in exchange and T1a relaxation have a much smaller effect, due to the very rapid relaxation that occurs when spins are in the ÒbÓenvironment. The problem of achieving complete selective saturation is further magniÞed in dilute solutions of ÒbÓin Òa.ÓThis is expected, as a large amount of nonsaturated ÒaÓmagnetization is available to be transferred into the much smaller ÒbÓpool, tending to drive the ÒbÓout of the saturation condition. An example of this effect is shown in Figure 21. Magnetization transfer imaging, in contrast to the ÒdoubleresonanceÓand related techniques, attempts to achieve contrast between samples with different exchange characteristics. In a qualitative sense, this does not require a particular degree of selective saturation. However, it must be borne in mind that any results are dependent on the choice of experimental method, and that they do not provide an absolute measure of a tissue characteristic. The foregoing analysis suggests that experimental conditions that yield the highest degree of selective

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saturation will provide the most contrast, and further, that the no-exchange estimate of optimal experimental conditions may be a good approximation of the desired conditions in the presence of exchange. IV. Magnetization Transfer Imaging Magnetization transfer imaging (MTI) techniques exploit exchange processes to develop contrast in magnetic resonance images. A logical extension of the double resonance technique (Forsen and Hoffman, 1963a, 1963b, 1964), MTI provides a window into exchange and relaxation behavior in a sample of interest. The Þrst application of a magnetization transfer imaging technique in vivo was accomplished in 1989 (Wolff and Balaban, 1989). With a continuous RF

Figure 20. (a) Dependence of longitudinal magnetization (Mza /M0 ) on exchange rate and T1a in a two-spin exchanging system. Assumed relaxation and exchange parameters are T2a = 0.05 s, T1b = 2.3 s, T2b = 40μs, M0a /M0b = 5. Saturation offset frequency is 2000 Hz and ω1 = 100 Hz. (b) Dependence of longitudinal magnetization (Mzb /M0 ) on exchange rate and T1a in a two-spin exchanging system. Assumed relaxation and exchange parameters are T2a = 0.05 s, T1b = 2.3 s, T2b = 40μs, M0a /M0b = 5. Saturation offset frequency is 2000 Hz and ω1 = 100 Hz.

54

JOSEPH C. McGOWAN

Figure 20. (Continued)

Þeld that was applied off-resonance as referenced to the free water protons, a degree of selective saturation of the bound protons was achieved. The method included an auxiliary RF transmitter channel to provide continuous irradiation to the sample at a frequency several kilohertz removed from the free water resonance, while the main RF channel was available to excite and acquire the signal in the usual manner. The saturated spins were observed to transfer magnetism to the observable free water pool, resulting in a net signal decrease (Ceckler et al., 1992; Edzes and Samulshi, 1978; Eng et al., 1991). Thus, images were formed where a signal reduction (hypointense area) was associated with the presence of an exchange process. The differential effect on the free water resonance was assumed to result from magnetization transfer, whether as a consequence of dipoleÐdipolemediated cross-relaxation or as a result of chemical exchange processes. Along the lines of the double resonance technique, the longitudinal relaxation time with and without saturation was measured in addition to the reduction of longitudinal magnetization in the presence of saturation, allowing the calculation of an exchange constant (Forsen and Hoffman, 1963a). This calculation was done on a pixel-by-pixel basis, allowing generation of images that reßected the calculated exchange rate. Later, this part of the methodology

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Figure 21. The departure from saturation of the ÒbÓpool under conditions of continuous RF irradiation (ω1 = 100 Hz, ω = 2000 Hz) as a function of exchange rate (kab ) with various ratios of Ma0 /Mb0 (from lower curve, 1, 10, 102, 103, 104) As the system becomes more dilute the mechanism of exchange is more effective at driving the ÒbÓpool out of saturation.

was called into question in a letter by Yeung. He pointed out that unless the experiment is conducted with complete selective saturation, that is, complete saturation of the bound spins along with zero saturation of the water proton spins, the reaction rate calculation is not accurate. Yeung further asserted that these conditions were unlikely to be achieved in vivo (Yeung, 1993). YeungÕs argument establishes that the ForsenÐHoffman reaction rate, like the magnitude of the MT effect (difference between the experiments with and without saturation), is an experiment-dependent parameter as opposed to an absolute measure of a tissue property.

A. Pulsed Off-Resonance Magnetization Transfer Techniques An alternative experimental approach to magnetization transfer imaging employs pulsed, off-resonance irradiation. Like the continuous saturation method

56

JOSEPH C. McGOWAN

discussed above, this method applies RF irradiation at a frequency removed from the free water resonance. Saturation pulses are interspersed throughout the imaging sequence so that no additional RF transmitter channel is required (McGowan et al., 1994). The MTC achieves a steady-state magnitude (analogous to the steady-state magnetization associated with the repetition of onresonance pulses) which depends on the sample characteristics as well as the acquisition parameters. Pulsed off-resonance MTI has the advantage of ease of implementation, requiring no additional hardware or pulse designs over and above those required for conventional MR imaging. In early work the behavior of the pulsed MT effect was investigated under conditions of variable saturation, achieved by varying the duty cycle (or number of pulses) of saturating irradiation, as well as by varying the saturation frequency offset. SpeciÞcally, a gradient echo sequence was modiÞed by placing 19-ms sinc-shaped pulses at intervals within each repetition time period (McGowan et al., 1994). These pulses replaced pulses used for fat saturation and consisted of the central lobe of the {sin(x)/x} function. The sinc function has been ubiquitous in MR pulse design because its Fourier transform (or the pulse shape in the frequency domain) is a square wave. Contrast from T1 and T2 weighting in the MT images was minimized by the use of short (5 ms) echo times and ßip angles of 5Ð7◦ . The average amplitude of the saturation pulses was 3.7 × 10−6 T, resulting in an ω1 value of 156 Hz. Saturation pulses were applied at varying frequency offsets up to 15 kHz from the free water resonance. In order to measure the MTC effect, a reference image was acquired without saturation (that is, with the amplitude of the saturation pulses adjusted to zero) for each plane of interest. The average pixel intensities of representative homogeneous volumes of tissue were then compared with the average intensity of the identical pixels in the reference image to arrive at the fractional signal reduction. Magnetization transfer ratio images were obtained by dividing (pixel by pixel) the magnetization transfer image by the reference (no saturation) image. As a control, a vial of a 0.1 mM solution of MnCl in water (T1 = 900 ms and T2 = 81 ms) was imaged simultaneously with animal samples. This phantom was not expected to exhibit magnetization transfer. The data from the magnetization transfer experiments may be represented as a Z-spectrum (Grad and Bryant, 1990; Grad et al., 1990), where the water signal reduction due to the presence of saturation RF power is plotted as a function of the saturation frequency offset from resonance, ω. For each Z-spectrum, one reference image is acquired with saturation set to zero, and an MT image is acquired for every desired point. Magnetization transfer experiments to acquire the Z-spectra in animal tissue were carried out in vivo (piglets) and in vitro (bovine muscle) with from one to six saturation pulses per TR. Figure 22 shows the magnetization transfer ratio, deÞned as average signal intensity in

Figure 22. Magnetization transfer contrast (Z-spectra) in piglet brain, corresponding to 1, 3, 5, and 6 saturation pulses of 19 ms duration per TR of 140 ms. Sinc-shaped saturation pulses were applied with external Þeld B1 at 156 Hz. Imaging parameters included TE = 5 ms, ßip angle = 7◦ , four excitations, Þeld of view 12 cm, matrix 128 × 192.

a region of interest of the MT images, plotted as a percentage of the reference (zero saturation) image intensity, against the offset frequency of saturation pulses. The amount of MTC increases nonlinearly with increasing duty cycle at constant power, similar to the effect predicted by numerical simulations of Bloch equations (McGowan, 1993) and by the analytical models discussed earlier. Of concern in the evaluation of MT experimental results is the decrease in longitudinal magnetization due to direct saturation, deÞned as the signal loss that would occur in a homogeneous sample having identical relaxation characteristics to the free spins in the absence of magnetization transfer. The Z-spectrum of such a sample is of Lorentzian form, as predicted by the steadystate solution of the Bloch equations. It is clear that at offset frequencies close to resonance accompanied by high saturation power levels, the direct effect is dominant, and the points on the Z-spectrum reßect primarily characteristics of the free spins. In order to ensure that the pulsed off-resonance MT experiment was indeed sensitive to the exchange of magnetization between free and bound spins, results were compared with data from the MnCl sample. Figure 23 shows two Z-spectra of the MnCl phantom compared with two spectra from piglet brain (obtained with three and Þve saturation pulses per TR, respectively). One observes that the MnCl2, phantom experienced much less

58

JOSEPH C. McGOWAN

Figure 23. Comparison of MTI Z-spectra in piglet brain and MnCl2 phantom. Acquistion parameters as in Figure 22. The MnCl2 lineshape is Lorentzian as predicted by the steady-state solution of the Bloch equations for a single spin. These data represent direct saturation. The displacement of the piglet spectra results from the differential saturation of two pulse sequences with different duty cycles.

signal reduction due to the MT pulse sequence than did the tissue. In addition, the tissue spectra are separated by approximately 10% for their entire length, whereas the MnCl2, spectra are essentially the same for offset frequencies of greater that 2 kHz. The lineshapes of the MnCl2, Z-spectra are demonstrated to be of Lorentzian form by Þtting by an equivalent to Eq. (71), as shown in Figure 24. The piglet spectra cannot be Þt by Lorentzian lines and therefore do not primarily reßect direct saturation. Their displacement from one another reßects the differential saturation from the two pulse sequences with different duty cycles. These results indicated that direct saturation contributes to but does not dominate the Z-spectrum, and further that the Z-spectrum obtained with pulsed off-resonance saturation could be used to investigate magnetization transfer.

B. On-Resonance Pulsed MT Another method for generation of MTC employs a net zero degree, or Òtransparent,ÓRF pulse which is applied at the frequency of the free water resonance.

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Figure 24. The reduction in longitudinal magnetization in a 0.1 mM MnCl2 phantom in the presence of off-resonance saturation (diamonds) Þt to a Lorentzian (solid line). This represents the Z-spectrum of a nonexchanging system, which results entirely from direct saturation.

This is simply a short RF pulse modulated by a function that causes the free water spins to nutate away from, and then back to, the equilibrium position. The simplest example of such a sequence is the Òjumpand returnÓor binomial pulse. The theory of this method is reminiscent of the selective hydration inversion technique (Edzes and Samulski, 1977, 1978) and may be described as follows. If the total RF irradiation time (per repetition) is short compared to the T2 of the free water and long compared to the T2 of bound protons, this results in a net rotation of the bound proton spins. Once again, this appears to be primarily sensitive to the absolute T2 values of the two spin pools. However, the effect can alternately be viewed in light of the Fourier transform of the applied pulse. For example, consider the constant on-resonance pulse modulated by the function m(t) = cos(ωs t)

(80)

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JOSEPH C. McGOWAN

According to the modulation theorem, if the function f(t) has the Fourier transform F(ω), then the function f(t) cos ωst has the transform 1 1 F(ω − ωs ) + F(ω + ωs ) 2 2

(81)

That is, the modulation of the constant pulse by the binomial or similar function simply shifts the effective frequency of the ÒtransparentÓpulse by an amount ωs either side of resonance. Therefore, the combination of the short constant pulse Òon-resonanceÓand the modulation by the binomial function may have the same effect as that of applying pulses of saturating RF on both sides of the resonance frequency. Since the Bloch equation model is insensitive to the direction of the offset frequency, the effect on spin pools of different T1 /T2 ratios can be predicted by calculating the effective power as the sum of the individual components. From this point, the mechanism of saturation transfer operates identically to the CW experiment, and a similar effect is observed in the free water signal (Hu et al., 1992; Pike et al., 1992; Yeung and Aisen, 1992). C. A Relationship between Magnetization Transfer Contrast and T2 Since the Þrst magnetization transfer images were obtained it has been noted that MTC images are very similar in appearance to T2 weighted images. Therefore one is prompted to question to what extent MTC is a novel of contrast. A further question might be whether a relationship exists between T2 and magnetization transfer. Use of the simpliÞed models (Grad et al., 1990; Wu, 1991) as has been proposed ( Caines and Schleich, 1991; Caines et al., 1991; Grad et al., 1990; Wu, 1991) suggests that there is no relationship between the intrinsic T2 and magnetization transfer. Correlation is predicted, however, by the inclusion of the exchange of transverse magnetization, and this is consistent with some empirical evidence. D. Correlation in Images of Biological Tissue Magnetic resonance images were acquired as described earlier in brain tissue of human volunteers, piglets, and cat. Magnetization transfer images were compared with spin-echo images obtained during the same examination using single slice acquisition of identical slices. Scatter plots were constructed by plotting pixel intensity in the MT image against pixel intensity in the T2 (or T1) weighted image. Figure 25 shows the results of a correlation study in cat head. There are two obvious regions of correlation in this Þgure, which empirically correspond to brain and nonbrain (some muscle as well as noise).

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Figure 25. Pixel-by-pixel correlation between magnetization transfer contrast and T2 weighted contrast in cat head. The two regions of the graph correspond to brain tissue (r = 0.88) and nonbrain (r = 0.66). The lower correlation coeffecient of the nonbrain tissue is attributed to the inclusion of image points external to the tissue.

Similar results were obtained in piglets. In human volunteers the correlation was somewhat less apparent, in part because of the lower signal-to-noise ratio that results from a larger receive coil and lower saturation power, and in part because of a relatively small range of T2 or MT contrast. (A signiÞcant portion of the observed contrast in T2 weighted brain images obtained for this study was due to differences in proton density, consistent with the observations of others; (Ernst et al., 1987; Hennig et al., 1986). However, in a patient with multiple sclerosis (in a brain section demonstrating a broad range of T2 contrast), the correlation was easily visualized. This plot is shown in Figure 26. In that study there was no apparent correlation with T1. E. Correlation in Images of Agarose Gel Phantoms Further evidence of a possible relationship between observed T2 and MTC in a model system exhibiting magnetization transfer was obtained from agarose gel phantoms of graded concentrations, with variation of magnetization transfer effect via manipulation of the ratio of free spins to bound. The phantoms

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JOSEPH C. McGOWAN

Figure 26. Pixel-by-pixel correlation between magnetization transfer contrast and T2 contrast in the brain of a patient with multiple sclerosis. The pixels in the upper region comprise the whole brain.

were constructed as aqueous mixtures of agarose, a puriÞed linear galactan hydrocolloid isolated from agar or agar-bearing marine algae (Sigma Chemical Company, St. Louis, MO). The concentration of the phantoms varied from 12.5% to 0.45% by weight. These phantoms were imaged along with a phantom of plain water (identical to that which was used to mix the gels) using a standard pulsed off-resonance MT protocol. In addition, conventional relaxation time studies were done and maps of observed relaxation times were constructed using three-point pixel-by-pixel monoexponential Þtting. Maps of magnetization transfer ratios were constructed in the normal way by dividing (pixel by pixel) the MT image by an image obtained without saturation. Correlation between magnetization transfer ratio (MTR) and absolute observed T2 is demonstrated by Figure 27, and there was a relative lack of T1 versus MTC correlation. We note that Eq. (60) describes the relationship between concentration (1/ f ) and observed T2 in a system with fast exchange, suggesting a linear relationship between 1/T2obs and concentration, with the slope determined by the relaxation rate of the bound spins and the intercept equal to 1/T2a. Figure 27 demonstrates that the relationship is approximately linear, and the intercept is (within

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Figure 27. Pixel-by-pixel correlation between magnetization transfer ratio (deÞned as pixel intensity with saturation divided by pixel intensity without saturation) and absolute T2 as determined by three-point monoexponential Þtting. Six gel phantoms are shown with graded concentrations from 12.5% by weight to 0.45% by weight. The phantom corresponding to the upper right concentration of points contains water. Observed correlation corresponds to that observed in tissues.

experimental accuracy) approximately equal to the measured value of T2a (in the plain water phantom). This does not lead to a conclusion that the system exhibits fast exchange, however, as a similar argument could apply using Eq. (62) (slow exchange) if the exchange constant kab varies proportionally with concentration. As these two situations could not be differentiated with the experiment as performed, it was concluded only that there appeared to be a relationship between observed MT effects and T2, consistent with the idea that magnetization transfer may contribute to transverse relaxation in this model. In the pulsed MTI experiment, the repetitive application of saturation power forces the exchanging system into a steady state with regard to the observable longitudinal magnetization. The magnitude of this steady state reßects both tissue parameters and the degree of saturation achieved, which is a function of saturation power level and frequency offset. It is not necessary to achieve full saturation of the bound proton spins in order to obtain useful MTI data.

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Analysis of the Z-spectrum, which may represent a gradient of saturation effect, may prove to be useful in characterization of tissue.

F. Solving the Inverse Problem: Elucidation of Fundamental Model Parameters from the Z-Spectrum The binary spin-bath model based upon Bloch equations was used for much of the development of magnetization transfer experimental methodology. As noted, in that model a Lorentzian line shape is characteristic of both the free and bound spin pools. Because of the number of degrees of freedom involved with even the simplest two-site model, it is possible to achieve reasonable Þtting of experimental data (Figure 28). However, Yeung and Swanson observed that the Bloch model was unlikely to be representative of the bound spin compartment, which was postulated to exhibit more solid-like behavior. It was subsequently established that Þtting of large sets of experimental data could be improved by assuming non-Bloch behavior in the bound spin pool. The work of Henklemen, Swanson, and collaborators used substitution of a Gaussian lineshape (a lineshape that yields a reasonable approximation for RF absorption in many solids) for the absorption in the bound spin pool obtained good agreement between theory and experimental data in an agarose gel sample. The steady-state saturation equation derived by these investigators was as follows (Henkelman et al., 1993): Mza = 

Rb R M0b + Rr f b Ra + Rb Ra + Ra R  Ra + Rr fa + R M0b (Rb + Rr fb + R) − R 2 M0b

Mza is the magnetization of the free spins, Rx the relaxation rate of the x pool, R the exchange rate between a and b, and M0b the bound pool concentration normalized to the free pool. Rr f a is the Lorentzian governing saturation of the free pool, as predicted by the Bloch equations, Rr fa =

ω12 T2a 1 + (2πT2a )2

where T2a is the transverse relaxation time of the free spins, ω1 is frequency of precession corresponding to the amplitude of the off-resonance irradiation, and  is the frequency offset of the applied Þeld. In order to use a Gaussian lineshape for the bound spin pool, the following relationship was used:  2 2 π T2b e−(2π T2b ) /2 Rr fb = ω1 2

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Figure 28. Pulsed magnetization transfer (MT) imaging to obtain Z-spectra in piglet brain. Acquisition parameters included TR (repetition time) 140 ms, TE 5 ms, ßip angle 7◦ , four excitations/phase encode. MT saturation was applied with one, three, or six (top to bottom, respectively) sinc-shaped pulses of 19 ms duration per TR. Fitting to Òsix-pulseÓdata was performed by nonlinear least squares using the two-site Bloch model, which was constrained to positive parameter values which also Þt the one- and three-pulse data. Exchange and relaxation parameters for this Þt are T1a = 1.1 s, T2a = 45 ms, T1b = 0.4 s, T2b = 23 μs, kab = 9.3, f = 5.3. Predicted observed T1 is 870 ms. The root mean square error estimate of the six-pulse Þt is shown as dotted lines above and below the lower curve.

For application to tissue, it was subsequently found that a super-Lorentzian lineshape was advantageous, and further work has continued to reÞne this model (Li et al., 1997; Morrison, 1995). Still, solution of the inverse problem remains elusive as an in vivo technique. The requirement for acquisitions of MT data at a number of offset frequencies and with a range of saturation power levels is highly demanding of time. Current improvements in scan speed and reÞnements of the model have not yet proved sufÞcient for the design of a reasonable diagnostic study.

V. Application in Human Studies The quantitative endpoint of any of the experimental MT techniques is a value representing the difference in spin magnetization of the observed nuclei

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between the baseline condition and the saturation condition. For example, one might add MT saturation pulses to a standard MR imaging sequence in order to study a sample exhibiting the MT effect and containing water and macromolecular spins. Assuming that the saturation is perfectly selective, the water-spin magnetization will be maintained at a reduced value in the steady state. Moreover, the reduction will be larger in regions where the exchange of magnetization is more ÒefÞcient,Ówhere efÞciency is potentially a function of any of the six model variables introduced earlier. In practice, despite the fact that selective saturation is never perfectly selective in vivo, contrast between areas exhibiting varying degrees of MT effect is developed and superimposed upon the intrinsic contrast of the baseline image, be it proton-density weighting, T1 weighting, or some combination. In such an image, areas with highly efÞcient MT are dark, demonstrating that the saturation, or the reduced magnitude of longitudinal magnetization of the macromolecular spins, has been transferred to the water spins. Magnetization transfer contrast is used in a qualitative manner for applications including magnetic resonance angiography (MRA) (Edelman et al., 1992; Pike et al., 1992). Angiography refers to the imaging of blood vessels. MT is useful in this application because the MT effect is generally efÞcient in tissues and relatively ineffective in ßuids. In MRA, this translates into reduced tissue intensity, while blood remains bright. Other diagnostic studies use injected gadolinium-based contrast agents to modify the relaxation behavior of certain tissues. In these studies, incorporation of MT pulses into the imaging sequence can provide additional tissue suppression to allow the contrast-affected tissues to appear brighter. Appropriate control studies must be used in this case to ensure that the two independent effects are not confused. These applications of MT are now well established and implemented on many commercial scanners, in conjunction with both gradient-echo and spin-echo pulse sequences.

A. Quantitative MTI Magnetization transfer imaging pulse sequences are now available to some degree on state-of-the-art clinical MR scanners, although they may not be optimized for the acquisition of quantitative data, and certainly are not optimized for all possible applications. Further, there is great variability in the number of parameters that can be manipulated on any given scanner. Magnetization transfer saturation pulses can be added as a preparation to many pulse sequences used in clinical protocols. In order to make reasonable choices in acquisition parameters, it is useful to consider the pulsed off-resonance method of achieving MT saturation. Recall that continuous application of RF energy at the resonance frequency will lead to saturation of the overall spin magnetization.

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That is, the spin magnetization will be near zero magnitude and thus examining the magnetization with an MRI sequence will yield zero signal. Similar results are obtained in the steady state under conditions of pulsed application of RF. As the RF excitation is moved off-resonance, the saturation effect diminishes and goes to zero for large offset frequencies. However, the saturation effect is dependent upon the relaxation times of the affected spins, in such a way that solid-like macromolecular spins still experience some degree of saturation at relatively high offset frequencies (McGowan and Leigh, 1994). It is also observed that, for any given offset frequency, a larger magnitude of applied RF energy results in a greater degree of saturation, for all spins. The two essential parameters needed to describe the application of saturating RF energy are then effective offset frequency and effective saturation amplitude. The modiÞer ÒeffectiveÓis added to generalize the description and will be assumed in the following discussion. It is useful because in techniques other than pulsed offresonance MT, including those where pulse trains are given on-resonance to provide selective saturation, it is possible to establish the analogous equivalent frequencies and amplitudes to permit direct comparison with the continuous off-resonance RF case (McGowan, 1993). On MRI scanners where MT is implemented using off-resonance pulses the offset frequency can usually be read directly and modiÞed as a machine or control variable. The amplitude, on the other hand, may be given in degrees as would be a ßip angle, calculated by predicting the angle that spins would rotate if the pulse were given on resonance. Although this ßip angle does not have a physical basis, it can be used to calculate the strength of the MT saturation in more conventional units. Other possible variables are saturation pulse shape and duration, which also must be included in the calculation of effective saturation amplitude. When these variables are taken into consideration it is possible to generalize quantitative results that are from different groups and were acquired with different scanning equipment. An additional consideration is that it is possible to approach FDA limits on power deposition in humans with MT. Under these circumstances it may be useful to decrease the effective saturation amplitude along with the offset frequency, recognizing that to compare studies directly, both parameters must match. There is reason to consider using a baseline scan which minimizes relaxation time weighting, in order to avoid competing or canceling effects. For relatively rapid acquisition some centers have opted for gradient-echo based imaging with low ßip angle to minimize T1 weighting, and the shortest possible TE. The saturation effects on both water and macromolecular spins continuously decrease as the offset frequency is increased and saturation amplitude is held constant. There is no sharp boundary between regions where direct on-resonance saturation of the water spins is important, as opposed to where

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the transfer of saturated magnetization from macromolecular spins dominates the observation. Rather, both effects are likely to be present in any envisioned experiment. It follows, as has been noted, that the observed MT effect is highly dependent on the experimental parameters. On the other hand, theoretical predication and experimental observation conÞrm that the technique is robust and reproducible when proper attention is given to the acquisition parameters. The intensity of a region in an image obtained with MT contrast reßects the proton density in that region as well as relaxation times, which always inßuence image contrast to some degree depending upon the image acquisition parameters. For this reason it is desirable to normalize the MT data, with the object of calculating an index of MT effect which is to a degree independent of other measurable tissue parameters. By far the most common practice is to calculate an MT ratio (MTR), given here in a form equivalent to that of the originators (Dousset et al., 1992).  Ms · 100% (82) MTR = 1 − M0 In this equation Ms refers to the intensity of a region of interest (ROI) or a pixel under conditions of MT saturation and M0 refers to the intensity of the same region or pixel as measured on the control study. The ratio of intensities is historically subtracted from 1 so that the MTR increases with the MT effect. Pixels with near-zero intensity values (including regions without tissue present) are excluded from the analysis to avoid the problems of division of very small numbers. In practice, the MTR is not an absolute measure. It is rather a function of the amplitude of the effective saturating RF as well as of its frequency offset (Grad and Bryant, 1990; McGowan and Leigh, 1994). The MTR can be examined with a variety of techniques including region-of-interest analysis (Dousset et al., 1992). For example, contour mapping (Kasner et al., 1997; McGowan et al., 1998, 1999) of MT ratios has been demonstrated to be useful. An additional application has been the analysis of groups of MTR pixel values with histogram techniques (van Buchem et al., 1996, 1997). It has been noted that the measured or observed value of T1 differs from the intrinsic value because of the mechanism of exchange. This variation is understood and indeed was a fundamental part of the original double resonance methodology. The effect of the application of saturation pulses is to shorten the observed T1, as was observed and explained by Mann in 1977. Clearly, the measurement of T1 in the presence of saturation (T1sat ) provides another potential MR parameter for study. Such data can be obtained through a conventional inversion recovery experiment modiÞed to include MT saturation. Consideration should be given to the T1-shortening effects of MT saturation when protocols are designed, in order to avoid mixed contrast that is difÞcult

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to interpret. For example, in a study with heavy MT weighting and heavy T1 weighting, a region with effective MT would tend to be darker as a result of the MT weighting, but brighter as a result of the T1 shortening. In tissue, the former effect would be expected to dominate, but the latter would tend to bias a quantitative result to lower MT. For this reason quantitative MT studies are often designed to minimize relaxation time weighting of the acquired images. Region of interest analysis is, again, the most common technique used to evaluate quantitative imaging results. To characterize the MT effect, it is desirable to obtain two images: an image acquired in the presence of selective saturation of the macromolecular spins, and a control image identical in all respects except for omission of saturation. The images should be acquired sequentially or at the same time with no subject motion between acquisitions. There may be some advantage to registering the images to one another via rotation and translation operations, although this is not always essential. If a particular structure is of interest, a simple ROI analysis of a deÞned area may be most useful. Homogeneity of the structure and clearly deÞned physical boundaries, which ideally will not overlap the boundaries of the ROI, will maximize the precision of the measurement. The selected pixel locations on both images should then be used to Þnd corresponding intensities, which will be used to determine the MTR of the region. If, on the other hand, the region is not well deÞned or the whole image must be examined, it is useful to compute a pixel-by-pixel MTR map. To do so one must exclude pixels of low (near-zero) intensity, which could cause the computed MTR to be a very large number. As an example, this can be done via segmentation of brain parenchyma prior to analysis. Alternatively, a simple hard threshold can be applied, perhaps to exclude any pixels which have intensities lower than 10% of the maximum intensity of the proton-densityweighted control image. Both techniques will exclude pixels corresponding to noise whether external to the body being studied or within voids such as sinuses. With either technique, an index of pixels subsequently analyzed can be maintained so the eventual results can be put back into image format and viewed as a map of MTR. An example of an image with MT weighting, together with its corresponding MT map, is given in Figure 29. Compare the appearance of the lateral ventricles (central dark region on the MT map, light on the MT-weighted image). The irregular appearance of the borders of these cerebrospinal ßuid (CSF) spaces on the MT-weighted image is characteristic if disease, which is seen to extend outward from the ventricles. The quantitative MT values shown on the MT map highlight the heterogeneity of the disease. Note that in ßuids such as CSF the MT effect is small as predicted by theory.

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Figure 29. MT-weighted image and map of calculated MTR values (MTR map) in a patient diagnosed with multiple sclerosis. Average numerical values in regions of interest are depicted and demonstrate heterogeneity of the disease in this instance.

B. Example: Applications of Magnetization Transfer to Multiple Sclerosis and Diffuse Brain Disorders Multiple sclerosis (MS) is a diffuse brain disorder characterized by myelin loss resulting from a recurrent or chronic angiocentric inßammatory process. Hallmarks of the disease include multifocal inßammatory lesions characterized by lymphocytes and macrophage inÞltration, demyelination, and gliosis. In some cases remyelination is observed (Prineas and McDonald, 1997). Multiple sclerosis is considered by many to be a disease of white matter, as a result of the noted effects on myelin, but that premise has been challenged by evidence suggesting that axonal transection may be both widespread and to a degree responsible for neurologic impairment (Trapp et al., 1998). This observation is consistent with earlier Þndings that axonal damage was highly associated with permanent disability characterizing the later stages of MS (Allen and McKeown, 1979). The impact from an imaging perspective is that this observation could argue for shifting the focus of investigation to gray matter rather than white, where most MS lesions are found. State-of-the-art magnetic resonance imaging is not in isolation diagnostic for MS, which is conclusively identiÞed only in conjunction with a clinical (neurologic) examination. However, MRI is highly sensitive for detecting MS lesions and has rapidly become the standard methodology for conÞrming the diagnosis of MS. The lesions of

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MS in the brain and spine are bright on T2-weighted images, and some but not all lesions enhance with gadolinium contrast agents when viewed with T1-weighted imaging. However, as noted with regard to diagnosis, MRI is not speciÞc for the disease. By that is meant that MRI cannot at present distinguish the various classiÞcations of the disease (e.g, relapsingÐremittingvs chronic progressive) or prognosticate outcomes. There also exists a paradox in that clinical and cognitive status may not be closely correlated with imaging results. To be fair, the same shortcomings are present with any competing imaging modality. An attraction for the use of MRI in MS is that there is a need for sophisticated measures which are surrogate markers for the disease in order to document progression and to assess the efÞcacy of treatment protocols. This explains the current interest in quantitative imaging techniques such as magnetization transfer, whereby distinctions can be made within and among images when contrast differences are too subtle to be appreciated via subjective evaluation of the image. Contributing to this subjectivity is that when images are read by a radiologist or Þlmed by a technician for off-line evaluation, the image contrast is adjusted for optimal viewing via dynamic range control. The decision of the technician regarding what is optimal must be a compromise and may be called into questions by the radiologist, but the fact remains that typically only a single combination of contrast adjustment parameters is used for viewing the entire image or group of images. Longitudinal application of quantitative techniques allows ÒtrackingÓof disease processes in a way that is not possible when image contrast is adjusted in this way. Appropriate quantitative imaging techniques should be robust and reproducible, and ideally would reveal information about the underlying histopathologyÑthe microscopic state of the disease. Early experience with MT, as well as the theory that was advanced to describe the technique, suggested that MT might be a noninvasive probe for pathological study in vivo. This is especially desirable in MS, since patients often develop Þrst symptoms of the disease when relatively young and may expect to live for many years with MS. Investigations employing MT techniques are ongoing in a variety of animal models of disease and are expected to provide new insights that will allow investigators to differentiate the various pathological aspects of the disease. MT imaging as it is currently implemented is well suited to explore the natural history of MS, as has been demonstrated by a wide range of studies and centers employing the technique, and analyses exploiting the MT effect may play a role in upcoming pharmaceutical treatment trials. As noted, region-of-interest analysis refers to the averaging of data from a region of tissue (such as a portion of a white matter structure in the brain) that is expected to be homogeneous. This type of analysis is most frequently used in MT studies of MS and of animal models for the disease. One of the Þrst studies employed MTR to characterize inßammatory lesions in a guinea pig

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model of experimental allergic encephalomyelitis (EAE) without demyelination (Dousset et al., 1992). Their results were compared with data from human volunteers and MS patients using essentially identical techniques. The initial observation was that MTR was reduced in all lesions, where a lesion was deÞned as a relatively bright area on T2-weighted imaging. There was a suggestion in this study that MT might differentiate between inßammation and demyelination by virtue of smaller observed MT changes in the inßammatory lesions. Additionally, it was noted in MS patients that some areas of tissue where lesions were not detected (and thus were read as ÒnormalÓby a radiologist) appeared to be abnormal by MT Þndings. The observation of ÒoccultÓ white matter disease tended to conÞrm previous histopathological Þndings of microscopic damage due to MS in macroscopically-normal tissue (Allen and McKeown, 1979). Findings of reduced MT in MS as well as the presence of abnormal MT in NAWM were in turn conÞrmed by other investigators who noted areas of lowered MT in regions adjacent to lesions (Hiehle et al., 1994) and in frontal lobe NAWM (Filippi et al., 1995). It has also been suggested that changes in NAWM detectable via MTR analysis preceded by several months development of new MS lesion (Filippi et al., 1998), although those results were in apparent conßict with those of other investigators who found no signiÞcant MTR reduction prior to the appearance of lesions in three patients studied weekly (Silver et al., 1998). Further study is warranted (and ongoing) to resolve the apparent discrepancy. The natural history of MS lesions has been probed by examining MTR in lesions differentiated by enhancement pattern, where lesions were divided into groups characterized by radiological Þnding. Highest MTRs corresponded to homogeneously enhancing lesions, lower values to nonenhancing lesions, and the lowest values were found in the central portion of ring-enhancing lesions (Petrella et al., 1996). These results suggested a pattern whereby homogeneously enhancing lesions, representing early inßammatory lesions, might evolve to ring-enhancing or -nonenhancing lesions. Subsequent deactivation of the center portion of the ring-enhancing lesion could change the lesion to nonenhancing status. Resumed activity in the lesion might return it to a ringenhancing presentation, but as the tissue became essentially dead there would eventually be no return to enhancement (Petrella et al., 1996). Region-of-interest analysis was used in a study testing the sensitivity of MT to known histopathological changes in a feline model of Wallerian degeneration (Lexa et al., 1994). MT was found to provide a reliable indication of structural changes at a time when such changes were not detected via conventional imaging or with light microscopy, but were seen on electron microscopy. The time course of the MT Þndings corresponded to known histologic phases of Wallerian degeneration (Lexa et al., 1994), supporting the idea that quantitative MT imaging could provide a window on tissue structure.

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Drawbacks of ROI analysis include difÞculties in reproducibility as a result of the drawing technique and the human dependence upon placement of the ROI. Hand-drawn ROIs are particularly subject to human error, and ROIs of Þxed shape may necessarily contain tissue outside of the structure of interest. If the ROI is near a boundary, partial volume averaging may inßuence the results. Since MS is a disease characterized by focal lesions, small errors in placement of the ROI may result in relatively large errors in terms of average MTR. Finally, regions of different size may introduce statistical complications due to differences in variance with contributions from both the heterogeneity of the tissue and the number of pixels included. In MS, analysis of MT data by ROIs yields information about disease progression and extent. However, ROI analysis may not be the best technique for global characterization of MS, where both macroscopic and microscopic pathology is known to be present. Histogram analysis has been explored as an alternative to using a series of ROIs to describe the global state of disease in a region, tissue type, or whole brain (van Buchem et al., 1996, 1997, 1998). By constructing an MTR histogram, one trades spatial information present in an image for insight into the distribution of MTR values. Histograms thus provide a means of estimating the relative volumes of tissues characterized by speciÞc ranges of MTR, and allow conclusions to be drawn regarding both focal and diffuse aspects of the disease. Consider a histogram with the value of a parameter (such as MT ratio) on the horizontal axis and the prevalence of that value (number of pixels with that value) on the vertical axis. The range of horizontal axis values is divided into ÒbinsÓwith ÒbinsizeÓused to describe the interval of values corresponding to a single bin. Selection of an appropriate bin size is an important consideration as it will inßuence the appearance of the histogram as well as the value of the numerical parameters used to describe it. A large number of bins will produce a histogram with peak characteristics diminished and with excessive noise. Too few bins results in loss of the distribution information that the histogram is intended to provide. The optimal size may be the smallest number of bins that produces a smooth appearance of the histogram and is related to the noise present in the raw data. Common numerical indices which may be extracted from histograms, including peak height and peak location, may demonstrate effects modulated by bin size. Magnetization transfer histograms in brain are designed to depict a weighted distribution of disease in tissue. In studies to date the histograms are typically characterized by a single relatively sharp peak, which is asymmetric and has a preponderance of pixels with lower MTR. The location of the peak in normal control subjects corresponds to normal MTR values in white matter. The weighting of the distribution refers to the fact that the histogram will reßect the presence of a few pixels with sharply lowered MTR, such as would describe an

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MS lesion, and also the presence of many pixels with slightly lowered MTR, corresponding to MR ÒoccultlesionÓin NAWM. In the Þrst study to apply this technique (van Buchem et al., 1996), MT histograms were constructed using MTR maps from the Þve consecutive MRI slices rostral from the anterior commissure. Thus, a slab with a total thickness of 2.5 cm was examined from a brain region where a relative minimum of extracerebral tissue was present. The chosen brain volume also contained the periventricular area, including the corona radiata and centrum semiovale, sites of predilection for MS lesions. A bin size of 1% MTR was chosen and results were normalized to account for differences in brain volume/slice area. Results of this study indicated that the peak height of the histogram was a highly signiÞcant indicator of the presence of disease. Another observation was that, although the location of the peak was not different between control subject and MS patients, the distribution of pixels in the MS group was signiÞcantly shifted toward lower values. A longitudinal component of that study showed that peak height also decreased over time in a subgroup of seven patients, and that there did not appear to be a relationship between the peak height change and Kurtzke expanded disability status scale (EDSS) or ambulation index (AI) (van Buchem et al., 1996). Another study was conducted using more sophisticated image analysis that allowed precise and highly reproducible segmentation of brain parenchyma from other cerebral tissue. The brain tissue thus ÒisolatedÓwas subjected to MT histogram analysis in order to develop histograms limited to brain tissue (van Buchem et al., 1997). However, results were similar to those of the earlier study and were primarily limited to peak height changes with disease. Shape differences in the histograms were further quantiÞed by the introduction of the parameter MTRx, deÞned as the xth percentile of the histogram, or that MT value where the integral of the histogram was equal to x% of the total. In that study MTR25 and MTR50 were different in patients compared with controls, whereas MTR75 was not different. These results suggested a role for MTR histograms in monitoring disease progression, with potential application to therapeutic trials. On the other hand, the paradox that disease severity by MT was not strongly correlated to disease severity by clinical parameters remained. A larger study in 44 patients probed the relationships among MTR histogram parameters, clinical status, and neuropsychological test results (van Buchem et al., 1998), Þnding some apparent correlation. A new measure was added: unnormalized histrogram peak height (Hap). The reason for including this raw number was to attempt to include effects of atrophy, known to be present in the disease. The Hap was found to exhibit signiÞcant correlation with disease duration. This result was in apparent contrast with previous studies showing minimal correlations between duration of disease and MRI lesion load (Edwards et al., 1986; Huber et al., 1988) and was consistent with the idea that the course of MS is characterized by long-term progression. Results also

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suggested that increasing physical disability was accompanied by an increasing shift of the MTR distribution in the direction of lower values, highlighted by changes in MTR50 and MTR25. These correlations were weak, but this may be explained by the exclusion of spinal cord tissue from analysis combined with the relative high weighting of the EDSS and AI tests toward motor pathways. Neuropsychological testing comparisons indicated that the unnormalized histogram peak height was the most sensitive indicator of clinical status, and that it appeared to discriminate among patients classiÞed as normal, moderately impaired, and severely impaired. This suggested that the clinical neuropsychological manifestations of MS were a function of both atrophy and tissue disruption. Another study that employed the histogram analysis technique was designed to examine the effects of treatment with a pharmaceutical agent, interferon β-1b (Richert et al., 1997). In this case there was not an apparent treatment effect. The impact of MS disease in the spinal cord is acknowledged but less understood or studied by comparison with cerebral lesions. In general, MRI in the spine is more difÞcult because of geometry, complications of RF coil design, and magnetic susceptibility and motion artifacts. However, there is promise of improvements on all of these fronts, and some studies in spinal cord have been carried out successfully in models of diffuse disorders. For example, in a rat model of spinal cord injury using a procedure where a standard weight was dropped from a prescribed height, MTR histogram parameters were used to probe the extent of injury and correlation with microscopic damage as detected with histopathology. In this study another new parameter was introduced, which was the area of the histogram corresponding to statistically ÒnormalÓ white matter as determined by control studies. Compared with other measures previously employed, it was found to be most highly correlated with weight drop height in the model. All histogram parameters were found to be correlated with each other to some degree, and all were found to be highly correlated with histopathology, indicating the potential in this model for noninvasive measures of the extent of tissue injury. Interestingly, MTR-based parameters were noted to be slightly better than pathology at predicting weight drop height from the data. Figure 30 is derived from this study and represents a composite MTR histogram from the study population, demonstrating the global shift that accompanied different drop heights (McGowan et al., 2000). A relatively novel method for viewing MTR data is to display the MTR values as an overlay on the MTR or other image, using contour mapping techniques. Additional visibility may be provided by using colored contours over a grayscale image. The objective of such a display is to enable the detection of gradients and boundaries of abnormal MTR too subtle to be detected by conventional reading of the image. The technique was developed in an animal study of diffuse axonal injury (DAI) and was employed to explore correlation between MTR and histopathologic characteristics in a well-controlled animal model of diffuse axonal injury. Brain MRI of the injured animals was normal both

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Figure 30. Composite histograms from a study of spinal cord injury via standard weight drop in rats, demonstrating differences between control animals and two groups of injured animals where the severity of the injury was varied by changing the height from which the weight was dropped.

immediately following the injury and 1 week later, excluding signiÞcant contributions from hemorrhage. Magnetization transfer ratio contours were used to identify areas of abnormal MTR that were statistically different from normal tissue (i.e., 2 SD from normal). Only regions far from tissue boundaries were examined in order to avoid contamination of the study by partial volume effects (the inclusion of two tissue types in a single voxel). This constraint tended to make the study more conservative, since it is known that damage in this model is more severe near boundaries. Results indicated that MTR analysis had a positive predictive value of 67% for pathology-positive lesions, rising to 89% if the MTR was abnormal on the acute MRI as opposed to the later study. Corresponding negative predictive values were 56% and 61% (McGowan et al., 1999). Contour plotting was Þrst employed in MS in a study designed to investigate the appearance of lesion boundaries in brain. The results of the study indicated that most or all MS lesions examined demonstrated a gradient of MTR at the boundaries, as opposed to a sharp delineation between diseased and normal tissue. This was in contrast to observations in human lesions of diffuse axonal injury, which were by comparison well circumscribed (Bagley et al., 1999, 2000). A Þnal observation suggesting that MTR might provide a window on tissue structure was found in a dog model of Krabbe disease. Here, known patterns of

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demyelination associated with the disease were clearly observed in an affected animal and were found to be in sharp contrast to diffuse damage due to radiation in a treated affected dog and a sham-irradiated animal (McGowan et al., 1998). Contour plotting revealed the characteristic inside-to-out demyelination in this model in a way that ROI analysis was unable to do, suggesting future application of Ònoninvasive histopathology.Ó

VI. Conclusions The study of magnetization transfer as a novel contrast mechanism for MRI offers potential value in a number of ways. It is one of the Þrst examples of the quantitative use of MR imaging results, and thus has opened new avenues of investigation. The MRI examination generates huge numbers of data, some of which are unused in the process of generating an impression of the radiologic Þndings. Subsequent MRI scans with different acquisition parameters or in different planes may provide redundant information. Modern techniques are reducing inefÞciencies, but the protocol for the most appropriate or advantageous examination is still the result of a subjective judgement. Further examination of the meaning and use of the numerical data will no doubt prove beneÞcial. Magnetization transfer studies follow earlier studies that focused on individual pixel or ROI intensities and performed calculations to extract, for example, T1 values. Although it is clearly possible to obtain this data reliably, it is not common practice as there is judged to be little added value over the conventional T1 weighted image. The difference with MT is that the numerical calculation of MTR has been shown to be easily accomplished at a variety of sites, reproducible, and robust. There appears to be added value in the MTR, even though it remains experiment dependent as opposed to being an absolute measure of a tissue characteristic. The MTR apparently distinguishes different types of normal and diseased tissues, and does so in some cases where conventional relaxation-weighted images fail to. Thus, it offers insight into the natural history of some diseases and disorders. With regard to the phenomenon of MT, its study may provide answers to unresolved issues such as the nature of T2 relaxation and the relationship between this relaxation and tissue structure. The solution of the inverse problem-tissue parameters arrived at via analysis of the complete Z-spectrum remains elusive because of experimental constraints as well as theoretical gaps, but might be expected to be at some point in hand. Magnetization transfer theory may provide a key to a fuller understanding of relaxation in tissue and as a result the design of more efÞcient, sensitive, and speciÞc MRI studies.

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Appendix I: Solution of the Complete Coupled Bloch Equations for Two-Site Chemical Exchange The solution is normalized to M0a = 1.0, with f = M0a /M0b , Rx = 1/Tx , and AÐ D deÞned after the solution. Mza represents the longitudinal magnetization of the free spins, that is, the spoins with the smaller ratio of T1 to T2. Delω is equal to ω in the text and is deÞned as the frequency offset for saturating RF energy.  ω12 R1b ω12 Delω4 R1b M za := − R1a − − 2 AD f A B C(−R2a − ka)D f −

ω12 Delω2 R2a R1b ω12 Delω4 ka 2 R1b + A2 B 2 C(−R2a − ka)2 D A2 B C D f

+

ω12 R2b Delω2 R1b ω12 Delω2 ka R1b + 2 2 A BC D f A C(−R2a − ka)D f

+

ω12 R2b Delω2 ka 2 R1b ω12 R2b R2a R1b − A2 C(−R2a − ka)2 B D A2 C D f

−

ω12 ka Delω2 R1b ω12 R2b ka R1b + A2 C D f A2 C(−R2a − ka)D

ω12 ka R2a R1b ω12 f ka 3 Delω2 R1b − A2 C(−R2a − ka)2 B D A2 C D  ω12 ka 2 R1b ka R1b − − R1a − ka − A2 C D D

+

−

ω12 f ka ω12 ka ω14 ω12 Delω2 f ka 2 + − − − D AC AD ADC ABC

−

ω12 Delω2 R2a ka ω12 Delω4 ka + A2 B C(−R2a − ka)D A2 B C D

− −

ω14 Delω4 ω12 Delω4 f ka 3 − A2 B C 2 (−R2a − ka)D A2 B 2 C(−R2a − ka)2 D

ω14 Delω4 f ka 2 ω12 R2b ω14 R2b R2a + − A2 B 2 C 2 (−R2a − ka)2 D AC A2 C 2 D

+

ω14 Delω2 R2a ω12 Delω2 ka 2 ω14 Delω2 ka + + A2 B C 2 D A2 B C D A2 B C 2 D

−

ω12 R2b Delω2 ka ω14 R2b Delω2 ω12 R2b ka 2 + 2 + 2 2 2 A CD A C(−R2a − ka)D A C (−R2a − ka)D

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+

79

ω12 R2b Delω2 f ka 3 ω14 R2b Delω2 f ka 2 + 2 2 2 2 A C(−R2a − ka) B D A C (−R2a − ka)2 B D

−

ω12 R2b R2a ka ω12 f ka 2 R2a ω14 f ka R2a − − 2 2 A CD A CD A2 C 2 D

−

ω12 f ka 2 Delω2 ω14 f ka Delω2 ω14 R2b ka + 2 + 2 2 2 2 A C D A C(−R2a − ka)D A C (−R2a − ka)D

ω12 f 2 ka 4 Delω2 ω14 f 2 ka 3 Delω2 + 2 2 2 C(−R2a − ka) B D A C (−R2a − ka)2 B D  ω12 f ka 3 ω14 f ka 2 f ka ω12 − − − A2 C D A2 C 2 D DC +

C=− − + +

A2

A = −R2b − f ka −

ka 2 f −R2a − ka

B = −R2b − f ka −

ka 2 f −R2a − ka

Delω4 Delω2 R2b Delω2 f ka + + (−R2a − ka)A B (−R2a − ka)A (−R2a − ka)A

Delω4 f ka Delω4 f ka 2 Delω2 f ka 2 R2b − + 2 2 (−R2a − ka)B (−R2a − ka) B A (−R2a − ka)2 B A Delω2 f ka 3 R2a Delω2 R2a R2b R2a f ka + − − 2 (−R2a − ka) B A AB A A

ka Delω2 ka R2b f ka 2 − − + f ka AB A A D = − R1b − f ka − −

ω12 Delω2 A C(−R2a − ka)

ω12 Delω2 f ka 2 ω12 (R2a + ka) + A B C(−R2a − ka)2 AC

Acknowledgment The author is grateful to Drs. John S. Leigh, John Schotland, and Robert Grossman for collaboration and valuable conversations. This work was supported in part by the United States National Institutes of Health via research grant NS34353.

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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 118

Noninterferometric Phase Determination DAVID PAGANIN AND KEITH A. NUGENT School of Physics, The University of Melbourne, Victoria 3010, Australia

I. Introduction and Overview . . . . . . . . . . . . . . . . . . . II. Methods of Phase Imaging . . . . . . . . . . . . . . . . . . . A. Phase-Sensitive Imaging . . . . . . . . . . . . . . . . . . . 1. Zernike Phase Contrast . . . . . . . . . . . . . . . . . . 2. Hoffman Phase Contrast. . . . . . . . . . . . . . . . . . 3. Schlieren Phase Contrast . . . . . . . . . . . . . . . . . 4. Differential Interference Contrast . . . . . . . . . . . . . . 5. Propagation-Based Phase Visualization . . . . . . . . . . . B. Phase Measurement . . . . . . . . . . . . . . . . . . . . . 1. The HartmannÐShackSensor. . . . . . . . . . . . . . . . 2. Curvature Sensing . . . . . . . . . . . . . . . . . . . . 3. Through-Focal Series . . . . . . . . . . . . . . . . . . . 4. Interferometry . . . . . . . . . . . . . . . . . . . . . . III. A New Approach to Phase . . . . . . . . . . . . . . . . . . . A. Generalized Radiance . . . . . . . . . . . . . . . . . . . . B. A New DeÞnition of Phase . . . . . . . . . . . . . . . . . . C. The Interaction of the Generalized Phase with a Potential . . . . . IV. Propagation-Based Phase Recovery. . . . . . . . . . . . . . . . A. General Case. . . . . . . . . . . . . . . . . . . . . . . . B. The Coherent Transport-of-Intensity Equation . . . . . . . . . . C. Solution of the Coherent Transport-of-Intensity Equation . . . . . 1. Uniqueness of the Phase Recovery. . . . . . . . . . . . . . 2. Well-Posedness of the Solution . . . . . . . . . . . . . . . 3. Uniform Intensity Solution. . . . . . . . . . . . . . . . . 4. A Rapid Algorithm for Nonuniform Intensity . . . . . . . . . 5. Numerical Stability of the Reconstruction . . . . . . . . . . 6. Simulated Example . . . . . . . . . . . . . . . . . . . . D. Coherence Requirements for Propagation-Based Phase Measurement V. Experimental Demonstrations . . . . . . . . . . . . . . . . . . A. Phase Retrieval with Visible Light . . . . . . . . . . . . . . . 1. Optical Microscopy. . . . . . . . . . . . . . . . . . . . 2. Optical Phase Tomography. . . . . . . . . . . . . . . . . 3. In-Line Holography. . . . . . . . . . . . . . . . . . . . B. Phase Retrieval with X-rays . . . . . . . . . . . . . . . . . C. Phase Retrieval with Electrons . . . . . . . . . . . . . . . . D. Phase Retrieval with Neutrons . . . . . . . . . . . . . . . . VI. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . .

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I. Introduction and Overview The effects of phase are familiar to us all. Witness phenomena such as the twinkling of a star, which is the result of turbulence-induced phase shifts on the light incident from space; the heat shimmer over a hot road; or the effects of a lens in a projector. Indeed, for many systems such as biological samples in electron and optical imaging and almost all objects when illuminated with neutrons, the effects of phase are more important than the effects of absorption. Phase visualization and measurement is therefore a key enabling capability throughout science. It is a part of undergraduate physics that all waves are characterized by an amplitude and a phase. In more advanced treatments based on the theory of partial coherence, it is recognized that waves are always a superposition of waves with different frequencies and different phases. The result is that the concept of phase tends to lose meaning unless strenuous efforts are made to retain the coherence of the wave. Phase measurement has therefore been traditionally taken to implicitly require light that is highly coherent (spatially and temporally). However, research into adaptive optics, which attempts to develop techniques that are able to remove phase aberrations on wavefronts in real time, has been driven by astronomical problems and so uses starlight, which has a very broad range of wavelengths and therefore does not itself have a well-deÞned phase. It is therefore clearly possible to sensibly perform phase measurement even though the conventional idea of phase has broken down. More recently, the development of third-generation synchrotron sources has led to the availability of X-ray sources with a very small spatial extent. This very quickly led to the observation of strong refractive effects and the evolution of a new mode of phase visualization. In parallel with this work, the realization was developing that the refractive effects in all of these forms of optics could form the foundation for a new approach to phase measurement. In this article we review the development of this approach and present a summary of the experimental results to date. The structure of the paper is as follows. In Section II we give a brief outline of various existing methods of phase imaging. These methods may be broadly separated into two categories depending on whether the information they provide is qualitative (phase-sensitive imaging) or quantitative (phase measurement). In Section III we reevaluate the concept of phase from the point of view of the generalized radiance, which amounts to thinking of a Þeld in terms of the ßow of energy rather than in terms of amplitude and phase. This leads us to a new deÞnition of the concept of phase which

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is well deÞned for a partially coherent Þeld, and which reduces to the usual deÞnition in the coherent limit. More importantly, in Section IV we use this new conception of phase to show how one may perform quantitative noninterferometric phase imaging using partially coherent radiation. The phase retrieval algorithm thus obtained is rapid, deterministic, and robust in the presence of noise, and it returns a unique value for the phase map without the unwrapping problem associated with interferometry. This phase may be meaningfully related to the structure of a sample under investigation, even when the radiation is very far from being coherent. In Section V, we consider the application of these ideas to a wide range of experimental systems. These include two- and three-dimensional imaging using polychromatic visible light in microscopy; solving the twin-image problem of in-line holography; phase retrieval using hard X-rays; quantitative phase imaging of magnetic materials in Lorentz electron microscopy; and neutron phase imaging. In Section VI we make some concluding remarks.

II. Methods of Phase Imaging Complex scalar Þelds are completely speciÞed by their modulus and phase at each point of space-time. Because of the rapidity of Þeld oscillations at optical and higher frequencies (Born and Wolf, 1993), only the mean-square modulus of such Þelds (averaged over many cycles) is directly measurable. Consequently, phase-imaging methods for radiation at optical and higher frequencies are necessarily indirect. Whether for the purposes of qualitative or quantitative phase visualization, the primary aim of all methods of phase imaging is somehow to convert phase variations into intensity variations, which may then be directly observed. In this section we review a number of standard methods of phase imaging. Phase imaging may be split into two broad categories. The Þrst describes systems for which the phase is rendered visible but does not yield quantitative data. For the purposes of this review, we refer to this as Òphase-sensitive imaging.ÓThe second class of techniques has the capability to yield quantitative data and we term this Òphasemeasurement.Ó The methods of phase-sensitive imaging that we discuss include Zernike phase contrast, differential interference contrast, Hoffman phase contrast, Schlieren phase contrast and its variants, and propagation-based phase visualization. We also include some other methodologies that we regard as being able to be subsumed under these headings. The four methods of phase measurement we describe are the HartmannÐShacksensor, curvature sensing, through-focal series methods, and interferometry.

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A. Phase-Sensitive Imaging 1. Zernike Phase Contrast The principle of Zernike phase contrast, published in a Nobel prizeÐwinning paper of 1942 (Zernike, 1942), is elegant and simple. Suppose a thin, transparent, weak phase object is brought into focus by a near-perfect imaging system. Then the resulting complex disturbance ψ(r⊥ ) will be: ψ(r⊥ ) ≈ exp(iφ(r⊥ ))

(1)

where r⊥ denotes a two-dimensional position vector in the plane perpendicular to the optic axis z, and φ(r⊥ ) is the phase of the wave. The observed intensity will be very close to unity over the in-focus image, and so the transparent object will be essentially invisible. Since the phase object is weak, that is, |φ(r⊥ )| ≪ 1, we may approximate the complex exponential by its Þrst-order binomial expansion to give: ψ(r⊥ ) ≈ 1 + iφ(r⊥ )

(2)

The essence of ZernikeÕs idea was to note that the constant term of this expression can be changed from a 1 to an i (unit pure imaginary number) by placing a glass slide at the back focal plane of the objective lens of the imaging system where the slide has sufÞcient additional thickness at the optic axis to shift the constant term by π/2 radians. This Þeld is then reimaged so that the resulting wave is described by: ψ(r⊥ ) = i(1 + φ(r⊥ ))

(3)

With the Zernike phase-contrast Þlter (i.e., the slide) in place, the intensity of the resulting in-focus image is given by: I (r⊥ ) ≡ |ψ(r⊥ )|2 = (1 + φ(r⊥ ))2 ≈ 1 + 2φ(r⊥ )

(4)

Thus one obtains an image that has a linear dependence on the phase distribution. The use of more elaborate Þlters and optical conÞgurations is possible and has some advantages, but these will not be discussed here. 2. Hoffman Phase Contrast Hoffman phase contrast∗ is designed to achieve contrast in transparent and semitransparent specimens by converting phase gradients into variations of intensity. We have already seen that Zernike phase contrast relies on altering the phase of the radiation in the back focal plane of the objective lens. ∗

See, for example, http://micro.magnet.fsu.edu/primer/techniques/hoffmanindex.html.

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In the most basic form of Hoffman phase contrast microscopy, the amplitude of the radiation in the back focal plane is ÒmodulatedÓusing a Þlter that has a gray partially transmitting rectangular strip through its center, to one side of which all light is blocked, and to the other side of which all light is transmitted. The presence of a spatially varying phase distribution causes the image to acquire structure and so the phase is rendered visible. 3. Schlieren Phase Contrast The basic setup of Schlieren phase contrast (see, for example, Meyer-Arendt, 1992) differs from the basic setup of Zernike phase contrast in only one respect, namely that the Þlter at the back focal plane is replaced by a knife edge which blocks out half of the Fourier spectrum of the wave. Alternatively, Schlieren phase contrast can be seen as a limiting case of Hoffman phase contrast whereby the gray transmitting strip of the Hoffman modulator is reduced to zero width. Once again, the effect of perturbing the Fourier-transformed waveÞeld at the back-focal plane of the objective renders phase variations visible as intensity modulations. Schlieren phase contrast is extensively used in studies of ßuid dynamics. The crystal-based methods of X-ray phase contrast Þrst developed by Ingal and Beliaevskaya (1995) and further developed by Wilkins and co-workers (Davis et al., 1995a,1995b; Gao et al., 1995), and others (for example, Zhong et al., 2000), are analogous to schlieren imaging. The Foucault mode of electron microscopy, often used in the visualization of magnetic Þelds (see, for example, De Graef, 2001), is also a form of Schlieren imaging. 4. Differential Interference Contrast Nomarskii and Weill described the basic principle behind the differential interference contrast (DIC) microscope in 1955. Without going into details, the essence of the technique is to achieve phase contrast by splitting a waveÞeld into two copies, slightly displacing one of the copies transversely with respect to the other, applying a constant phase bias to one, and then recombining the waveÞelds. Assuming the phase bias to be π/2 radians, the transformation applied to the Þeld is: ψ(r⊥ ) → ψ(r⊥ ) + iψ(r⊥ − δr⊥ )

(5)

where δr⊥ denotes a small relative transverse displacement of the waveÞelds. If we now make the assumption of a weak pure phase object and

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take the square modulus of the result, then we end up with the following expression for the intensity: I (r⊥ ) = 2(1 + φ(r⊥ ) − φ(r⊥ − δr⊥ )) ≈ 2[1 + δr⊥ · ∇⊥ φ(r⊥ )]

(6)

where ∇⊥ denotes the gradient operator in the plane perpendicular to the optical axis. For small relative displacements δr⊥ of the waveÞeld with respect to itself, we conclude that the phase contrast observed in DIC is proportional to the gradient of the phase along the direction of δr⊥ . Thus, DIC achieves contrast in transparent and semitransparent specimens by converting phase gradients into variations of intensity. 5. Propagation-Based Phase Visualization All of the methods of phase-sensitive imaging which have been described so far have relied on the use of specialized optical elements to render phase variations visible as intensity variations. However, the observation of the twinkling of stars, or the light patterns on the bottom of a swimming pool, afÞrm that phase effects can be rendered visible without any optics at all. Moreover, the fact that phase can be rendered visible via a simple defocus of an imaging system is well known and was pointed out in ZernikeÕs original paper on phase contrast, where he stated, ÒEvery microscopist knows that transparent objects show light or dark contours under the microscope in different ways varying with defocus.Ó(Zernike, 1942). The use of propagation-based phase visualization has gained some importance in the last few years where it has been realized that it is possible to perform phase visualization with energetic X-rays without the need for specialised optics. The effect was Þrst noted with third-generation X-ray sources and has subsequently been used quite extensively to image samples (Snigirev et al., 1995), and even to create qualitative tomographic reconstructions (Spanne et al., 1999) where the three-dimensional Laplacian of the phase is recovered. Interestingly, Wilkins et al. (1996) have shown that phase visualization is possible even with laboratory X-ray sources, and this has raised the possibility of clinical applications of phase X-ray radiography (see, for example, Kotre and Birch, 1999).

B. Phase Measurement 1. The HartmannÐShack Sensor The HartmannÐShacksensor (for a good review of this Þeld, consult Tyson, 1991) provides a means of phase measurement that is widely used by both the astronomical adaptive optics (see, for example, Rigaut et al., 1997) and

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ophthalmology (see, for example, Roorda and Williams, 1999) communities. In both cases the device is used to determine phase perturbations induced by the passage of light waves through an aberrating medium. The HartmannÐShacksensor is an array of lenslets, each of which brings the incident Þeld to a focus. When normally incident plane waves are shone onto the sensor, each lenslet brings the light to a focus at the center of its associated detector. If an aberrated wave is incident onto the sensor, then the location of each spot will be displaced by a vector proportional to the average phase gradient over the lenslet. This displacement may be sensed by, for example, a quadrant detector. The resulting signals may then be used to directly feed into a ßexible (i.e. adaptive) optic to correct for the phase aberration, or it may be processed to create an estimate of the phase distribution. The advantage of the HartmannÐShacksensor is its simplicity and its ability to interface directly into a real-time correction system. The disadvantage is that the spatial resolution over the wavefront is limited by the size and number of the lenslets, and it cannot easily be incorporated into imaging systems such as microscopes. 2. Curvature Sensing The method of curvature sensing was pioneered for use in the astronomical adaptive optics community by Roddier and colleagues at the University of Hawaii (Roddier, 1988, 1990; Roddier and Roddier, 1993). It can be shown, building on the previous quotation from Zernike, that when a wave of uniform intensity, but nonuniform phase, is imaged, then the difference between a positively and negatively differentially defocused image is proportional to the Laplacian of the phase: that is, to the curvature of the wave. The essence of curvature sensing is to record these two images and to use the difference between them as a signal to feed into an adaptive optic. Typically the curvature sensing is done in real time in order to correct an aberrated image. The phase itself is therefore not normally recovered, but this is certainly possible. The explicit determination of the phase using this method amounts to a requirement to solve the uniform transport of intensity equation and this topic is covered in detail in Section IV.C.3. 3. Through-Focal Series The discussion of curvature sensing reinforces the idea that the intensity of an out-of-focus image formed by a given optical system will be inßuenced by the phase distribution of the radiation over the in-focus image. More generally, the intensity of the radiation over any out-of-focus plane is a function of both the intensity and phase of the in-focus radiation over the plane of interest.

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The idea of the through-focal-series approach to phase determination is to acquire a set of images at a range of valus of defocus. The data set will include the in-focus image as well as data with a very large defocus. This data are then entered into an algorithm that Þnds the in-focus phase distribution that is consistent with the complete set of data. The problem of phase determination using through-focal series has been a major issue in high-resolution transmission electron microscopy and workers in this Þeld have developed a range of techniques to recover phase. Of particular interest in this context is the work of Van Dyck and co-workers (Van Dyck and Coene, 1987; Coene et al., 1992; Op de Beeck et al., 1996) where iterative techniques have been extensively applied to the determination of the phase of an electron waveÞeld at the exit surface of a crystal. Cloetens et al. (1999) have applied these techniques to X-ray phase determination. They start with a unique initial estimate for the phase which is obtained from the focal series, and then recursively optimize this estimate using a full description of the image formation process based on Fresnel diffraction. This procedure was used to image a polystyrene foam sphere, an object that is essentially transparent at the hard X-ray energies used in the experiment. Cloetens et al. (1999) took the additional step of incorporating such a focal-series based phase retrieval into a tomographic algorithm, allowing them to demonstrate quantitative three-dimensional phase tomography using hard X-rays. This technique has been dubbed ÒholotomographyÓto indicate a synthesis of holographic and tomographic techniques. The question of the uniqueness of the phase retrieval from a through-focal series of data has yet to be answered deÞnitively, although the work discussed in later sections of this paper demonstrates that the phase is uniquely deÞned by the intensity under the assumption that the phase is continuous. 4. Interferometry Interferometry is an extremely well known technique and has an extensive literature (for a review, see Hariharan, 1992). However, it has some features that, perhaps because of its familiarity, are sometimes neglected. The basis of interferometry is to overlay one coherent beam with another. The resulting coherent superposition results in interference fringes described by  (7) I (r ) = I1 (r ) + I2 (r ) + I1 (r )I2 (r ) cos (r )

where I1 (r ) is the intensity of the reference wave and I2 (r ) is the intensity of the wave of interest (the object wave), and (r ) is the phase difference between the two waves.

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Consider an object wave potentially containing both spatial intensity and phase variations. Inspection of Eq. (7) conÞrms that identical modulations in the third, interference, term may arise from variations in either the phase or the intensity of the wave of interest, or a combination of both. Consequently, the foregoing expression does not enable the effect of phase and intensity variations to be decoupled using a single interferogram. In general, then, the determination of cos  requires at least two independent measurements. Secondly, note that only the cosine term is measured and this only yields the phase modulo 2π. This does not present a problem in one dimension, but the unwrapping of this effect is two dimensions is a complex task particularly if the phase distribution is not continuous (for a review of phase unwrapping, see Strand and Taxt, 1999) or the data contain signiÞcant levels of noise. We make these points here as we later wish to compare the limitations in the propagation-based phase recovery techniques with those of interferometry. In summary, we can see that there is a wealth of approaches to phase visualization and to phase measurement and we have only touched on the literature in the preceding paragraphs. In what follows, we consider the issue of phase in terms of describing the directions of energy ßow. We note that the work on Schlieren imaging, curvature sensing and the HartmannÐShacksensor all implicitly associate phase with the direction of energy ßow. It seems, therefore, that association of phase with propagation is a very natural perspective to adopt. Given the foregoing, the next section discusses how the idea of phase may be generalized and then adapted to make phase measurements in a very broad range of applications.

III. A New Approach to Phase A. Generalized Radiance A number of the phase measurement techniques described earlier implicitly associate the phase distribution with the direction of energy ßow. In this section we introduce the Wigner formulation so as to show how this relationship may be made more formal. The Wigner function (Wigner, 1932) is widely used in quantum mechanics and is increasingly being recognized as a powerful analytical tool in optics (Bastiaans, 1986; also see the December 2000 issue of the Journal of the Optical Society of America A for a special issue entitled ÒWigner Distributions and Phase Space in OpticsÓ).

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Consider a partially coherent quasi-monochromatic paraxial wave described via its mutual optical intensity function, J (r1 , r2 ) (Marathay, 1982), deÞned by J (r1 , r2 , z) ≡ ψ(r1 , z)ψ ∗ (r2 , z)

(8)

where r1 and r2 are two-dimensional position vectors perpendicular to the optic axis z, and the angle brackets denote an ensemble average. Introduce a new coordinate system r⊥ ≡

1 (r1 + r2 ) 2

x ≡ r1 − r2

(9a) (9b)

If we write the mutual intensity function with these coordinates, then the Wigner function for a wave may then be written as    W (r , p) = J (r , x ) exp −2πi x · p/λ d x (10) The standard results of paraxial partial coherence theory may then be applied to show that the resulting function propagates according to W (r⊥ , p⊥ , z) = W (r⊥ − z p⊥ / p, p⊥ , 0),

(11)

where p is the average momentum of the wave. It also follows that the probability distribution is obtained from  ρ(r⊥ , z) = W (r⊥ , p⊥ / p, z)d p⊥ (12) A momentary reßection on the preceding expressions leads one to be struck by their very geometric form. If we interpret the vector p⊥ as the direction cosine of a traveling Þeld, then Eq. (11) simply describes geometric projection and Eq. (12) says that the intensity at a point is the integral of the Wigner function (or, as it is known in this context, the generalized radiance) over all angles of propagation. It is also easy to show that the Wigner function is always real (though not always positive). We thus have a partially coherentÑor coherentÑw aveÞeld simply described in terms of a real function of two variables, position and direction, which obeys simple geometric propagation laws. It is tempting to take this picture a long way; the simple interpretation breaks down for nonparaxial waves, and the possibility of the generalized radiance becoming negative makes a rigorous and self-consistent physical picture rather difÞcult. However, this observation leads one to consider the possibility of

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recasting our conventional ideas of phase in terms of propagation. It is to this thought that we now turn.

B. A New DeÞnition of Phase In the remainder of this paper we will use a notation appropriate for quantum mechanics. Note, however, that with a suitable translation of notation, the ideas apply to all forms of complex scalar waves. Consider a random statistically stationary waveÞeld with an associated probability ßow vector deÞned by h ∇ψ(r , t)] (13) im In a region of space that is free of sources and sinks, the principle of energy conservation implies that the ensemble-averaged ßow vector must obey the continuity equation:  r ) ≡ Re[ψ ∗ (r , t)  j(

 r ) = 0 ∇ ·  j(

(14)

If the waveÞeld is stationary-state (in the quantum-mechanical sense) or, equivalently, coherent (in the optical sense), then its spatial part may be written as  ψ(r ) = ρ(r ) exp[i S(r )], where ρ(r ) is the probability density and S(r ) is the phase. In this case the probability current is time-invariant and assumes the form (Messiah, 1961)  r ) = h ρ(r )∇ S(r ) (15) j( m where m is the mass of the particles of the matter wave. Evidently, the phase and probability density determine the probability current. Since both the current and probability distribution are observables (Aharonov et al., 1993), we conclude that the phase may be deÞned in terms of observables, without any reference to interferometry. Paganin and Nugent (1998) have taken these observations and created a meaningful and very general deÞnition of phase that may be based entirely on the concept of the current vector. While the current vector is considered to be the fundamental quantity, the approach of Paganin and Nugent differs signiÞcantly from other optical frameworks that are based on observables and use correlation functions as their starting point (Wolf, 1954; Gabor, 1961). In general, the probability current associated with a given radiation Þeld will be a function of time. Since we assumed the Þeld to be statistically stationary, (Born and Wolf, 1993), we could meaningfully introduce the ensemble aver r ). This is a age of the probability current for a partially coherent Þeld,  j(

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well-deÞned vector Þeld and will be used as the basis for our formulation of phase. This notion remains well deÞned for partially coherent Þelds and we will show that it reduces to the conventional deÞnition of phase in the coherent limit of vortex-free waves (for the case of a scalar Þeld). Paganin and Nugent deÞned the normalized probability current in terms of the ensemble average current using: ö r ) ≡  j(

 r )  j( m lim+ h ε→0 ρ(r ) + ε

(16)

Over regions of nonzero time-averaged probability density, Eq. (16) describes a well-deÞned vector Þeld which may therefore be Helmholtz decomposed into a potential and a rotational component in the usual way (Morse and Feshbach, 1953). Performing this decomposition, the authors were able to rewrite the probability current in the following form, which is analogous to the expression for the electromagnetic current vector in the presence of both scalar and vector electromagnetic potentials (Messiah, 1961): 1 ρ(r ){∇φ S (r ) + ∇ × φV (r )} (17) m Equation (17) is regarded as deÞning the scalar phase, φ S , which is single valued, and the vector phase, φV , which is divergence-free, in terms of the  r ) and the ensemble-averaged probensemble-averaged probability current  j( ability density ρ(r ). Equation (17) may be inverted to express the phase  r ) (Paganin and Nugent, components in terms of the probability current,  j( 1998). This decomposition is unique up to a vectoral constant that may ßoat between the two components; we place this vectorial constant in the gradient term. The phase so deÞned obeys the following Poisson-type differential equations:  r ) =  j(

ö r ) ∇ 2 φ S (r ) = ∇ ·  j(

(18a)

ö r ) ∇ 2 φV (r ) = −∇ ×  j(

(18b)

These may be solved for the phase via the following integrals (Morse and Feshbach, 1953):  ö r ′ ) 1 ∇ ·  j( d 3r′ φ S (r ) = − (19a) 4π |r − r′ | φV (r ) =

1 4π



ö r ′ ) ∇ ×  j( d 3r′ |r − r′ |

(19b)

In our work, we regard these expressions as the deÞnitions of phase.

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We emphasize that these deÞnitions are valid and meaningful even when the wave is partially coherent. We conclude this section by showing that the deÞnition of phase given in Eqs. (19a) and (19b) reduces to the usual deÞnition of phase in the coherent limit, provided of course that one is dealing with a scalar waveÞeld. In the  r ) = ρ(r )∇ S(r )/m (Messiah, 1961), Eqs. (19a) and coherent limit, where j( (19b) reduce to  1 ∇ 2 S(r ′ ) 3 ′ d r (19c) φ S (r ) = − 4π |r − r′ | φV (r ) =

1 4π



∇ × ∇ S(r ) 3 ′ d r |r − r′ |

(19d)

 Thus, in the Note that the vector phase φV (r ) will vanish if ∇ × ∇ S(r ) = 0. coherent limit, the vector phase will vanish if the conventional phase of the wave is single valued and continuous. In the case of a coherent Þeld, then, the vector phase is only nonzero if the phase of the waveÞeld is discontinuous, or multiply valued, and so corresponds to a topological phase (Dirac, 1931). It is also apparent from Eq. (19c) that the scalar phase reduces to the conventional phase when the Þeld is coherent and the phase is continuous (Paganin and Nugent, 1998). We see, then, that the phase of the waveÞeld as we deÞne it reduces to the traditional deÞnition when the Þeld is coherent and vortex free. However phase measurements are usually used to probe the properties of a sample, such as in phase-contrast microscopy. In the next section we explore whether our deÞnition is meaningful in the context of such measurements even though the object will be probed using a partially coherent Þeld.

C. The Interaction of the Generalized Phase with a Potential Nugent and Paganin (2000) have considered the behavior of the generalized phase in terms of its interaction with a potential. In this review, we brießy summarise their argument. In their paper, they considered a general partially coherent scalar wavefunction of the form  aν ψν (r )e2πiνt (20) (r , t) = ν

where aν denote the amplitudes of the component wavefunctions, ν denotes the corresponding frequencies, t is time, and ψν denotes the spatial part of

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each monoenergetic component of the wavefunction. Note that this is closely related to the coherent mode formulation of coherence theory developed by Wolf (1982). They showed that the Wigner function of this wavefunction is  |aν |2 Wν (r , p) (21) W (r , p) = ν

where

Wν (r , p) ≡

1 (2πh)3



    ψν r + x /2 ψ ∗ ν r − x /2 e−i p · x /øh d x

It follows that the average probability ßow vector is given by    r ) = 1 |aν |2 pWν (r , p)d p  j( m ν

(22)

(23)

This states that the partially coherent average probability ßow vector can be written as the incoherent sum of the ßow vectors for the component wavefunctions. The interaction with a potential can therefore be described as the sum of the interactions with the component wavefunctions and this is the argument adopted. Without going into the details of the calculation, the result is that the outgoing ßow vector can be written as the sum of the incoming vector plus the effect of the potential:  (0)  1  |aν |2 ρν (r )∇ SV (r , ν) (24) j (r ) =  j0 (r ) + m ν

where  j0 (r ) is equal to the incident probability current at z = 0, and SV is the phase induced by the presence of the potential. Assume the potential term may be factorized into the form SV (r , ν) = (r ) f (ν) where we deÞne f (ν) to be such that f (ν) = 1, where ν is the average frequency of the incident  wavefunction. In this case, the sum in Eq. (24) can be written as ∇(r ) ν |aν |2 f ν ρν (r ), where f ν are the values of f (ν) evaluated at the frequencies in the sum. Nugent and Paganin (2000) then make a key assumption. They assume that f ν ≈ 1 over the spread of frequencies in the wavefunction. This assumption implies that dispersion is negligible over the frequency width of the wavefunction. In this case, using the properties of the Wigner function, we obtain   |aν |2 f ν ρν(0) (r ) ∼ |aν |2 ρν(0) (r ) = ρ(r ) (25) = ν

ν

so that Eq. (24) may be written 

 1 (0) j (r ) ∼ =  j0 (r ) + ρ(r )∇(r ) m

(26)

NONINTERFEROMETRIC PHASE DETERMINATION

This allows us to write 

 1 (0) j (r ) ∼ = ρ(r )∇[S(r ) + (r )] m

99

(27)

Thus, we come to the conclusion that the probability current leaving the potential has a form identical to that of the coherent probability current where the generalized phase, deÞned in Eqs. (19a) and (19b), acts precisely as would the conventionally deÞned phase. We therefore have the strong deduction that propagation based phase determination techniques can be applied even though the incident wave does not have a conventionally deÞned phase. In other words, a determination of the probability current will allow the accurate probing of the phase modiÞcation of the wavefunction by the medium in precisely the same way as would a phase-based coherent measurement.

IV. Propagation-Based Phase Recovery A. General Case We begin by considering the case of a coherent nonparaxial wave, examples of which might include a monoenergetic beam of electrons or an atom laser (Helmerson et al., 1999). Substitution of Eq. (15) into Eq. (14) yields (Madelung, 1926) ∇ · (ρ(r )∇ S(r )) = 0

(28)

This equation can be shown to have a unique solution for the phase S provided that the probability distribution is known and is always greater than zero,∗ except for some special cases of great symmetry and little practical importance. Thus, given these conditions, the phase of a wave is uniquely determined by its distribution of probability density in three dimensions. This is an interesting observation; however, we do not pursue if further in this review. Instead, we specialize our considerations to the paraxial case. ∗ An analogous uniqueness proof for the uniqueness of determination of the time-dependent phase of scalar quantum-mechanical waves from the time-varying probability density was obtained using the hydrodynamic formulation of the time-dependent Schr¬ odinger equation. See E. Feenberg (1933), The scattering of slow electrons in neutral atoms, doctoral thesis, Harvard University. FeenbergÕs proof is given on pp. 71Ð72of E. C. Kemble (1937), The Fundamental Principles of Quantum Mechanics, with Elementary Applications, New York: Dover Publications, New York, and on pp. 49Ð50of I. Bialynicki-Birula, M. Cieplak, and J. Kaminski (1992), Theory of Quanta, New York: Oxford University Press.

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B. The Coherent Transport-of-Intensity Equation All of the quantitative date presented to date have been under the paraxial approximation and this is what we now consider. Equations (11) and (12) may be combined to write  (29) ρ(r⊥ ) = W (r⊥ − z( p⊥ / p ), p⊥ , 0)d p⊥

It is easily shown from this that

∂ρ(r⊥ ) = −∇⊥ · ∂z



p⊥ W (r⊥ , p⊥ , z)d p⊥

(30)

Gureyev et al. (1995a) have shown that a substitution of a coherent wave into this expression, along with the paraxial approximation, leads to the transportof-intensity equation: h ∂ρ(r⊥ ) = − ∇⊥ · (ρ(r⊥ )∇⊥ S(r⊥ )) ∂z p

(31)

This equation can also be readily obtained by writing the paraxial version of Eq. (28). It is the solution of this equation that is the topic of the remainder of this review. Before considering the solution and application of this result, we note that the transport-of-intensity equation was, to our knowledge, Þrst derived in the context of phase retrieval by Teague in 1983. Teague derived the expression in a rather different manner that is perhaps simpler than the approach developed here, but has less generality. Note, moreover, that earlier published instances of the equation (such as that of Rytov et al. in 1978) certainly exist. Further, special cases of the transportof-intensity equation may be traced as far back as BremmerÕs paper of 1952, and the work of Lynch et al. in 1975 (also see Spence, 1981). It should also be noted that Eq. (31) is at the quantitative heart of all propagation-based phase visualization techniques in the sense that all of them can be simply understood in terms of this propagation expression. C. Solution of the Coherent Transport-of-Intensity Equation 1. Uniqueness of the Phase Recovery The derivative of the probability along the z-axis and the probability distribution in that plane are both observable quantities and so Eq. (31) offers a direct approach to the quantitative measurement of phase from noninterferometric measurements of probability density. This approach will require two

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consecutive measurements of the probability distribution and so we require that the waveÞeld be statistically stationary. We also point out that we do not measure the amplitude and phase of a single particle and so the approach does not violate the uncertainty principle. The emphasis on this work, however, is on the solution of (31), which has elsewhere been proven (Gureyev et al., 1995a) to have a unique solution for the phase, up to a physically meaningless additive constant, provided the probability distribution is always strictly positive. This additive constant is meaningless as the wave equation is invariant under a shift in the origin of time. The requirement of strict positivity is an important one. The presence of an intensity zero heralds the possible presence of a dislocation in the phase of the wave and it can be shown that, in this case, two different phases may have identical probability distributions in space. In all of the experimental results presented here we make the assumption that there are no phase dislocations in the Þeld. It has been pointed out that this is a restrictive assumption. Allen et al. (2001) and Nugent and Paganin (2000) have explored these issues in some detail. Moreover, it has been found that iterative approaches can deal with phase dislocations in a more coherent manner, but the fact remains that in many circumstances interferometry is the only method by which the phase may be unambiguously recovered. Before proceeding, following the work of Nugent and Paganin (2000), we will just explore this matter a little further. It can be shown that any multivalued phase distribution may be written as a sum of edge and screw dislocations (Voitsekhovich et al., 1998). It follows (Nugent and Paganin, 2000) that the general coherent transport equation (31) may be written in the form  m j ∂  p ∂ρ(r⊥ ) = −∇⊥ · (ρ(r⊥ )∇⊥ φ S (r⊥ )) − ρ(r⊥ ) (32) h ∂z r j ∂θ j j Here, m j is the topological charge of the jth dislocation and r j is the distance between the jth dislocation and r⊥ . The physical picture implied by this equation is that the effect of the scalar phase is a lateral translation of probability density as it moves along the axis z. The effect of the screw dislocation is a rotation as we move along z. It follows, therefore, that the presence of screw dislocations has a characteristic signature in the propagation of the probability distribution. It is thought that it is this feature that enables the iterative phase recovery techniques to be more successful, whereas the algorithms that are the primary subject of this paper explicitly exclude such structures. The approach to this problem using our direct techniques is the subject of further work, although we note that Nugent and Paganin

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(2000) have proposed the rudiments of an algorithm. In the remainder of the paper we exclude consideration of the reconstruction of phase discontinuities by assuming a priori that the phase is single-valued and continuous. 2. Well-Posedness of the Solution Before even considering the question of appropriate algorithms for solution of the transport-of-intensity equation for continuous phase maps, we need to address the three questions of solubility, uniqueness, and stability. These questions were considered by Gureyev et al. in a pair of papers published in 1995 (1995a; 1995b). The upshot of their investigation, which will not be presented in detail here, was to show that, subject to certain conditions, there exists a unique solution to the transport-of-intensity equation which is stable in the precise mathematical sense of the said solution depending continuously on the input data. Stated differently, the inverse problem of the retrieval of continuous phase maps using the transport-of-intensity equation is ÒwellposedÓin the sense of Hadamard (1923; Kress, 1989, p. 221). By this we mean that the problem is well posed in the space of singlevalued continuous phase maps when the probability density of the radiation is strictly positive over the simply connected region that constitutes the region of interest, and when one has knowledge of either Dirichlet or Neumann boundary conditions on the phase. One typically works with zero Dirichlet boundary conditions (sample is surrounded by a primary unperturbed beam of plane waves) or periodic boundary conditions (appropriate for imaging of perfect crystals). A certain nonclassical boundary condition is also permitted (probability density of radiation ÒbeamÓvanishes outside the region of interest). Without going into the detailed interpretation of these conditions in this paper, the conclusion is that the phase recovery problem is mathematically well posed and that we should be able to Þnd a stable solution to the problem with realistic data. It is to the Þnding of this solution that we now turn. 3. Uniform Intensity Solution Gureyev and collaborators (Gureyev et al., 1995b; Gureyev and Nugent, 1996) explored the solution of the transport-of-intensity equation by expanding the unknown phase as a weighted sum of orthogonal basis functions and solving the resulting system of linear equations for the weighting coefÞcients. This early work was particularly concerned with solving the problem in terms of orthogonal Zernike polynomials (Wang and Silva, 1980) but it was also recognized that the same broad approach could be adapted to Fourier series

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expansions (Gureyev and Nugent, 1996). It was quickly appreciated that the numerical requirements for Þnding the solution with this approach made the required matrix inversion impractical, even when fast Fourier transforms could be employed to perform the decompositions. However, this was not the case if the probability (intensity) distribution could be assumed to be uniform over the in-focus plane. Under this assumption, the transport-of-intensity equation clearly reduces to the form h ∂ρ(r⊥ ) = − ρ0 ∇⊥2 S(r⊥ ) ∂z p

(33)

where ρ0 is the uniform probability (intensity). In this case, the Gureyev and Nugent solution reduces to a very simple Fourier-based technique (Gureyev and Nugent, 1997) that may be written as follows: S(r⊥ ) =

p −1 1 ∂ρ(r⊥ ) F F ρ0 h ∂z k⊥2

(34)

where F is the Fourier transform operator and k⊥ is the position vector in Fourier space. This technique was immediately applied to the reconstruction of an X-ray phase distribution (Nugent et al., 1996), which represents the Þrst quantitative reconstruction of phase using propagation-based techniques. This experiment is discussed in more detail in Section V.B. We also note in passing that Eq. (33) summarizes the mathematical basis for curvature sensing and also for the defocus method of phase visualization. Note also that Eq. (33) will permit phase measurement using only one plane of data. Clearly, in general, an estimate of the longitudinal derivative along the optic axis requires two measurements so that the rate of change may be determined. However, if one of the planes is known a priori, then only one plane is required. In the case described by Eq. (33), one data plane is uniform by assumption and so can be deemed to be known. This observation has a parallel in interferometry where, as was pointed out in Section II.B.4, a nonuniform wave requires multiple data sets. However, in the case of a uniform (or known) intensity distribution, only one interferogram is required. 4. A Rapid Algorithm for Nonuniform Intensity The method we review here, due to Paganin and Nugent (1998), can be based on the fast Fourier transform (Teukolsky et al., 1992; Brigham, 1974) and can, in some ways, be considered a generalization of the Gureyev and Nugent (1997) approach as it reduces to that form in the case of nonuniform intensity. An

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alternative method of solution, based on the so-called full multigrid algorithm, has also been published (Gureyev et al., 1999) and we note that Teague also proposed a solution method based on GreenÕs functions (Teague, 1983), although, to our knowledge, this has never been implemented. By making use of the Helmholtz decomposition theorem for vector Þelds (Morse and Feshbach, 1953), we may decompose the quantity under the divergence sign of (31) in terms of the gradient of a scalar potential (x, y, z)  and the curl of a vector potential (x, y, z):  y, z))⊥ ρ(x, y, z)∇⊥ S(x, y, z) = ∇⊥ (x, y, z) + (∇ × (x,

(35)

Following (Teague, 1983), we discard the vector potential: ρ(x, y, z)∇⊥ S(x, y, z) ≈ ∇⊥ (x, y, z)⊥

(36)

an equation which may be substituted into (31) to arrive at a Poisson type equation (Jackson, 1998): ∇⊥2 (x, y, z) = −

p ∂ ρ(x, y, z) h ∂z

(37)

This has the formal solution ∂ p (38) (x, y, z) = − ∇⊥−2 ρ(x, y, z) h ∂z If we apply ∇⊥ to both sides of (38), make use of (36) to eliminate the scalar potential (x, y, z), divide through by ρ(x, y, z), and take the two-dimensional divergence of both sides of the resulting equation, we arrive at a second Poissontype equation:   p 1 ∂ ∇⊥ ∇⊥−2 ρ(x, y, z) (39) ∇⊥2 S(x, y, z) = − ∇⊥ · h ρ(x, y, z) ∂z This has the formal solution   ∂ 1 p ∇⊥ ∇⊥−2 ρ(x, y, z) S(x, y, z) = − ∇⊥−2 ∇⊥ · h ρ(x, y, z) ∂z

(40)

It is the application of this result that is at the heart of the experimental work that we present in Section V. 5. Numerical Stability of the Reconstruction In general, the transport of intensity equation phase retrieval algorithm equation (40) is numerically unstable. We consider this issue in the context of the uniform intensity algorithm.

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The principal issue lies in the noise-exacerbated Òdivision by zeroÓinstabilities which occur as k x , k y → 0 in Fourier space. Such instabilities manifest themselves as signiÞcant low-frequency artifacts, which are a strong function of the noise-induced perturbations which will inevitably be present in the data. These instabilities are avoided by suitable regularisation (Tikhonov, 1963; also see, e.g., Kress, 1989, pp. 224Ð225)of Eq. (34). For example, Tikhonov regularization (Bertero et al., 1990; Piana and Bertero, 1996) molliÞes the division-by-zero instability of an expression such as 1/u via the prescription 1/u → u/(u 2 + α), where α is the Òregularization parameterÓwhich tames the singularity at u = 0. Tikhonov regularisation of the transport-of-intensitybased phase retrieval algorithm (34) leads to the expression S(r⊥ ) =

p −1 k⊥2 ∂ρ(r⊥ ) F F 4 ρ0 h ∂z k⊥ + α

(41)

where the value of the nonnegative real regularization parameter α depends on the level of noise in the data. Equation (41) reduces to (34) when α = 0. The application of the ideas underlying this scheme, when applied to Eq. (40), lead to a phase retrieval algorithm which is deterministic, rapid, and stable with respect to noise and which yields a unique solution for the phase from noninterferometric intensity measurements alone. Further, the full phase map is recovered, rather than the modulo-2π phase map furnished by conventional interferometry (Strand and Taxt, 1999). 6. Simulated Example We close this section with an example of the action of the transport-of-intensity phase retrieval algorithm on simulated noise-free coherent data, as shown in Figure 1. Diffraction patterns for monochromatic scalar electromagnetic waves are calculated using the angular-spectrum formalism, making use of the fast Fourier transform. Dimensions of all images are 1.00 cm square = 256 × 256 pixels. The wavelength of the radiation was taken to be 632.8 nm (visible-light HeNe laser), with defocus distance ±2 mm from the central plane. The intensity in the in-focus plane z = 0, which varies from 0 to 1 in arbitrary units, is given in Figure 1a. Within the area of nonzero illumination, the minimum intensity was 30% of the maximum intensity. (The black border around the edge of the intensity image corresponds to zero intensity.) The input phase, which varies from 0 to π radians, is shown in Figure 1b. The negatively and positively defocused images are given in Figures 1c and 1d, respectively, and have respective maximum intensities of 1.60 and 1.75 arbitrary units; the propagation-induced phase contrast is clearly visible in each of these images. We see that this propagation-induced phase contrast is

Figure 1. Computer simulations for TIE-based phase retrieval using noiseless coherent radiation. (a) Aperture plane intensity; (b) aperture plane phase; (c) negatively defocused intensity; (d) positively defocused intensity; (e) intensity derivative, estimated from the difference of (d) and (c); (f) recovered phase, obtained using TIE processing. (Panels (a) and (b) are courtesy of Public Domain Images, http://www.PDImages.com/.)

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reversed when one moves from positive to negative defocus.∗ The two defocused images are subtracted to form the intensity derivative, which is given in Figure 1e. We notice that the intensity derivative is a much stronger function of the phase than the intensity in the in-focus plane. The images in Figures 1a and 1e were then processed according to a computer implementation of Eq. (40), with a computation time of a few seconds, in order to yield the recovered phase map given in Figure 1f. Note that Figures 1b and 1f are plotted on the same grayscale levels, indicating that the recovered phase is quantitatively correct. The success of this computational test conÞrms that we have a rapid and reliable approach to phase recovery.

D. Coherence Requirements for Propagation-Based Phase Measurement In this section we summarize the argument of Nugent and Paganin (2000) to estimate the coherence requirements to obtain noninterferometric phase measurements using the transport-of-intensity equation. In practice, a measurement of the longitudinal spatial derivative of the probability density will entail a measurement via the approximation ∂ρ(r⊥ ) ∼ ρ(r , +z/2) − ρ(r , −z/2) = ∂z z

(42)

This requires a measurement of the probability over two closely spaced planes separated by z. However, the momentum distribution in the probability current will blur the measurement of these distributions even though the current at a point deÞnes the phase precisely (compare Eqs. (19a) and (19b)). This blurring will limit the precision of the measurement. Nugent and Paganin therefore estimated the coherence requirements on the waveÞeld for this effect to be experimentally negligible. The blurring will have two components: (i) that due to the distribution in transverse momentum in the incident beam, and (ii) the additional transverse momentum distribution produced by dispersion (frequency-dependent phase shift) in the potential. We consider each of these. ∗ This is a consequence of the reciprocity law of diffraction, which implies that the intensity of a negatively defocused image is the same as the intensity for the forward defocused image of the complex conjugate of the wave. See E. Wolf (1980). Phase conjugacy and symmetries in spatially bandlimited waveÞelds containing no evanescent components. J. Opt. Soc. Am. 70, 1311Ð1319,Eq. (2.3). Also, cf. F. Zernike (1942), which speaks of Òtheinversion of the image from positive to negative when passing from inside to outside focus.Ó

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By modeling all spatial and wavelength distributions as Gaussian, Nugent and Paganin found that the lateral spatial coherence length should obey ℓlat >

λ 1 √ 2π 2 γmin

(43)

where γmin ≡ q/z is the minimum deßection angle to which the experiment is sensitive. This result implies essentially that the blurring in the image is a result of the spatial convolution of the data with the appropriately magniÞed (or demagniÞed) source size. This is an unsurprising result and implies a reasonably stringent spatial coherence that is limited by the spatial scale of interest. This compares with holographic image formation where the required spatial scale of the source is determined by the minimum fringe spacing to be resolved. The second source of degradation arises through the frequency dependence in the phase shiftÑthat is, by the longitudinal coherence. By making some simple assumptions about the frequency dependence of the phase shift, Nugent and Paganin come to the remarkable conclusion that the longitudinal coherence requirement is ℓlong ≫ λ

(44)

This condition is extremely lax as it implies very little limitation on the longitudinal coherence. Of course, this extreme conclusion is not correct and contains implicit assumptions about the nature of the dispersion in the sample. However, it does imply that the propagation approach to phase imaging is very forgiving in terms of monochromaticity and this has important implications for phase measurement with low brightness sources. We also point out that this conclusion is consistent with elementary observations. For example, the refractive effects seen at the bottom of a swimming pool, or the twinkling of stars, typically show very few chromatic effects and therefore permit a propagationbased phase measurement with white light. Indeed, white light measurements are at the heart of most adaptive optics systems. In summary, then, we have an approach to phase recovery that has very forgiving requirements on the coherence of the radiation. We are therefore in a position to consider the results of the experimental implementation of this work.

V. Experimental Demonstrations The propagation-based approach to phase has been used widely for phase visualization, but the transport-of-intensity phase measurement has been largely restricted to the University of Melbourne group and its collaborators. In this section, we describe the experimental results published to date.

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A. Phase Retrieval with Visible Light 1. Optical Microscopy The inspection of small transparent structures such as cells and optical Þbers is of fundamental importance to the biological and materials sciences. Various modiÞed forms of optical microscopy, such as those that employ Zernike phase contrast or Nomarskii differential-interference contrast, yield qualitative images of such objects. The quantitative analysis of these images has, to our knowledge, only been performed under the assumption of a weak, pure phase object (see, for example, van Munster et al., 1997, 1998 and references therein). We note that interference microscopes exist and are able to perform quantitative phase extraction (Barer, 1952; Davies and Wilkins, 1952; Barer and Smith, 1972). However, it is important to remind ourselves that high resolution microscopy is critically dependent on the use of partially coherent, or even incoherent, illumination (Born and Wolf, 1993). Thus, interferometric microscopes are unable to combine high resolution with phase measurement. Barty and co-workers (1998) have used the propagation phase retrieval approach to demonstrate quantitative two-dimensional imaging of a transparent object using an unmodiÞed conventional visible-light microscope. In the Þrst experiment, published in 1998, Barty et al. imaged a 3M F-SN3224 single-mode optical Þber,∗ which had been independently characterized using both atomic force microscopy (Huntington et al., 1997) and commercial proÞling techniques.† This independent characterization of the refractiveindex structure of the Þber served as a reference standard against which to assess the quantitative nature of the technique. The sample was illuminated in transmission mode with an incandescent bulb as a source. Light from this thermal source was passed through an interference Þlter, which had its passband centered at 625 nm, with a spectral width of 10 nm. The spectral width of the Þlter was sufÞciently large to allow Barty et al. to demonstrate the use of the phase retrieval algorithm with partially coherent illumination of insufÞcient coherence for interferometric phase measurement. The in-focus (bright Þeld) image of the optical Þber is shown in Figure 2a. Images overfocused and underfocused by 2 μm are shown in Figures 2b and 2c, respectively. The propagation-induced phase contrast so well known to optical microscopists is visible in both of these images, together with the contrast reversal experienced as one goes through focus. The phase recovered from transport-of-intensity processing of the intensity data is shown in Figure 2d, with Figure 2e giving a comparison of the measured phase proÞle (dashed ∗

Sourced from 3M Optical Fibers, West Haven, CT, USA. P102 analyzer, York Technologies, ChandlersÕFord, UK.

†

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line) and the phase proÞle obtained from the independent analyses mentioned earlier (solid line). This comparison between the measured and known proÞles demonstrates the quantitative correctness of the technique. The work described by Barty et al. used a condensor numerical aperture of 0.2. This does not provide the highest resolution possible. However, more recent work has indeed demonstrated that this approach to microscopy can be

Figure 2. TIE-based phase recovery of a single-core optical Þber. Panel (a) shows the intensity distribution in the plane of best focus; (b) and (c) show respectively the intensity at plus and minus 2 μm defocus either side of best focus. Panel (d) shows the phase image recovered from the images in panels (a), (b), and (c) using the phase-retrieval algorithm described in Section IV.C.4. Note that the Þber is clearly visible in the recovered phase image, but that only regions of strong phase change are visible in the bright-Þeld intensity images (a)Ð(c).Panel (e) gives a comparison of the measured phase proÞle (dashed line) and the phase proÞle obtained from an independent method (solid line).

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(e) Figure 2. (Continued)

performed with a condensor numerical aperture equal to the objective numerical aperture (Barone-Nugent et al., in preparation). Thus, very high resolution phase microscopy is possible. 2. Optical Phase Tomography In 2000, Barty and co-workers extended this work in a paper that achieved quantitative phase tomography using the transport-of-intensity equation (Barty et al., 2000). A tomographic dataset was collected for 200 equally spaced angular orientations of a twin-core optical Þber, through 180◦ in 0.9◦ steps. The axis of rotation ran through the center of the Þber. Transport of intensity processing of this tomographic dataset yielded a set of 200 phase maps in less than 10 minutes on a DEC Alpha 600 au workstation. These 200 phase images were then aligned to correct for the effects of precession, and the resulting dataset fed into a conventional tomographic reconstruction using the Þltered back-projection algorithm (see, for example, Barrett and Swindell, 1981). The result is shown in Figure 3. The application of conventional tomographic reconstruction techniques implicitly assumes that multiple scattering can be ignored in the sample. The

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Figure 3. Three-dimensional reconstruction of the refractive index distribution of a twincore optical Þber.

measured phase was indeed found to be in agreement with the known properties of the Þber and conÞrmed that the assumption of small scattering was valid. However, most samples of interest will scatter very strongly and it is not clear that it will be possible to usefully apply simple tomographic techniques in general. It is likely, however, that the phase and amplitude information will provide an excellent input to diffraction tomography algorithms. This is a subject for further research. 3. In-Line Holography GaborÕs pioneering paper on holography (Gabor, 1948) formulates a Ònew microscopic principleÓ which Òallows one to dispense altogether with . . . objectives.ÓAs an example of the contemporary application of holographic ideas, the numerical reconstruction of digitised in-line holograms yields a promising lensless technique for X-ray (Jacobsen et al., 1990) and electron imaging (Silverman et al., 1995; Vu et al., 1995). However, such an approach suffers from the well-known contamination due to the twin image (Nieto-Vesperinas, 1991). This twin image remains

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one of the basic issues that need to be addressed in making in-line holography a practical technique for short wavelength imaging (Nugent, 1991). Various approaches have been developed to deal with this, some of which rely on taking two or more holograms to aid the analysis. For example, one can take two holograms and employ a variant of the famous GerchbergÐSaxton (1972) phase-retrieval algorithm to eliminate the twin image (Kodama et al., 1996). The use of such an algorithm is not successful when the object signal is too strong (Lindaas et al., 1996). Another method is to take holograms at distance z and 2z behind the object and process them appropriately; this only partially compensates for the twin-image problem (Xiao et al., 1998). Some further strategies, reviewed by Spence (1997; see also Spence et al., 1995), also rely on taking multiple holograms in order to eliminate the twin image. Tiller et al. (2000) have presented an alternative strategy for twin-image elimination that utilizes the Paganin and Nugent approach. The experimental setup differs from the usual setup of inline holography only insofar as three images are required rather than one (see Fig. 4). The strategy is simple: perform

Figure 4. Generic experimental setup for TIE-based inline holography which eliminates the presence of the twin image in the reconstruction. Setup with point-source illumination (pointprojection microscopy system). Inline holograms are measured over the planes A, B, and C, allowing TIE-based phase retrieval in the central plane B. Since both the amplitude and phase of the waveÞeld disturbance in the plane B are known, this may be numerically backpropagated to give the reconstructed amplitude and phase of the radiation at the exit surface of the object.

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a phase retrieval using the three closely spaced holograms shown in Figure 4, and then inverse-diffract the associated complex disturbance from the hologram plane back to the exit surface of the object.∗ This yields the decoupled amplitude and phase of the radiation at the exit surface of the object, without the twin-image contamination of conventional in-line holography. The experiment was performed using HeNe laser light of wavelength 632.8 nm, in the point-projection geometry shown in Figure 4. The distance ρ c from the point source to the object was 15 cm, and the distance z from the sample to the central image plane B was 12 cm. The sample was a segment of human hair. In-line holograms were measured over the three closely spaced planes. The intensity of the hologram over the central plane B is shown in Figure 5a. A conventional holographic reconstruction of the sample is given in Figure 5b, showing the usual twin-image contamination. The retrieved phase is shown in Figure 5c. Together with the measured intensity in Figure 5a, this was used to propagate the radiation back to the object plane. The resulting intensity distribution is shown in Figure 5d. It can be seen that the phase determination has largely eliminated the twin-image contamination.

B. Phase Retrieval with X-rays The Þrst demonstration of quantitative phase retrieval using hard X-rays was published by Nugent and co-workers in 1996. A schematic of their experimental setup is given in Figure 6; the radiation energy was 16 keV and the sample of interest was a carbon TEM calibration grid of period 330 microns. At 16 keV the X-ray radiation at the exit-surface of the carbon grid had negligible intensity modulation. The transport-of-intensity equation was inverted using the Fourier transform technique of Gureyev and Nugent [1997; see Eq. (11)] and the effects of Þnite source size compensated using a simple deconvolution algorithm, to yield the retrieved quantitative phase map shown in Figure 7. This retrieved phase map produced an average phase shift imprinted by the sample on the wave that was in good agreement with an independent calculation from an absorption experiment at a different X-ray energy, thus demonstrating the quantitative nature of the technique. A more recent demonstration of quantitative phase retrieval using the transport of intensity equation was published by Gureyev et al. in 1999. Employing a method of solution based on the so-called Full Multigrid algorithm, ∗ The use of such a strategy was suggested, in the context of phase retrieval using twodimensional scalar electromagnetic waves, in M. R. Teague, Image formation in terms of the transport equation, J. Opt. Soc. Am. A. 11, 2019Ð2026(1985).

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Figure 5. Use of TIE-based phase retrieval and backpropagation to eliminate the twin image of in-line holography. (a) In-line hologram recorded 12 cm downstream of a single human hair; (b) conventional holographic reconstruction of the object, which is contaminated by the twin image; (c) phase in the detector plane which is recovered by solving the TIE; (d) backpropagated intensity at the exit-surface of the object which is obtained using the detector plane intensity (a) and the recovered phase (c).

they inverted the transport of intensity equation to achieve quantitative phase imaging of a polystyrene sphere using hard X-ray synchrotron radiation from a third-generation source. In Figure 8, we reproduce one of the phase maps so derived. Another example of phase retrieval using X-rays in the context of medical imaging was given in another paper by Gureyev et al. (2000). Cloetens and co-workers (1999) have used the iterative multiple defocus technique developed for electron microscopy to generate the technique they

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Figure 6. Schematic of experiment for quantitative TIE-based phase retrieval using hard X-rays.

term holotomography. This was discussed in Section II.B.3. Momose and colleagues (1998) have used interferometry to perform X-ray phase tomography. Allman et al. (2000a) have demonstrated phase recovery for much less energetic X-rays using both projection microscopy (which is essentially a form of in-line holography; cf. Section V.A.3) and using a zone plate soft

Figure 7. Results for quantitative TIE-based phase retrieval using hard X-rays from a second-generation source.

Figure 8. Results for quantitative TIE-based phase retrieval using hard X-rays from a third-generation source.

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X-ray microscope. This is the Þrst X-ray work for which absorption could not be ignored and also produced X-ray phase images with a very high spatial resolution, of the order of 100 nm. The results obtained in this work are perhaps not as striking as those obtained with more energetic X-rays. The reasons for the poorer results lies in the complex background created by the other diffraction orders in the zone plate, and the stringent alignment requirements implied by the need to remove the spurious apparent phases created by instability (at the 50-nm level) in the imaging system. We note that these results merely conÞrm that, no matter what technique is used, high resolution phase measurements place strict requirements on the experimental stability.

C. Phase Retrieval with Electrons The techniques described in the section discussing optical microscopy can also be transferred directly to transmission electron microscopy. To explore this possibility, Bajt et al. (2000) obtained phase images of a magnetic domain in a cobalt Þlm. The data acquired by these workers are summarized in Figure 9. The characteristic contrast change between positive and negative defocus is clearly evident in Figures 9a through 9c. Figure 9d shows the difference between the positive and negative defocus images. The phase calculated using processing via our solution of the transport of intensity equation is shown in Figure 10. Note also that absorption by the sample was very strong in this experiment and so, as with the experiments of Allman et al. (2000a), the uniform intensity approximation was very strongly broken and so the rapid approach of Paganin and Nugent (1998) was of critical importance. It is interesting to note that these data were obtained with a conventional electron microscope. The resulting phase distribution was compared with electron holographic measurements of the same sample and the phase distributions were found to be in excellent agreement. The direct technique demonstrated by Bajt et al. is thus rather more ßexible than electron holography principally through its applicability in any TEM with a good-quality detector, and the lack of any need for a reference beam. Lorentz microscopy and electron holography (Midgley, 2001) have been extensively used for the study of vortices in superconductors. This has been the subject of much work by Tonomura and colleagues (Harada et al., 1993; 1997; Tonomura, 1998; Tonomura et al., 1999) and it is interesting to speculate whether, in this context, direct approaches based on the transport-of-intensity equation offer any advantages over the more conventional approaches. Allen et al. (2001a). point out that the application of the phase-retrieval algorithm as described here is not suitable for high-resolution electron microscopy

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Figure 9. Raw data for quantitative TIE-based phase retrieval using an unmodiÞed electron microscope. (a) Positive defocus image of cobalt dot; (b) negative defocus image; (c) in-focus image; (d) difference between positive and negative defocus images.

of periodic objects as it is unable to properly handle phase discontinuities that inevitably arise. They go on to give an alternative method of phase retrieval which overcomes this problem, and is also able to correct for the aberrations of the microscope. We point out here that the handling of such phase discontinuities is an open question for propagation-based phase determination, but the work of Nugent and Paganin (2000) does give reason for optimism that the direct approaches will ultimately be able to characterize such Þelds.

D. Phase Retrieval with Neutrons The physics of the interaction of neutrons with matter enables neutron radiography (Schlenker and Baruchels, 1986) to be an effective complement to X-ray

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Figure 10. Surface plot of the retrieved phase, obtained using TIE-based processing of the data in Figures 9c and 9d.

radiography. Allman et al. (2000b) have published a simple quantitative method that makes available a new contrast mechanism for neutron radiography and allows samples to be imaged at reduced radiation doses. They were also able to accurately measure large phase shifts from detailed structures that are not amenable to conventional techniques due to the very large phase gradients. Experiments were carried out at the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR), Gaithersburg, MD, û with an approximately monochromatic wavelength of 4.43 A. The geometry of the experiments was that of simple projection, as was the case with the hard X-ray experiments. The divergence of the neutron beam was limited by a mask to around 6 mrad and neutrons illuminated a sample placed about 1.8 m from the beam guide exit. As in the other work, images were taken in two planes. The Þrst was a contact image and the second was a phase contrast image with the detector positioned 1.8 m from the object. A lead sinker was imaged and the measured phase was obtained by solving the transport-of-intensity equation. The measured phase is shown in Figure 11. The phase deformity (FM) is an artifact of a Gadolinium Þducial mark. The hollow core (HC) is clearly seen in the phase image. The phase image obtained is, to an excellent approximation, described by a convolution of the perfect image with the intensity distribution of the effective

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Figure 11. Phase retrieval using neutrons. (a) Quantitative phase map of the lead sinker showing the hollow core (HC) and Þducial mark (FM) artifact; (b) quantitative phase proÞle AA′ through the sinker (gray) and calculated proÞle (black). The calculated proÞle is based on the known shape, scattering length, and orientation of the sinker.

source. Allman et al. compensated for the source size using a deconvolution based on very conservative estimates. A proÞle of the recovered phase (AA′ ) is plotted in Figure 11b along with the predicted phase proÞle determined from the known sample geometry, scattering length, and orientation. The quantitative agreement between the two is excellent. Note the 1400 radian phase excursion

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of the sample occurring in a few hundred microns (fewer than 10 pixels). An interferometric experiment would require submicron resolution to accurately measure such a rapid phase excursion and is not a practical option. We note also that, although these experiments were not carried out using broadband polyenergetic neutron beams, this option does exist. This possibility offers considerable promise for sources such as nuclear reactors, which have a very low brightness; in such a context, ßux is at a premium and one would rather not lose orders of magnitude in intensity in the act of sufÞciently monochromating the beam for interferometric phase determination.

VI. Conclusion This paper has reviewed the modern state of phase-sensitive imaging in optics. We have touched on all the main areas of phase visualization but have by no means presented an exhaustive summary of all the excellent work that has been published. The main focus of this review has been on the work toward techniques for the determination of phase from the propagation of intensity. This is a burgeoning area with numerous applications that have yet to be fully explored. In this review, we have taken care to point out the advantages and limitations of the method of phase determination using propagation. Although it has not been explicitly stated, it is clear that the technique is most directly sensitive to the Þrst and second derivatives of the phase (phase gradient and curvature, respectively). This implies that very gentle phase variations are not particularly well recovered. That this is the case is brought out very clearly by the Fourier transformÐbased algorithms where it is readily seen that the lower spatial frequencies require greater ampliÞcation and will therefore be more noise sensitive. The neutron work summarized in the previous section clearly indicates, however, that large phase gradients are very effectively recovered even if the phase cycles through many radians between adjacent pixels of the image. It is also clear that, although uniform intensity Þelds may be recovered using data over one measurement plane, two are needed for the general (i.e., nonuniform intensity) case. In our discussion of interferometry, however, it was pointed out that this general requirement also applies to interferometry. We have also had cause to consider the matter of phase discontinuities in detail. This is an area in which the limitations are very clear and are in need of explicit consideration. However, we again point out that the familiar technique of interferometry also faces difÞculties in properly recovering a noncontinuous two-dimensional phase map. This problem is buried in the active research area of phase unwrapping.

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We note, however, that the real power of the approach reviewed here lies in its ability to perform phase measurement and phase imaging with radiation of very low coherence, particularly longitudinal coherence. We believe that the internally consistent picture of phase and phase recovery that we have developed here offers great promise in those many phase measurement problems that will beneÞt from a relaxation of the brightness, or coherence, requirements of the source.

Acknowledgments The authors acknowledge extensive support from the Australian Research Council in the development of the work described here and during the writing of this review. One of the authors (KAN) records his particular gratitude to the many excellent students and colleagues who have worked with him on the development of the ideas that are the principal subjects of this review.

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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 118

Recent Developments of Probes for Scanning Probe Microscopy EGBERT OESTERSCHULZE University of Kassel, Institute of Technical Physics, 34132Kassel, Germany

I. Introduction . . . . . . . . . . . . . . II. Atomic Force Microscopy. . . . . . . . . A. Working Principle. . . . . . . . . . . B. Mechanics of Cantilever Probes . . . . . C. Materials Available for Probe Fabrication . 1. Silicon . . . . . . . . . . . . . . 2. Gallium Arsenide. . . . . . . . . . 3. Carbon . . . . . . . . . . . . . . D. Concluding Remarks . . . . . . . . . III. Near-Field Optics . . . . . . . . . . . . A. Theory of Far-Field Optics . . . . . . . B. Introduction to Near-Field Optics . . . . C. Passive Probes . . . . . . . . . . . . 1. Aperture Probes . . . . . . . . . . 2. Coaxial Probes. . . . . . . . . . . 3. Bow-Tie Antenna Probes . . . . . . 4. Solid Immersion Lens Probes. . . . . 5. Scattering Tip . . . . . . . . . . . D. Active Probes . . . . . . . . . . . . 1. Light-Emitting Active Probes. . . . . 2. Light-Detecting Active Probes . . . . References . . . . . . . . . . . . . . .

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129 130 130 131 133 135 141 143 150 151 151 154 156 156 171 175 178 181 182 182 187 191

I. Introduction Scanning probe microscopy (SPM) has gathered enormous attention since the invention of scanning tunneling microscopy (STM) by Binnig and Rohrer (Binnig et al., 1981, 1982; Binnig and Rohrer, 1982) and in particular the introduction of atomic force microscopy (AFM) by Binnig et al. (Binnig et al., 1986). During the two decades of their existence, SPM methods and applications have ramiÞed in a tremendous variety of ways (Wiesendanger, 1994). To give an exhaustive overview of recent developments of SPM probes is thus a quite delicate task. In what follows we discuss some aspects of recent developments of SPM probes for scanning near-Þeld optical microscopy. 129 Volume 118 ISBN 0-12-014760-2

C 2001 by Academic Press ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright  All rights of reproduction in any form reserved. ISSN 1076-5670/01 $35.00

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The general operation principle of SPM methods is surprisingly simple. It comprises the precise scanning of a pointed probe and the detection of the local interaction between the probe tip and the sample surface. Obviously, data aquisition is done serially and is followed in almost all cases by electronic evaluation to obtain images of the desired sample property. Although STM was the Þrst scanning probe method, AFM found much more attention and application because it is not restricted to electrically conducting samples. This is the reason we turn our attention in the following discussion to some basics of atomic force microscopy (sometimes also called scanning force microscopy, SFM).

II. Atomic Force Microscopy A. Working Principle In AFM the sensor consists of a cantilever, i.e., a mechanical beam that carries a sharpened tip as shown schematically in Figure 1. The cantilever probe may be operated under various ambient conditions, such as air, ßuids, UHV, or cryogenic or elevated temperatures. Two general modes of operation were realized (Meyer and Heinzelmann, 1995). In the static mode the tip is in contact with the sample surface and the amount of the quasi-static cantilever bending, that is, the strength of the short-range repulsive force, is a measure of the sample topography. However, the tip locally exerts a nonnegligible force onto the sample. This may be exploited on one hand to study friction forces and thus the tribological surface sample properties (e.g., Schwarz et al., 1997). On the other hand the hazard of damage might occur in particular if very soft samples, such as biological and polymer materials, are investigated.

Deflection sensor Holder Deflection system

Cantilever Tip

Figure 1. Schematic view of a multifunctional AFM cantilever probe with a deßection sensor, an actuator, and various types of sensors integrated in the tip for the measurement of other surface properties than topography.

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Therefore, various types of dynamic operation modes were introduced where the cantilever tip is vibrated at or near to its resonance frequency. Typical vibration amplitudes are in the range of 1Ð100nm. In the slope detection method the amplitude or phase variation of the cantilever deßection is detected, keeping the vibration frequency of the tip Þxed (Martin et al., 1987). Frequency modulated (FM) AFM is the most popular at present because it offers the same sensitivity as the slope detection method but with an improved bandwidth (Albrecht et al., 1991). It is capable of achieving true atomic resolution on conducting as well as dielectric surfaces (Giessibl, 1995; Bammerlin et al., 1997). In all dynamic modes the cantilever bending is sensitive to the force gradient (in most cases of the long-range forces) rather than to the force itself. Depending on the separation distance between the tip and the sample and also their composition, the total force gradient may be dominated by various types of forces, such as van der Waals force, Coulomb force, or magnetic force. A detailed theoretical description is given, for example, in Meyer and Heinzelmann (1995) and Giessibl and Bielefeldt (2000). A noteworthy advantage of the dynamic mode in comparison to the static mode is the strong reduction of forces exerted onto the sample and the improved bandwidth. The AFM offers a huge potential to be upgraded to a multifunctional probe integrating sensors into the tip to become a versatile and powerful tool for high-resolution imaging of various surface properties in surface science. This is what we emphasize in this paper. Presently the AFM with its offsets covers multifaceted applications in physics, chemistry, biology, mechanical and electronic engineering, etc. (Wiesendanger, 1994).

B. Mechanics of Cantilever Probes In general the naked AFM probe proves to be in principle a quite simple sensor with well-known mechanical properties. The cantilever is theoretically described in terms of a thin elastic beam that is governed by the following equation of motion in Fourier space considering only small deßections z (Sarid, 1991): d 4 z(y) − κ 4 z(y) = 0 dy 4

(1)

y denotes the coordinate in the cantilever direction. A more precise description taking drag forces into account may be found in Blom et al. (1992) and Salapaka et al. (1997) and references therein. The resonant frequency ωn in our simpliÞed

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model is connected with the parameter κ n of the nth ßexural bending mode by  I E (2) ωn = κn2 A ρ and thus depends on the geometry given by the momentum of inertia I and the cross-sectional area A of the beam and on the material properties, that is, YoungÕs modulus E and the mechanical density ρ. In case of a mechanical beam of length l, constant width ω, and rectangular cross section A = tω, that is, a homogeneous thickness t, an eigenvalue equation is derived from Eq. (1) with the following Þrst Þve eigenvalues for κ n in ascending order of n: κn l = 1.875; 4.694; 7.855; 10.996; 14.137 With Eq. (2) the resonant frequency ω1 of the Þrst ßexural mode is

d E ω1 = 1.0149 2 l ρ

(3)

(4)

Applying the deÞnition of the compliance k of the cantilever (Sarid, 1991) I (5) l3 To the particular case of the rectangular beam just mentioned gives for the Þrst ßexural model k = 3E

1 bd 3 E (6) 4 l3 The theoretical description becomes increasingly complicated if single crystalline substrate materials, such as silicon or gallium arsenide wafers, have to be considered. Their mechanical anisotropy is described relating the stress tensor σ with the strain tensor ǫ via the stiffness coefÞcient matrix Ckl (Nye, 1985): k=

σk =

6  l=1

Ckl ǫl

with k = 1, . . . , 6

(7)

Assuming a crystalline solid of cubic symmetry leads to the following deÞnition of YoungÕs modulus for the [100] direction (Heuberger, 1991): E [100] =

(C11 − C12 )(C11 + 2C12 ) C11 − C12

(8)

However, in many fabrication schemes discussed in literature, anisotropic wet chemical etching processes are utilized for the processing of cantilever probes made of silicon or gallium arsenide wafers. This results in mechanical beams

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oriented in the [110] rather than the [100] direction. The corresponding YoungÕs modulus E in the desired direction is obtained by a proper coordinate transformation (Nye, 1985)   (9) 1/E = S11 − (2S11 − 2S12 − S44 ) · l12l22 + l22l32 + l32l12

where lj denote the direction cosine between the desired direction and the ith axis of the elementary lattice cell. The coefÞcient of the inverse elastic stiffness matrix Sij depend on the Cij, as follows (Nye, 1985): C11 − C12 = 1/(S11 − S12 )

C11 + 2C12 = 1/(S11 + 2S12 ) C44 = 1/S44

(10)

Our brief introduction to some theoretical aspects of the mechanics of AFM probes is concluded presenting the expression of the force gradient sensitivity in dynamic AFM with its respective frequency shift δω (Albrecht et al., 1991):

ω1 k B T B 4 k kB T B ∂ F

 2  and δω2  =   = (11)

∂z min ω1 Q A0 k Q A20

where kB T is the thermal energy of the cantilever, B the bandwidth, A0 the vibration amplitude, and Q the quality factor of the cantilever. Measurements in the dynamic mode with high force sensitivity thus require probes of low k, high quality factor Q, and high resonant frequency ω1 operated at low temperatures T. √ Bruland et al. demonstrated a force sensitivity of 10−17 N/ Hz for an ultrasoft silicon nitride cantilever with a compliance of 110 μN/m operated at 10 K (Bruland et al., 1998). Similar soft cantilevers were investigated by Berman and Tsifrinovich (1998) for the purpose of single-spin detection using magnetic force microscopy. Nevertheless, Giessibl et al. (1999) pointed √ out that the noise limited detection of topography features scales with 1/ k. They utilized a quartz fork of high compliance to receive true atomic resolution (Giessible and Bielefeldt, 2000). A further enhancement of δω was proposed, operating the cantilever at higher ßexural modes [see, e.g., (Minne et al., 1996a; Rabe et al., 1998; Hoummady and Farnault, 1998) and references therein].

C. Materials Available for Probe Fabrication The discussion of the mechanical behavior of AFM probes in the previous section already indicated that reliable AFM measurements require probes

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with reproducible mechanical properties. Nowadays most schemes for probe fabrication presented in the literature rely on microelectromechanical system (MEMS) technology that was adapted for the Þrst time by Albrecht and Quate (Albrecht and Quate, 1988; Akama et al., 1990) for this purpose. Meanwhile, both bulk materials, such as silicon and gallium arsenide (Heisig and Oesterschulze, 1998) and thin-Þlm materials, such as silicon dioxide and silicon nitride (Albrecht et al., 1990a), metals (Boisen et al., 1996; Beuret et al., 1998a), polycrystalline diamond (Niedermann et al., 1996; Oesterschulze et al., 1997), and polymer material (Pechmann et al., 1994), are utilized. The mechanical properties of the most important single crystalline materials are compiled in Table I. With reference to Eq. (4) it is striking to note that the resonant frequency of the Þrst ßexural mode of a cantilever can only be varied by a factor of about 3.8, if diamond with its extreme mechnical properties is used as probe material instead of GaAs. However, the same variation is much easier obtained by simply adapting the cantilever geometry, such as by increasing the cantilever length by a factor of 1.95. Thus the proper material choice is much more important in view of the environmental conditions to be fulÞlled in the desired AFM

TABLE I Mechanical Properties of Some Important Single Crystalline Materials Used for MEMS Fabrication of SPM Probes Material Mechanical properties Lattice constant (nm) Separation of nearest neighbors (nm) Mechanical density ρ (g/cm3) Stiffness coefÞcients (GPa) C11 C12 C44 YoungÕs modulus E [100] (GPa) YoungÕs modulus E [110] (GPa)  E [110] /ρ (m/s) PoissonÕs ratio ν[100] Torsion modulus G [100] (GPa) Hardness (kg/mm−2) (load in g) a

Si (Schulz and Blachnik, 1982)

GaAs (Blakemore, 1982)

Diamond (von Munch, 1982)

0.543 0.235

0.565 0.245

0.357 0.155

2.329

5.317

3.515

165.0 64.0 79.2 129.2 168.4 8,503.3 0.28 79.2 1,150 (25)

119.0 53.8 59.5 85.5 121.5 4,780.3 0.31 59.5 750 ± 40 (25)

1,076.4 125.2 577.4 1,050.3 1,163.6 18,194.5 0.10 577.4 10,300 (1,000)

Data are given at room temperature. SI denotes semiinsulating material. Values of the coefÞcient Cij are noted in the coordinate system of the elementary unit cell of single crystalline Si, GaAs, and diamond.

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experiment, such as chemical inertness, high (or very low) electrically or thermally conducting probes, or superhard tip material for invasive experiments. For the geometrical design of cantilevers some rules of the thumb may be considered. A small compliance of the probe requires, in accordance with Eq. (6), thin, long cantilevers. In the other extreme of an exceedingly large compliance, a short, thick cantilever is demanded. In this particular case it is important to have superhard materials with low abrasion for tip fabrication at oneÕs disposal because tip wear becomes an important issue (e.g., Bharat Bhushan, 1997; Khurshudov et al., 1997, and references therein). In the following, some of the most common materials are discussed in detail. 1. Silicon The unique mechanical and electronic properties of silicon in combination with the comprehensive spectrum of available technological processes made silicon the ideal material for MEMS fabrication. As mentioned earlier, Albrecht and Quate Þrst introduced AFM cantilever probes based on bulk crystalline silicon and thin Þlms of silicon dioxide and silicon nitride (Albrecht and Quate, 1988; Albrecht et al., 1990a). In the case of bulk silicon material, the cantilever structure is obtained by conventional lithography and wet chemical or plasma etching processes. Subsequently, sharp tips are fabricated by isotropic or anisotropic underetching of circular or square masking pads (Boisen et al., 1996; Wolter et al., 1991). The typical tip radius of curvature is in the range of 10Ð20nm. However, as can be seen from Figure 2, high-aspect-ratio tips with a radius of curvature of less than 5 nm are already commercially available.

Figure 2. Commercially available sharpened silicon tip integrated into a cantilever probe. The SEM image reveals a radius of curvature of the tip of less than 5 nm. (Courtesy of NanoSensors GmbH & Co. KG, Sensitec-Building, Wetzlar, Germany.)

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EGBERT OESTERSCHULZE

Furthermore, several authors have discussed the possibility of sharpening of silicon tips (Marcus et al., 1990), exploiting the complex rheological behavior of thermally grown silicon dioxide (see, e.g., Kao et al., 1987, 1988, and references therein). It comprises repeated thermal oxidation of the silicon tips at proper temperatures followed by subsequent wet silicon dioxide etching (Ravi and Marcus, 1991; Zhang and Zhang, 1996). Section III.C.1c is concerned with some details of the rather complex oxidation process of silicon. If single tip fabrication is appropriate, focused ion beam treatment of tips might also be an option for tip sharpening as was demonstrated by Hopkins et al. (1995). Although silicon probe fabrication works quite reliably, the cantilever thickness in case of bulk micromachined probes shows a dispersion due to the total thickness variation (TTV) of commercial silicon wafers of typically 3Ð5μm. This gives rise to a notable margin of probe properties according to Eqs. (4) and (6). Several technological approaches were conceived to reduce the thickness variation applying etch stop techniques (Collins, 1997). A p/n junction is introduced via doping the highly p-doped substrate with n material to obtain an etch stop layer prior to the etching process of the cantilever membrane. Similar results are obtained by ion implantation of oxygen (Yang et al., 2000), carbon, gold, and titanium (Nakano et al., 1999) to form an intermediate layer in the host material. Both techniques are capable of preparing the rather thin cantilevers that are inevitable for very sensitive FM AFM measurements in accordance with Eq. (11). Yang et al. (2000) achieved cantilevers of only 60 nm thickness and 9.6 μm length with a calculated force sensitivity of only 10−17 N. Their use is obviously restricted to the dynamic AFM mode because they would be instantly destroyed by attractive forces during the jump into contact to the sample surface. Another approach to realize thin cantilevers with homogeneous thickness makes use of silicon-on-insulator (SOI) substrates (Itoh et al., 1995; Hosaka et al., 2000). McCarthy et al. applied the focused ion beam (FIB) method for single fabrication of silicon probes with very precise geometry (McCarthy et al., 2000). An alternative solution exploits thin deposited layers with homogeneous thickness as cantilever material. In particular, silicon dioxide and silicon nitride have found widespread use. To integrate a sharp tip molding is the preferred method. Albrecht involved pyramidal etch pits obtained by anisotropic etching of (001) oriented silicon wafers for this purpose (Albrecht et al., 1990a). In a similar approach the evaporation of material through an oriÞce is used to fabricate conelike tips (Albrecht et al., 1990a; Spindt et al., 1976). However, to mechanically handle the exceedingly fragile cantilever membrane with the already molded tip, one faces the difÞculty of mounting a mechanical holder on each probe chip in a batch process. Various types of mounting processes have been presented in the literature, such as silicon to silicon bonding (Mihalcea et al., 1996), anodic bonding (Albrecht et al., 1990a), glueing (Scholz et al.,

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1997), and soldering (Hantschel et al., 2000). In Section II.C.3c the projection mask technique is introduced, which works without the necessity to mount a mechanical holder and offers some additional advantages. Several approaches have been discussed in the literature to facilitate the handling of silicon-based AFM probes during their operation. As shown schematically in Fig. 1, this may include the integration of an intrinsic, that is, electronic, mechanism to detect the cantilever bending and/or the integration of a microactuator necessary for the tip positioning in the feedback loop of the AFM. a. Piezoresistive AFM Probes. For the detection of the cantilever bending in almost any case external optical and electrical detection methods were applied, such as the beam deßection method (Meyer and Amer, 1988), interferometry (Erlandsson et al., 1988; Rugar et al., 1989; Putman et al., 1991; Ruf et al., 1997) or capacitive or piezoelectric readout (Goddenhenrich et al., 1990; Itoh and Suga, 1994; Giessibl, 2000). Most of them are easy to implement, need little instrumentation, and offer a rather high sensitivity (Sarid, 1991). However, they demand a careful adjustment of the optical or electrical components that is most undesirable in some particular experimental environments, such as vacuum, low or elevated temperatures, restricted geometry of the experimental setup, and parallel probe operation. Tortonese et al. exploited the piezoresistive effect of silicon to get rid of any external detection scheme (Tortonese et al., 1993). In the piezoresistive effect the mechanical stress σ gives rise to a relative change R of the electrical resistance R due to a deformation of the electronic band structure (Smith, 1954; Kanda, 1982): R = σ R

(12)

 denotes the tensor of the piezoresistive coefÞcients. In the Þrst experiments presented by Tortonese et al. an external Wheatstone bridge was used to improve the sensitivity. However, it proved to be advantageous to integrate the complete Wheatstone bridge next to the supporting point of the cantilever to reduce the inßuence of environmental temperature changes (Jumpertz et al., 1998; Su et al., 1999; Gotszalk et al., 2000; Thaysen et al., 2000). Brugger et al. have adapted the piezoresitive sensor arrangement to yield a highly sensitive probe for torque measurements in magnetometry (Brugger et al., 1999). An enhancement of the sensitivity was achieved exploiting higher ßexural modes as shown by Volodin and van Haesendonck (1998). Next to the sensitivity, the operational bandwidth of the displacement detector is an important issue. Su et al. have optimized the piezoresitive displacement sensor to realize scan rates of 1 mm/s, sufÞcient for most applications (Su et al., 1999).

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EGBERT OESTERSCHULZE

b. Microactuator. In single AFM probe conÞgurations the control of the tipto-sample distance is in almost any case accomplished using an external and bulky piezoelectric stack actuator. However, if parallel operation of an extended array of probes is intended, this is not adequate. Therefore, cantilevers with integrated actuators for positioning the tip have been developed by different groups. In most cases reported in the literature zinc oxide (ZnO) (Albrecht et al., 1990b; Fujii et al., 1995) or lead zirconate titanate (PZT) (Miyahara et al., 1999; Lee et al., 1999a) were used as actuator material. Nevertheless, these materials also show some notable disadvantages. Typical deßections of the very cantilever of some microns require control voltages of typically some 10 to 100 V. Furthermore, technological processes for structuring the mentioned materials are not easy to control. Polymer material, such as polyaniline, might be an interesting low-cost alternative not mentioned thus far for this purpose. It is interesting enough to note that the adequate control voltage of polyaniline is in the range of 0.5Ð0.7V, that is, lower by two orders of magnitude than that for piezoelectric materials. c. High-Speed and Parallel AFM ConÞgurations. In AFM-related techniques the serial image formation process obviously restricts the temporal bandwidth. High-speed imaging requires cantilevers with high resonant frequencies operated in the FM AFM mode. Garcia et al. (1995) were among the Þrst to present nanocantilevers with a resonant frequency of the Þrst ßexural mode in the range of GHz, and they discussed the impact of the cantilever geometry on the latter. Various other groups succeeded in fabricating silicon nanocantilever probes for the same purpose (Walters et al., 1996; Stowe et al., 1997; Wago et al., 1997; Paloczi et al., 1998). Kawakatsu et al. (2000a) introduced nanometric oscillators consisting of a silicon tip of tetrahedral geometry acting as a concentrated mask. The latter has a size of 100Ð1000 nm with a compliance of typically 0.1Ð100N/m and resonant frequencies in the range of 0.01Ð1GHz (Kawakatsu et al., 2000b). In contrast to the conventional cantilever geometry the orientation of these nanometric oszillators is perpendicular to the subtrate surface (Saya et al., 2000). A parallel arrangment of cantilever probes is an additional option to improve the bandwidth of the AFM. Batch fabrication on a base of silicon substrates provides an adequate method to accomplish arrays of cantilever probes as was demonstrated by Minne and co-workers (1995a). In their pioneering work they demonstrated the individual control of at least 50 cantilevers, integrating both a piezoresitive readout and a micractuator based on thin Þlms of ZnO. The schematic setup as well as a close-up of an array of 50 cantilevers is depicted in Figure 3. The parallel arrangement of probes was exploited not only for high-speed parallel imaging (Manalis et al., 1996), but also for various other

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(a)

(b) Figure 3. (a) Schematic drawing of a single cantilever with integrated piezoresistive readout (sensor region) and thin Þlm ZnO actuator (actuator region). (b) SEM image of 18 cantilevers of an array of 50 cantilevers which span 10 mm. (Reproduced with permission from Minne, S. C., Adams, J. D., Yaralioglu, G., Manalis, S. R., Atalar, A., and Quate, C. F. (1998). Centimeter scale atomic force microscope imaging and lithography. Applications of Physics: Letters 73: 1742Ð 1745; Wilder, K., Soh, H. T., Minne, S. C., Manalis, S. R., and Quate, C. F. (1997). Cantilever arrays for lithography. Naval Research Reviews XXIX:35Ð48).

path-breaking high-speed applications, that is, ultra-dense lithography (Minne et al., 1995b, 1996b) and data storage (Chui et al., 1996). Figure 4 shows a prototype of a sliding headÑcalled a millipedeÑwith a 32 × 32 array of silicon AFM cantilevers presented by Despont and coworkers (2000). Each cantilever is provided with a piezoresistive readout and an electrical resistor for heating the tip. In accordance to the Þrst experiments done by Mamin and Rugar (1992), data recording is accomplished by

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EGBERT OESTERSCHULZE

(a)

(b) Figure 4. SEM image of the millipede: an array chip of 32 × 32 AFM cantilevers used for thermomechanical ultrahigh density data storage. (Reproduced with permission from Vettiger, P., Despont, M., Drechsler, U., D¬ urig, U., H¬ aberle, W., Lutwyche, M. I., Rothuizen, H. E., Stutz, R., Widmer, R., and Binning, G. K. (2000). The millipedeÑmore than one thousand tips for future AFM data storage. IBM Journal of Research and Development 44(3):323Ð340).

thermomechanical indentation of the tip into a thin polymer Þlm on a rotating dish. The plastic deformation of the polymer material gives rise to recorded pits of yet about 30Ð40 nm dimension and approximately the same pitch, that is, storage densities of 500 Gbit/inch2 (Vettiger et al., 2000). It is worth to note that no feedback loop is required to keep each sensor in track with the spinning disk surface. Meanwhile, Kawakatsu et al. (2001) introduced an array of a million of cantilevers made from SOI substrates. The cantilevers with the integrated tips are obtained by an intricate technological process (Kawakatsu et al., 2001). SEM images of the cantilever array and the cantilever in detail are given in Figure 5. The same group also presented an array of the nanometric oscillators just mentiond, exhibiting a resonant frequency of approximately 1 GHz, which might be an option to further increase the operational bandwidth of array arrangements (Kawakatsu et al., 2000a).

141

RECENT DEVELOPMENTS OF PROBES

10 m (a)

(b)

Figure 5. (a) SEM image of millions of single crystalline silicon AFM cantilevers with 10 μm spacing. (b) Zoom image showing two facing rows of cantilevers, each with a tip of 2 μm height. (Reproduced with permission from Kawakatsu, H., Saya, D., Fukushima, K., Hashiguchi, H., Toshiyoshi, G., and Fujita, H. (2001). Millions of cantilevers for simultaneous atomic force microscopy presented during MEMS 2001, Interlaken, Switzerland).

2. Gallium Arsenide Gallium arsenide and silicon both show fcc crystal lattice symmetry. However, there are considerable differences in their etching behavior. The ionic/covalent bonding in the GaAs crystal gives rise to a completely different etch rate distribution, as shown in Figure 6, in comparison to that of Si (Seidel et al., 1990a). One typical etchant employed for wet chemical etching processes consists of a mixture of H2SO4:H2O2:H2O of varying composition; that is, an oxidizing and reducing agent, and a complex-forming substance to get the sparingly soluble oxides of gallium and arsenic solved in aqueous solutions. The etch distribution reveals a minimum and maximum etch rate on (111)

[100]

mask geometry

[ 011]

etching

etched structure

[ 001] { 111 } A ( 011 )

(a)

[ 011]

[010] {111 } A

(b)

[ 011]

Figure 6. Anisotropic etching through square windows in an etch-resistant masking layer on a single-crystalline gallium arsenide substrate gives rise to a dovetail structure. Three-dimensional and top views of the resultant etch structure are given in (a) and (b), respectively. In the [011] direction, the etch pit geometry is identical to that obtained by anisotropic etching of (001)ø direction, two (111) A-planes emerge with a negative oriented silicon. However, in the [011] surface orientation that gives rise to a knife-edged structure.

142

EGBERT OESTERSCHULZE

Figure 7. SEM images of two GaAs tips fabricated by anisotropic etching with an aqueous solution of H2SO4:H2O2:H2O. (a) Obelisk-shaped tip deÞned by the intersection of four identical crystal planes. (b) The intersection of only three noncoplanar crystal planes guarentees a pointed, that is, a very sharp, tip. (Reproduced with permission from Heisig, S., and Oesterschultze, E. (1998). Optical active gallium arsenide probes for scanning probe microscopy. SPIE 3467:305Ð 312).

crystal planes terminated by Ga(ÒAÓ)and As(ÒBÓ)atoms, respectively. This is in contrast to the etching behavior of Si, which shows a relative minimum of the etch rate on all equivalent (111) crystal surfaces (Seidel et al., 1990b). This is illustrated in Figure 6 for the simple case of anisotropic etching through a square opening in a masking layer. In case of silicon the well-known pointed pyramidal etch pit develops, whereas in case of GaAs a knife-etched structure emerges. A detailed discussion of the rather complex etching behavior of GaAs is given in Howes and Morgan (1986). As a consequence, the prediction of structures obtained by underetching of masking pads on a GaAs substrate, as used, for example, for tip fabrication, is rather complicated. Nevertheless, inherent in the complex etching behavior is the capability to obtain a richer diversity of tip shapes in comparison to silicon, as was demonstrated by Heisig and Oesterschulze (1998). The two selected tips shown in Figure 7 exemplify the variation of tip geometries. We must mention that sharpening of GaAs tips by repeated oxidation is not feasible owing to the minor quality of the native oxides of Ga and As. The different etch behavior of (111)A- and (111)B-planes also demands a proper etching process for the fabrication of the AFM cantilever structure. Two methods have been proposed for this purpose. In the Þrst case thin burried layers of AlAs were used as on etch stop layer exploiting the etch selectivity with respect to GaAs (Miao et al., 1995). The technique allows a precise control of the probe beam thickness (Hascik et al., 1996; Mounaix et al., 1998), as was proven by Harris et al. (1996) in the case of 100-nm thin GaAs cantilevers. However, if conventional SPM probes with a cantilever thickness and a tip height of some micrometers are of interest, this method is too time-consuming and expensive

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becuase of the necessary epitaxial growth of the GaAs- and AlAs-layer. The second approach uses an atomizer to spray the etchant onto the substrate rather than dipping it into the etch solution (Chen et al., 1987; Tanobe et al., 1992). It avoids any kind of sumptuary epitaxial layer growth and offers some practical advantages over dip etching, that is, negligible temperature dependence, improved stability of the etching process, and substantially reduced surface roughness. Heisig et al. presented a GaAs cantilever with excellent thickness homogeneity over the entire cantilever fabricated by the spray etching technique (Heisig and Oesterschulze, 1992). 3. Carbon Various modiÞcations of carbon-containing material have attracted the attention for tip fabrication. Before embarking on the crystalline modiÞcation of sp3 bonded carbon, that is, diamond, other activities are discussed that are recently developing. a. Electron Beam Deposited Tips. As early as 1990, Akama et al. reported the Þrst results of carbon-containing STM tips produced by electron beam deposition from the gas phase in a scanning electron microscope (SEM), although the contamination lithography process had been invented earlier (Broers et al., 1976). This process is capable of fabricating high-aspect-ratio tips with sharp apexes (Okayama et al., 1988; Ichihashi and Matsui, 1988). However, another modiÞcation caused a much greater stir: the nanotube arrangement of carbon (and, since then, other materials). b. Nanotubes. In 1991 Iiiji reported for the Þrst time on the fabrication of needle-like tubes composed of graphitic carbon called nanotubes. Synthesis of carbon nanotubes relies in almost all cases on laser vaporization, discharge between carbon electrodes, or thermal decomposition of hydrocarbons (Dresselhaus et al., 1996; Harris, 1999). The impact of nanotube material on SPM applications can be derived from its unique structural composition as well as its extraordinary physical properties. Their geometry can be thought of as arising from the folding of a graphene sheet to form a seamless hollow cylinder composed of carbon hexagons (Iiiji et al., 1992). The diameter of single- or multiwalled nanotubes (SWNTs or MWNTs) is in the range of some nanometers, whereas lengths of some 10 micrometers or even more are easy to achieve (Harris et al., 1996). That is, nanotubes exhibit a very large aspect ratio and are therefore perfect for imaging deep surface structures (Dai et al., 1996). Furthermore, the radius of curvature of SWNTs is superior to those of commercially available silicon- or silicon nitrideÐbasedtips as was proven by AFM and STM measurements (Nagy et al., 1998; Wong et al., 1998a;

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EGBERT OESTERSCHULZE

Dai et al., 1998). As expected from the similar structure of carbon in the basal plane of graphite, nanotubes offer a tremendeous value of YoungÕs modulus of about 0.4Ð1.3TPa (Teacy et al., 1996; Wong et al., 1992; Salvetat et al., 1999) for rope diameters of 20Ð2 nm in the axial direction, rivaling that of diamond (see Table I). However, nanotubes are both stiff and gentle. Nanotubes maintain a remarkable resistance to fracture (Falvo et al., 1997, 1999; Nardelli et al., 1998; Ru, 2000). It they encounter a surface at near-normal incidence, they buckle if the force exceeds the Euler buckling force (Dai et al., 1996) FEuler = π 2 E I /l 2

(13)

In this context E denotes YoungÕs modulus, I the stress moment, and l, r the length and radius of the nanotube, respectively. Furthermore, buckling is accompanied by high strain resilience. Conventional solid AFM tips are destroyed in crashing into a sample surface, even in the case of a moderate force load. Nanotubes, however, evade fracture by bending; that is, they offer an intrinsic protection mechanism to avoid fracture. Additionally, this prevents damage to delicate materials, such as biological samples. Frictional and other mechanical properties of nanotubes residing on surfaces are discussed elsewhere (Nardelli et al., 1998; Falvo et al., 1999a, 1999b; Hertel et al., 1998; Kizuka, 1999). Nanotubes show also interesting electrical properties. Their carrier densities, susceptibilities, and conductivities resemble those of graphite. However, transport properties and ESR measurements indicate that carrier localization occurs (Tans et al., 1997; Bezryadin et al., 1998; Avouris et al., 1999; Poncharal et al., 1999; Gaal et al., 2000). Furthermore, nanotube Þlms serve as excellent Þeld emitters (Bonard, 1998, and references therein). They produce large current at low electric Þelds and offer a performance that is superior to the intensively studied diamond Þlms (e.g., Okano et al., 1994; Kang et al., 1996). This behavior may be explained in view of the particular electronic structure of nanotubes (de Heer et al., 1997). In this context it is noteworthy that nanotubes provide another feature of great interest in biological and chemical applications: the possibility of altering the very tip by chemical treatment with the aim of sensing and manipulating samples on a molecular level (Wong et al., 1998b; Terrones et al., 1998). Although this brief discussion has revealed the enormous potential of nanotubes, it is nevertheless not straightforward to get nanotubes attached to tips of cantilever probes. Dai et al. (1996) and Barwich et al. (2000) used carboncontaining adhesive to manually mount MWNTs to the tip of conventional AFM tips. Nishijima et al. (1999) and Akita et al. (1999) Þxed the MWNT by deposition of electron beam deposition of carbon in a SEM. To date no batch process has been presented for the fabrication of MWNT or SWNT tips.

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c. Diamond. Diamond with its very small bond length of only 0.155 nm between two neighboring carbon atoms shows the highest mechanical hardness (see Table I) of all materials known to date. Thus it is not surprising that diamond as substrate material is the most important candidate in applications where tip wear is an important issue. At Þrst AFM tips were formed by fracture, but the geometry and surface chemistry of such tips are not well deÞned (Binnig and Rohrer, 1986; Marti et al., 1987). In similar approaches tips for STM have been obtained by grinding and polishing of single-crystalline materials (Visser et al., 1992; Kang et al., 1996). However, all these techniques require expensive bulk diamond as starting material. Isolated diamond grains were deposited by CVD processes on W and Si tips (Germann et al., 1990; Liu et al., 1994). The missing control of orientational relationship between tip and crystal, the poor yield and reproducibility, and the preferred deposition of crystals at the side walls of the tip rather at its apex were the most substantial problems reported. Givargizov et al. (1995, 1996) presented a CVD technique capable of growing single diamond whiskers on (111) oriented Si pillars. However, the (111) orientation of the substrate is rather undesirable for batch fabrication owing to its etching behavior (Heuberger 1991; Givargizov et al., 1993). Conventional Molding. First approaches (Niedermann et al., 1996; Oesterschulze et al., 1997; Okano et al., 1994; Kang et al., 1996) for batch fabrication of CVD deposited diamond probes used the molding technique introduced by Albrecht et al. for the fabrication of silicon nitride tips (Albrecht et al., 1990a). Both diamond tips (Niedermann et al., 1996; Scholz et al., 1997) and all-diamond probes (Niedermannet al., 1998; Kulisch et al., 1997; Mihalcea et al., 1998) with a pyramidal tip obtained by molding from anisotropically etched pits in (001) oriented silicon wafers were realized, as can be seen from Figure 8a. Owing to the restricted accuracy during the lithographic deÞnition of the molds, in many cases knife-edged tips rather than pointed ones were obtained. Hantschel et al. (1999) introduced the tip-on-tip approach to overcome this problem. The accuracy was improved, reducing the window size for the deÞnition of the mould geometry to only 300 nmÐ1μm, a size that demands an advanced e-beam lithography process. The method is capable of fabricating pointed probes of both CVD diamond and various kinds of deposited metals. A typical diamond tip made by this technique is depicted in Figure 9. Technological efforts were substantially reduced by coating commericial high-end Si probes with thin CVD diamond Þlms but at the expense of a slightly enlarged radius of curvature (see Figure 8b) (Niedermann et al., 1998; Yuan et al., 1998; Trenkler et al., 2000). Niedermann et al. combined both methods to end up with sharp tips provided with an improved aspect ratioÑthe ratio of the tip height to its widthÑby introducing the double molding process (Beuret et al., 1998b). A comparison of all these diamond probes is given in Trenkler et al., (2000).

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EGBERT OESTERSCHULZE

Figure 8. (a) All-diamond probe fabricated by conventional molding (by courtesy of Niedermann) (Niedermann et al., 1998) and (b) commercial diamond-coated silicon tip (by courtesy of NanoSensors GmbH & Co. KG, Sensitec-Building, Wetzlar, Germany).

However, molding is accompanied by some noteworthy technological and application-oriented disadvantages. The interfacial diamond layer of lower quality forms the tip whereas the high-quality material resides on the cantilever rear. Its very rough surfaces add difÞculties with respect to the application of the optical beam deßection technique to detect cantilever bending. Furthermore, during AFM measurements the tip apex is not aligned perpendicular to the sample surface as would be desirable for nanoindentation experiments. The tip is shadowed by the cantilever beam, and thus addressing of certain spots on the sample by means of optical control is almost impossible. Last, but not

(a)

300 nm

(b)

Figure 9. SEM image of (a) a rugged diamond tip of 300 nm base width and (b) a metal tip made of a stack of 60 nm Cr, 20 nm W, and 5 μm Au. Each tip was fabricated by the tip-ontip technique. (Reproduced with permission from Hantschel, T., Trenkler, T., Vandervorst, W., Malav« e, A., B¬ uchel, D., Kulisch, W., and Oesterschultze, E. (1999). Tip-on-tip: A novel AFM tip conÞguration for the elecctricle characterization of semiconductor devices. Microelectronic Engineering 46:113Ð116).

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Figure 10. General fabrication scheme of the projection mask technique: The lateral cantilever structure deÞned on a conventional, that is, planar lithography, mask is transferred onto the already structured substrate. The later deÞnes the vertical geometry of the cantilever with the tilted tip. (Reproduced with permission from Malav« e, A., Leinhos, T., and Oesterschultze, E. (2001). Projection mask technique for the fabrication of cantilever probes, submitted).

least, two double-sided polished (001) oriented Si wafers including an intricate bonding process are necessary to accomplish diamond probes. In conclusion, fabrication of conventionally molded probes is awkward and quite expensive. Projection Mask Technique. A new technique called projection masking was introduced by Oesterschulze et al. to overcome most of the previously mentioned problems (Malav« e et al., 2001). Figure 10 shows the general principle of the method, which separates the deÞnition of the lateral cantilever structure from its vertical geometry. A one-sided polished (001) oriented Si wafer is structured by a conventional lithography and subsequent wet chemical or plasma etching process to obtain tubs that deÞne the vertical structure of the cantilever beam. The lateral structure of the probe is deÞned in a subsequent step by a projection mask lithography process that gives the name to this fabrication process. The mechanical holder of the probe is made afterward from the substrate by conventional MEMS processes. The projection mask technique is not restricted to polycrystalline diamond because any material that can be deposited as a thin Þlm is useful for the deÞnition of AFM probes. Nevertheless, we Þrst discuss results obtained for all-diamond probes in the following. Figure 11 comprises the fabrication process of all-diamond probes by the projection mask technique. In step (a) a (001) oriented silicon substrate is anisotropically etched with KOH to obtain 5- to a 30-μm deep etch tubs with (111) side walls and (001) oriented bottom. The thermally oxidized Si wafer is subsequently spin-coated with a photoresist layer and the lateral

148

EGBERT OESTERSCHULZE

projection mask

(001) Si photo resist silicon dioxide diamond

(a)

projection mask

(b)

(d)

h t (c)

holder

(e)

Figure 11. Fabrication scheme of the projection mask technique: (a) Top view and (b) cross section of the structured substrate and the already aligned lithography mask. The mask deÞnes the lateral outline of the cantilever via a proximity lithography and etching process while the vertical geometry is given by the structured substrate. The cantilever can be fabricated from the substrate or a thin deposited Þlm. (Reproduced with permission from Malav« e, A., Leinhos, T., and Oesterschultze, E. (2001). Projection mask technique for the fabrication of cantilever probes, submitted).

cantilever structure is transferred by optical lithography in step (b). Diffraction will distort the cantilever structure in the tub because of the proximity exposure. However, this happens reproducibly for all cantilevers and thus could be taken into account already during the deÞnition of the projection mask. It is important to note that the deÞnition of the tip is left almost unaffected because the distance between the projected tip geometry on the side wall of the tub and the projection mask can be reduced to less than 1 μm. In step (c) the structure in the photoresist layer is transferred into the oxide layer by BHF etching. Prior to selective HFCVD diamond deposition on the exposed silicon, the wafer is subject to ultrasonic pretreament with diamond powder immersed in pentane (Mihalcea et al., 1998). In the last two steps, (d) and (e), Þrst the outline of the mechanical holder is deÞned in the silicon dioxide layer on the other side of the wafer and Þnally the Si holder is etched, freeing the diamond cantilevers in the same step. Because of the coarse feature of the holder, an unpolished wafer surface is adequate and only single-sided polished wafers are necessary.

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As can be seen from the cross section in step (b) the tip geometry is deÞned by the structure of the tilted side walls of the tub. Both the shape and the orientation of the tip can easily be adapted by a certain etching process to compensate for the alignment angle of typically 4Ð10◦ between the cantilever and the sample surface in the AFM. Perpendicular orientation of the very tip on the sample surface can easily be achieved. In contrary to the conventional molding process the tip material is made of high-quality diamond material, the surface of the cantilever is almost ßat as necessary for the application of the beam deßection method, and the tip position can easily be controlled because of the good optical access to the cantilever. The Þrst SEM images of all-diamond cantilever probes made by the projection mask technique are shown in Figure 12. A v-shaped and single beam

(a)

(b)

(c)

(d)

Figure 12. SEM images of all-diamond AFM cantilever probes made of polycrystalline diamond applying the projection mask technique: (a) Diamond membrane with a v-shaped cantilever probe mounted on a silicon holder, (b) single cantilever with a pointed tip, (c) bottom view of the cantilever in a) revealing the rough growth surface of HFCVD diamond, (d) sharpening of the diamond grains by plasma treatment. (Reproduced with permission from Malav« e, A., Leinhos, T., and Oesterschultze, E. (2001). Projection mask technique for the fabrication of cantilever probes, submitted).

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EGBERT OESTERSCHULZE

cantilever are depicted in Figures 12a and 12b, respectively. Both images reveal the ßat cantilever surface as well as the good selective growth of the diamond layer. The angle between the tip and the cantilever in Figure 12c corresponds to that between the (001) and (111) silicon surface; that is, 125◦ . Figure 12d shows the same tip after oxygen plasma treatment. It reveals the sharpening of the grains constituting the diamond Þlm.

D. Concluding Remarks In the last sections most of the conventional techniques used for AFM probe fabrication were summarized. However, as already indicated in Figure 1, the intriguing potential of the SPM method relies on the idea of integrating additional sensors into the sampling tip of a conventional AFM probe. Most important next to a proper sensor concept for this purpose is the question of a material with appropriate properties for the desired application. Therefore, a few additional

TABLE II Some Physical Properties of Single-Crystalline (001) Oriented Materials Used for Probe Fabricationa Material Silicon (Schulz and Blachnik, 1982) Thermal properties Heat capacity cP (J/kgK) Conductivity kth (W/mK) Melting point (K) Thermal expansion coefÞcient (1/K) Seebeck coefÞcient (μV/K) Optical properties Refractive index (λ = 633 nm) Static dielectric constant ǫ Electrical properties Band gap (eV) Electron mobility (cm2/Vs) Hole mobility (cm2/Vs) SpeciÞc resistivity ( cm)

GaAs (Blakemore, 1982)

Diamond (von M¬ unch, 1982)

690 156 1685 2.56

327 45.5 1513 6.86

515.8 600Ð2000 Graphitized 1.0

−1600Ð1500

−680Ð130

3.4

3.878

2.41

11.8

13.18

5.70

1.12 (indirect) 120Ð1500 70Ð500 <105

1.42 (direct) 9200 (SI) 400 (SI) <109 (SI)

5.48 (direct) 1800 1600 <1017

a Data are given at room temperature (T = 296 K). In the case of GaAs, the abbreviation SI denotes semi-insulating material.

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151

properties of the single crystalline materials already mentioned in Table I are compiled in Table II. In the following section, a comprehensive summary of sensor activities in case of scanning near-Þeld optical microscopy will be given.

III. Near-Field Optics Starting in 1984, almost at the same time, Pohl and co-workers (1984), Durig et al. (1986), Lewis et al. (1984), and Fischer (1985) presented an innovative offset of the SPM method: scanning near-Þeld optical microscopy (SNOM) (sometimes also called near-Þeldscanning optical microscopy, NSOM). However, near-Þeld optics traces back to the pioneering work of Synge in 1928 and was resumed by OÕKeefe in 1956. The principal idea of SNOM is to circumvent the diffraction limit of conventional far-Þeld optics detecting the near-Þeld components, that is, the high-frequency components of the Þeld distribution of the object. Ash and Nicholls (1972) conÞrmed the working principle of near-Þeld optics in the microwave range. Before we embark on our discussion of recent developments in proper near-Þeld probes in the optical range, a short introduction to the basics of far-Þeld optics is Þrst given to determine its restrictions. It is followed by a brief introduction to some aspects of near-Þeld optics.

A. Theory of Far-Field Optics Optical imaging of objects relies on the propagation of Þeld energy from the object plane through the imaging system onto the image plane. The respective  r , t) as well as the magnetic Þeld H (r , t) time-dependent local electric Þeld E( have to obey the Helmholtz equation (Born and Wolf, 1959):     r , t) 1 ∂2 E( =0 (14) − 2 2 H (r , t) c ∂t with c = c0 /n the velocity of light in the respective medium. The latter might be expressed in terms of the light velocity c0 in free space and the refractive index √ n of the medium deÞned as n = ǫ, with ǫ the frequency-dependent dielectric function of the medium. If we assume the sample to have the local transmittivity TS (rÿ) in the specimen plane illuminated by the electric Þeld distribution E0 (rÿ) of the light source, then the Þeld distribution EI (rö) in the image plane is expressed in terms of two successive diffraction processes, one at the sample and the other one at the imaging lens system. This was conjectured by Ernst Abbe in 1873. The application of the FresnelÐKirchhoff diffraction

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EGBERT OESTERSCHULZE

theory allows a more detailed mathematical description. The diffraction processes are expressed by two surface integrals, the Þrst across the specimen plane with coordinates rÿ = (xÿ, yÿ) and area σ S and the second across the lens plane with coordinates rø = (xø, yø) and area σ L (Goodman, 1968):   ÿ ø ø ö e−ik(|r−r||r−r|) i2 ø ö E I (r) = 2 d xÿ d yÿ E 0 (rÿ)TS (rÿ) d xø d yø TL (r) (15) λ |rÿ − rø||rø − rö| σL

σS

λ denotes the wavelength of light, k = 2π/λ the absolute value of the propagation wave vector, and i the complex unit. In correspondence with HuygensÕprinciple, the last term of the integral is the GreenÕs function of a point source, that is, a spherical wave. Equation (15) is substantially simpliÞed in the Fraunhofer approximation abbrevating E 0 (rÿ)TS (rÿ) as E S (rÿ):    ÿ ö r  r  1 E I (rö) = d xÿ d yÿ E S (xÿ, yÿ)τ L − (16) mg 2 λ2 bλ gλ σS

1 E S (ÿ r ) ⊗ τL = mg 2 λ2



rÿ gλ



(17)

where τ L denotes the spatial Fourier transform of TL, also termed the point spread function (PSF), ⊗ the convolution operator, and g, b the image and object distance, respectively. Applying the convolution theorem on Eq. (17) allows to interpret image formation with a lens as spatial low-pass Þltering of the Fourier transform of the object transmission function F (E S (rÿ)) with TL. In the simple case of a circular lens of radius R and unit transmittivity,  1 for |rø| ≤ R ø (18) |TL (r)| = 0 else The Fourier transform of TL, that is, the PSF, is the well-known Airy function: √ ø2 + vø2 ) 2 2J1 (2π R u ø ø vø) = F (|TL (r)|) = π R (19) τ L (u, √ 2π R uø2 + vø2 where J1 is the Bessel function of Þrst order. The corresponding spatial cutoff frequency kLimit is obviously obtained from the Fourier transform of Eq. (17) to be kLimit =

nR 1 = dmin gλ0

(20)

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153

This differs only by a factor of 1/0.61 from the value obtained from the arbitrarily deÞned Rayleigh criterion (Hecht, 1989): 1 nR 1 NA ≈ 0.61 gλ0 0.61 λ0

kRayleigh =

(21)

with NA the numerical aperture of the imaging lens system. The Rayleigh criterion states that two point sources separated by a minimum distance dMin in the specimen plane can be distinguished in the image plane if the maximum and minimum of the corresponding diffraction patterns, that is, the PSF given in terms of Eq. (19), coincide. However, this is not the only restriction of the imaging capability of far-Þeld optics. This can easily be seen by writing the absolute value of the propagation vector k in terms of its components: 2= |k|



2

2π λ

= k x2 + k 2y + k z2

(22)

Obviously propagation through free space is only possible in the case of spa2 + k z2 = tial frequency components of the object inside the Ewald sphere kObject 2π 2 2 2 2 ( λ ) with kObject = k x + k y . For higher spatial frequencies kz becomes imaginary and propagation is restricted to evanescent Þelds, that is, exponentially damped Þelds with a characteristic damping length μ of

μ

−1

=

2

kObject −



2π λ

2

(23)

Owing to the strong damping of evanescent Þelds, far-Þeld optics is not capable of collecting high-frequency components of the object irrespective of the numerical aperture NA of the applied lens system. In conclusion, the two consecutive diffraction processes give rise to a double low-pass Þltering process where the Þrst is of fundamental physical origin that cannot be circumvented using far-Þeld optics. The improvement of spatial resolution obviously demands the detection of the high-frequency spatial components, that is, the evanescent waves originating at the sample in the specimen plane. This is exactly what near-Þeld optics deals with. Various schemes of pointed probes were invented to convert the evanescent Þelds into propagating ones that can easily be detected by conventional far-Þeld optics.

154

EGBERT OESTERSCHULZE

Link

Detector

Near-field probe

Light source

Sample

Light source

Detector

Figure 13. The general scheme of a near-Þeld microscope might be thought of as a conventional far-Þeld microscope setup with the addition of a pointed probe to pick up part of the near-Þeld components of the Þelds in the vicinity of the sample surface. The latter are converted by the near-Þeld sensor in propagating modes that can be detected using conventional optics and detectors.

B. Introduction to Near-Field Optics The general scheme of a near-Þeld microscope shown in Figure 13 is derived from a conventional far-Þeld setup introducing a pointed probe as is usual in SPM methods. It is used to convert part of the near Þeld into propagating modes that are accessible by far-Þeld optics via a link section. The latter has an important impact on the efÞciency of NFO probes as is discussed later. The arrows in the schematic view indicate the ray paths in the various operation modes of SNOM, such as illumination or collection mode in transmission, internal, and external reßection modes (Fischer, 1998). The characteristics of near-Þeld optical detection might be illustrated by discussing the simplifying analogy of a near-Þeld optical conÞguration with two interacting electric dipoles. As is well known from classical electrodynamics the Þeld distribution of a single electric dipole with dipole moment p observed in the direction of the unit vector n is given as (Jackson, 1982): far-Þeld

near-Þeld   ikr ik 1 e e 2  r = r n ) = k ((n × p ) × n − [3n (n · p ) − p ] − 2 (24) E( r r r r ikr

eikr 1 eikr − k 2 n × p r ikr r The Þrst expressions on the right side are the far-Þeld contributions expressed in terms of a spherical wave emerging with a certain angle dependence with respect to p . However, in the near Þeld the terms 1/r3 and 1/r2 dominate. In a more  r = r n ) = B(

k 2 n × p

RECENT DEVELOPMENTS OF PROBES

155

realistic approach a scattering sphere of radius R is considered as near-Þeld probe with the following total scattering cross section (Jackson, 1982): σ total ∼ k 4 R 6

(25)

and thus shows a substantial dependence on the radius R, that is, the geometry of the scattering particle. From these simplifying arguments the following rules in near-Þeld optics can be derived: r

r

r r

r

r

r

r

r

Owing to the strong distance dependence, the spacing between the nearÞeld tip and the sample surface has to be kept very small, that is, some nanometers (Hecht et al., 1997; Carminati et al., 1997; Valle et al., 1999). A strong far-Þeld component with 1/r dependence is superimposed on the near Þeld. For accurate measurements the tip has to be screened to avoid side-to-side cross-talk (Bozhevolnyi et al., 1994). A pointed tip is favorable to facilitate the approach to the sample surface. The dimension of the aperture tip should be at least of the same size as the object to be imaged (Ohtsu, 1998). The R6-dependence of the cross section [see Eq. (25)] on the near-Þeld probe dimension means that for very small dimensions the scattered and thus detectable Þeld energy drastically decreases; that is, one must Þnd a compromise between attainable lateral resolution and an acceptable signal-to-noise ratio. In particular cases modulation techniques are advantageous to improve the signal-to-noise ratio (Zenhausern et al., 1994). This very low efÞciency demands superior shielding of the tip and the link area (see Fig. 13) to avoid cross-talk via far-Þeld contributions of other sample locations. In contrast to far-Þeld optics, the wavelength in near-Þeld optics plays only a minor role. Because of the very small separation between the sample and the probe, a retroaction is expected that results in a complex inßuence of the near-Þeld probe on the measured signal (Ohtsu, 1998). The optical transfer function stays linear, but there is no way to deÞne a translation-invariant point spread function, although this was suggested at the beginning of near-Þeld optics (Courjon et al., 1990; Massey, 1984; Vigoureux and Girard, 1992). Thus image interpretation in near-Þeld optics is quite delicate (Carminati et al., 1997; Bozhevolnyi, 1997). Most of the near-Þeld probes might also be operated in reverse order; that is, as near-Þeld emitters or detectors.

For a general classiÞcation of near-Þeld probes they might be divided as shown in Table III into passive and active probes. The Þrst group of probes only directs the light coming from an external light source to an external

156

EGBERT OESTERSCHULZE TABLE III Classification of Near-Field Probesa Detector

Emitter

Passive

Active

Passive

Aperture Scatterer Immersion lens

Nanoscopic detector Integrated detector

Aperture Scatterer Immersion lens

Active Nanoscopic emitter Integrated emitter

a Integrated probes means in this context that rather large detector or emitter is integrated in the vicinity of the working tip of sub-wavelength dimension. In contrast, nanoscopic probes denote tips with sub-wavelength-sized features.

detector. Active probes, however, are used to detect or emit light directly in the pointed tip.

C. Passive Probes Passive probes have the main function to guide the luminous ßux that is made up of near- and far-Þeld components in near-Þeld optics. Irrespective of the nature of the near-Þeld probe, for example, a sub-wavelength-sized aperture or a scattering tip, its optical behavior might be described in Þrst approximation in terms of a scatterer that follows the geometrical as well as spectral dependence expressed in Eq. (25). Although the size of this scattering probe determines the attainable lateral resolution, the link between the far-Þeld optical components and the interacting near-Þeld probe has an important impact on its overall efÞciency. Thus we embark in all concepts of passive probes discussed in the following upon the energy transport properties before discussing recent developments of probes in detail. 1. Aperture Probes Aperture probes consist of a sub-wavelength-sized aperture in an opaque screen. If the aperture is brought into close vicinity to the sample, diffraction is negligible in Þrst approximation and only the aperture size rather than the wavelength determines the attainable resolution. Aperture probes are powered in almost all cases by a tapered hollow-pipe waveguide structure. a. Energy Transport in Hollow-Pipe Waveguides. A survey of the unfavorable waveguiding properties of tapered hollow-pipe waveguides can be obtained that discusses the corresponding mode spectrum in dependence of the

RECENT DEVELOPMENTS OF PROBES

157

10

α a, β a

8 6 4 2 0

a)

0

b)

2

c)

d)

4

6

e)

f) g)

8

10

k0 a Figure 14. Spectral mode distribution of a hollow-pipe waveguide with rectangular cross section of dimensions a × b (with a = 1.5 b) Þlled with a dielectric material (ǫ = 1): (b) TE01, (c) TE10, (d) TE11 and TM11, (e) TE02, (f) TE12 and TM01, and (g) TE20. Continuous lines correspond to the phase term βa and dashed lines to the exponential damping term αa. The later are separated by the respective cutoff frequency of each mode deÞned for βa = αa = 0. (a) is the light line for a wave propagating in free space.

waveguide geometry (Fillard, 1996; Paesler and Moyer, 1996). Although most of the tapered probes in near-Þeld optics exhibit a circular geometry, we will emphasize a rectangular one because it offers favorable polarization properties. Figure 14 comprises the low indexed modes of a hollow-pipe rectangular waveguide with a Þxed cross section of dimension a = 1.5 b. The compo = 2π/λ was separated in the waveguide nent kz of the propagation vector |k| direction into its real and imaginary parts: k z = iα + β

(26)

Obviously, each mode undergoes a transition from pure propagation to pure evanescent transport if the lateral dimension falls below the respective cutoff dimension. The corresponding critical lateral component of the propagation vector is expressed as follows:  mπ 2  nπ 2 mn + (27) kc = a b m, n are integers, at least one of which is different from zero. For sufÞciently small dimension all modes run into cutoff and the predominant part of the evanescent transport is obviously restricted to the lowest mode. This particular behavior allows control of the polarization state of the evanescent mode by varying the ratio a/b (Werner et al., 1998). Figure 14 also indicates that a further reduction of the cross-sectional dimensions gives rise to an enormous damping of the evanescent Þelds. This problem

158

EGBERT OESTERSCHULZE

becomes even more crucial if tapered waveguides are considered. Novotny et al. (1994) and Novotny and Pohl (1995) pointed out that the transmission of tapered waveguides drops more than exponentially with decreasing crosssectional dimensions. b. Field Distribution of Aperture Tips. First calculations of the Þeld distribution in sub-wavelength-sized apertures in an opaque planar screen consisting of an idealized metal were presented by Bethe (1944), Bouwkamp (1950), Bouwkamp and Casimir (1954), Leviatan (1986), and Roberts (1991a, 1987). They anticipated Þeld distributions to be adequately described by magnetic and electric dipoles of proper orientation and amplitude in the aperture plane. However, experimental results of the characterization of Þber probes presented by Oberm¬ uller and Karrai (1995) and Oberm¬ uller et al. (1995) indicated some deviation from the just-mentioned theory. Improved models were discussed taking the tapered region rather than a planar screen into account (Roberts, 1991b; Grober et al., 1996). A more realistic description was given in the papers of Novotny (1996) and Novotny et al. (1995) considering additionally a realistic dielectric function of the opaque metal layer. However, probe geometries discussed were restricted to Þber probes of circular symmetry. The theoretical description of near-Þeld probes of arbitrary geometry has been proposed to be done by numerical methods, such as Þnite integration methods (Rudow et al., 2000). This technique was applied to the study of the exemplary Þeld distribution of tapered aperture probes consisting of pyramidal aperture tips with a fourfold symmetry. The aperture tip is assumed to be made of a 60-nm thin silicon dioxide layer that is coated from the inside with a 65-nm thin, that is, opaque, aluminum layer. The particular tip structure will be introduced in Section III.C.1c in more detail. Figure 15 shows the electrical Þeld energy density in the two symmetry planes of the tip assuming that it is powered from the left-hand side via a waveguide structure not shown here. As indicated the polarization of the incoming mode is oriented in y-direction. The energy density is encoded on a logarithmic gray scale with a factor of 10 between successive lines. Thick black lines were added to identify the tip structure. Figure 15 (left image) reveals that the energy density drops by about Þve orders of magnitude along the propagation direction, the z-direction. Nevertheless, owing to the large opening angle of 70.5◦ this tip offers a rather huge transmission efÞciency because the geometrical cutoff region is restricted to about λ/3 of the taper. In contrast, conventional Þber probes with typically 20Ð30◦ opening angles sustain an even higher loss of transmission. The exponential damping of the electric Þeld inside the metal layer√ is attributed to the skin effect with a typical attenuation length given by δ = 2/(σ ωμ0 μ) with σ denoting the electric conductance, ω the frequency and μ the relative and

159

RECENT DEVELOPMENTS OF PROBES

E

100 nm

E

100 nm

Figure 15. Electric Þeld energy density distribution in the two symmetry planes (x,z) and (y,z) of a pyramidal aperture tip made of a 60-nm thin silicon dioxide layer that is coated with a 65-nm aluminum layer. The calculation was performed assuming ǫ Alu = −23.2 + 8.1i and ǫ Si O2 = 2.4 at a wavelength of 633 nm. The energy density is encoded on a logarithmic gray scale with a factor of 10 between successive lines. The tip is powered from the left (z-direction) via a waveguide structure not shown here. The polarization direction (y-direction) of the incoming mode is marked in both images.

μ0 the vacuum permeability (Collin, 1991). The energy density distribution in the aperture plane shows the well-known Þeld enhancement on the rim of the metal aperture in the polarization direction whereas only a single maximum occurs on the centerline in the perpendicular direction (Novotny et al., 1994). This saddle-shaped Þeld distribution in the aperture plane is also evident from the proÞles and the linear gray-scaled energy density distribution shown in Figure 16. The outmost drop of the Þeld energy density of both the electric and magnetic Þeld along the propagation direction is clearly seen from the proÞles in Figure 17. In agreement with BetheÕs theory, the magnetic Þeld prevails over the electric one, which is disadvantageous for imaging because optical contrast relies on the polarization of the sample induced by the electric Þeld. In conclusion, tapered aperture tips show an intricate Þeld distribution in the aperture plane that adds some difÞculties to the interpretation of near-Þeld optical images. The exceptional low transmission efÞciency raises tremendeous troubles, in particular in the case of experiments with low signal-to-noise ratios, such as Raman or single-molecule spectroscopy.

160

EGBERT OESTERSCHULZE

Energy density [a.U.]

3.0

y 2.0

104nm 1.0

x 0 -200

57nm Aluminium

-100

0

100

200

x,y [nm]

(a)

(b)

SiO2

Figure 16. (a) ProÞles of the electric Þeld energy density parallel (y-axis) and perpendicular (x-axis) to the polarization direction of the incoming mode that is used to power the tip. (b) The complete distribution in the aperture plane on a linear gray scale. It clearly reveals a saddle-shaped distribution.

c. Probes. In the beginning of near-Þeld optics, metallized fragments of glass slides were utilized as sensors (Durig et al., 1986; Pohl et al., 1988a). A miniaturized aperture was generated, locally imposing mechanical pressure. At almost the same time a cover glass provided with small apertures in its metallization layer was proposed as a near-Þeld probe (Fischer et al., 1988; -11

10

-12

10

Energy/m [J/m]

-13

10

ε 0 |E | 2

2

μ 0|H| 2

2

-14

10

-15

10

-16

10

-17

10

-18

Cut-Off

10

-300

-200

Aperture -100

0

100

200

300

z [nm]

Figure 17. Energy density of the electric and magnetic Þeld energy density along the centerline of the aperture tip of Figure 15. For the sake of clarity two dash-dotted lines were added to identify the location of the cutoff (z = −150 nm) for the lowest mode and the aperture area (z = 0).

RECENT DEVELOPMENTS OF PROBES

161

Pohl et al., 1988b; Fischer 1989a). However, the invention of metallized tapered Þbers (Betzig and Trautman, 1992; Betzig et al., 1992; Isaacson, 1991a, 1991b) and pipettes (Harootunian et al., 1986; Betzig et al., 1986, 1988; Cline et al., 1991; Shalom et al., 1992) with a small aperture leveraged what is today known as near-Þeld optical microscopy. In the following an overview of most of the conventional passive, that is, waveguiding probes used for scanning near-Þeld optical microscopy are discussed. Fiber Probes. Various fabrication schemes were invented for tapering of Þbers. In the beginning thermal pulling proposed by Betzig and Trautman (1992) gathered much attention. In particular Valascovic et al. (1995) and Islam et al. (1997) demonstrated that a huge variety of taper geometries is feasible varying the pulling parameters. However, the opening angle of thermally pulled tips is typically less than 20Ð30◦ with the effect that the transmission losses owing to evanescent energy transport in the taper are exceedingly high (see Section III.C.1a for details). Meanwhile, chemical etching of Þbers is a serious alternative for tapering Þber probes. St¬ ockle et al. (1999) and Sayah et al. (1998) demonstrated that in particular the tube etching method offers superior results. The SEM image in Figure 18a shows a typical Þber probe made by the tube etching method. It features a rather large opening angle that guarantees an improved transmission efÞciency. Furthermore, the very ßat and smooth surface of the etched Þber ensures low scattering and a low density of imperfections in the metallization layer. Another successful and elaborated etching method was presented by Ohtsu et al. using Ge-doped Þbers (Yatsui et al., 1998; Saiki et al., 1996).

(a)

(b)

Figure 18. (a) SEM image of a tube etched Þber probe with large taper angle (by courtesy of Zenobi) St¬ ockle et al., 1999. (b) SEM image of a triple tapered Ge-doped Þber probe for high transmission efÞciency. (Reproduced with permission from Yatsui, T., Kourogi, M., Tsutsui, K., and Ohtsu, M. (1998). Enhancing throughput over 100 times by a triple-tapered structure for near-Þeld optical Þber probe. SPIE Proceedings 3467: 89Ð98).

162

EGBERT OESTERSCHULZE

Their technique bears the potential to generate two or even three tapers of varying opening angles and tip shape (Mononobe et al., 1997). The enlarged opening angle guarantees a high transmission efÞciency in accordance with the discussion in Section III.C.1a. An example of a triple-tapered Þber tip is shown in Figure 18b. A detailed description of both the etching process and its characterization is given in Ohtsu (1998). The deÞnition of the aperture in the metal coating of tapered Þber probes is a problem we have not addressed thus far. Deposition of the metal under an oblique angle with respect to the rotating Þber is the common method to obtain apertures. However, this shadow mask method gives no reproducible results and in particular a selective variation of the aperture geometry is almost impossible. Therefore, focused ion beam (FIB) milling, that is, physical etching with a focused gallium ion beam of 5Ð20 nm diameter, was suggested as a superior method for aperture formation (Baida et al., 1993). Meanwhile, Pilevar et al. (1998) presented apertures of ca. 100 nm diameter and Heinzelmann et al. (1999) of about 50 nm, and the most recent results of Veerman et al. (1998) indicate apertures below 30 nm diameter. The SEM image of a typical FIB fabricated aperture tip in Figure 19 underscores the suitability of the method. The geometrically well-deÞned aperture shown has a diameter of ca. 35 nm with a very ßat surface which is also important with respect to image formation in both the topography and optical image. The aperture was sliced from a completely metal-coated pulled Þber. However, it should not be concealed that FIB cutting is an intricate and in particular serial process.

Figure 19. SEM image of a metallized Þber probe. The tip was sliced by focused ion beam (FIB) milling to obtain an almost ßat aperture with 35-nm diameter. (Reproduced with permission form Veerman, J.-A., Otter, A. M., Kuipers, L., and van Hulst, N. F. (1998). High deÞnition aperture proves for near-Þeld optical microscopy fabricated by focused ion beam milling. Application in Physics: Letters 72(24):3115Ð3117).

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163

As was already pointed out in Section III.B, the separation distance between the near-Þeld tip and the sample has an important impact on image formation. Shear-force detection is currently the most common tool for this purpose (Prater et al., 1991; Yang et al., 1992; Betzig et al., 1992; Toledo-Crow et al., 1992a). In shear-force detection the Þber vibrates laterally with respect to the sample surface and undergoes a damping owing to the interaction with the sample. A feedback loop is used to keep the vibration amplitude constant while scanning. Various techniques such as interferometry (Toledo-Crow et al., 1992b; Bozhevolnyi et al., 1993), triangulation (Betzig et al., 1992; Froehlich and Milster, 1995, Wei and Fann, 1998), capacitive readout (Decca et al., 1997), or piezoelectric readout (Brunner et al., 1997, 1999; Barenz et al., 1996) were invented to detect the Þber vibration. However, the physics of the shear-force mechanism is poorly understood and in particular the high longitudinal stiffness of the Þber brings with it the risk of damage to both the sample and the probe during unintentional contact. These disadvantages were alleviated by bending the Þber or pipette to obtain the cantilever probe known from AFM (Lieberman and Lewis, 1992; Ataka et al., 1996; Talley et al., 1996; Muramutsu et al., 1995, 1997; Taylor et al., 1997). Before concluding this section a brief discourse on the thermal properties of Þber probes will be given. In a simpliÞed model Kurpas et al. (1995) calculated the temperature distribution in tapered cylindrical waveguides as a function of the opening angle. For rather small angles, that is, a long taper that is subject to the cutoff effect, the maximum temperature is obtained far away from the Þber apex. However, the area of maximum temperature shifts to the apex for an opening angle of 20Ð30◦ that is typical for conventional tapered Þbers. In agreement with the mentioned theoretical results, St¬ ahelin et al. (1996) measured temperatures of about 470◦ C at the apex of metallized tapered Þbers for an input power of 10 mW. Similar results obtained by photothermal experiments were reported by Kavaldjiev et al. (1995). Although this temperature is well below the 660◦ C melting point of the aluminum used to coat the Þber, the tips are nevertheless damaged owing to the large difference of their thermal expansion coefÞcients (2.4 × 10−5 1/K for aluminum and 4.5 × 10−7 1/K for quartz), that is, the aluminum Þlm peels off in the vicinity of the apex. The thermal expansion of the near-Þeld probe has also an impact on the measurement even in the case of lower temperature changes as was demonstrated by Lienau et al. (1996). From the preceding discussion some rules are deduced for an optimum design of aperture near-Þeld probes. First of all the aperture size is expected to be below 100 nm. The taper of the waveguide should exhibit a large opening angle to assist in high transmission efÞciency and simultaneously to reduce the inßuence of heating effects. A pointed tip is favorable to achieve topography and optical resolution. To avoid mechanical damage to both the tip and the sample, the mechanical stiffness of the probe should be substantially lower

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EGBERT OESTERSCHULZE

than that of Þbers but large enough to avoid topographical artifacts. Of course a reliable near-Þeld optical measurement demands probes with highly reproducible optical and mechanical properties. MEMS Probes. Most of the preceding drafted demands for proper probes in near-Þeld optics can be fulÞlled utilizing cantilever-based probes with an integrated tip. They are capable of imaging topography in contact but also in noncontact mode. The compliance of AFM cantilevers can be easily adjusted by varying the geometry of the cantilever in accordance with Eq. (5). The remaining problem to solve is the integration of an optical waveguide into the tip with a miniaturized aperture at its apex. However, this technological task proved to be rather intricate. In the beginning Radmacher et al. (1994) used a molded silicon nitride AFM probe with a pyramidal tip that was completely covered with a metal layer. A thin photoresist Þlm was deposited to mask the tip base and Þnally an aperture was opened at the apex by a selective etching process (Ruiter et al., 1996). This serial method of aperture deÞnition was Þrst adopted by Abraham et al. (1998) for the reproducible batch fabrication of metal-coated quartz tips. An SEM image of a typical probe is shown in Figure 20a. It reveals that the aperture tip is proposed to be powered by an integrated optical waveguide structure on the cantilever top (Stopka et al., 2000). However, the realization of such an integrated AFM/SNOM probe adds tremendous technological demands on the simultaneous fabrication of the waveguide, the aperture tip, and an effective coupling mechanism between both. A similar technique for batch fabrication

(a)

(b)

Figure 20. (a) SEM image of an AFM cantilever probe with an integrated optical waveguide on the cantilever and an integrated metal-coated quartz aperture tip (by courtesy of Institut f¬ ur Mikrotechnik Mainz, Germany) (Stopka et al., 2000). (b) Close-up of an AFM probe with a coneshaped solid quartz tip. The tip is completly coated by a 60-nm thin aluminum layer. (Reproduced with permission from Eckert, R., Freyland, M., Gersen, H., and Heinzelmann, H. (2000). NearÞeld ßuorescence imaging with 2 nm resolution based on microfabricated cantilevered probes. Applications in Physics: Letters 77(23):3695Ð3697).

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165

of aluminum-coated solid quartz tips provided with an aperture and integrated in an AFM cantilever probe has been presented by Sch¬ urmann et al. (2000). It is surprising enough to note that even in the case of completely coated quartz tips the same authors were capable of demonstrating a lateral resolution of about 32 nm (Eckert et al., 2000). However, an exhaustive explanation of the contrast mechanism cannot be given here. The Þrst technological approach to a batch-fabricated aperture tip was presented by Prater et al. (1991), although its application was dedicated to scanning ion conductance microscopy. Aperture tip formation was accomplished by exploiting highly boron-doped silicon as etch stop layer for the KOH etching process. However, because of technological restrictions the achievable aperture dimension was in the range of 250 nm, which is not adequte for near-Þeld optics. Batch fabrication of aperture tips in the optical range was reported for the Þrst time by Mihalcea et al. (1996, 2000) exploiting the rheological behavior of thermally grown silicon dioxide. In this process an anisotropic etch process with KOH is used to deÞne pyramidal etch pits deÞned by the four (111)-oriented silicon crystal surfaces on a (001)-oriented silicon wafer. The opening angle of 70.5◦ of this tip mold is convenient in the light of the waveguiding properties as discussed in Section III.C.1a. Etch pits are subject to thermal oxidation to obtain a ca. 150-nm thin silicon dioxide layer on the wafer surface. However, the silicon dioxide layer shows a substantial thickness retardation at concave (and also convex) surfaces; that is, in the vicinity of the apex of the mold. This can be explained in view of the inßuence of the intrinsic mechanical stress in the oxide layer on the oxide growth. An adequate description of the rheological behavior of silicon dioxide is given in terms of a viscoelastic material, that is, a Maxwell liquid, assuming a nonlinear dependence of the viscosity of silicon dioxide on the intrinsic stress following the Eyring model (Hu, 1988; Senez et al., 1994). Vollkopf et al. (2001a) pointed out that oxide growth retardation in pyramidal etch pits depends strongly on the oxidation temperature. Figure 21a shows numerical calculations of the intrinsic stress distribution in case of two-dimensional oxidized silicon trench structures (Vollkopf et al., 2001b). Obviously, the intrinsic stress in the silicon dioxide layer peaks in the vicinity of the tip. The latter gives rise to a welldeÞned oxide growth retardation at almost the same location as was conÞrmed by SEM imaging of cross sections of oxidized trenches (Figure 21b). This retardation effect serves for the reproducible aperture deÞnition irrespective of the total thickness variation of the silicon wafer of about 3Ð5μm. Oxide growth retardation is more pronounced in the case of three-dimensional pyramidal etch pits because of the additional geometrical restriction that induces additional stress. FIB milling of released silicon dioxide tips was applied to study the thickness distribution of the oxide layer at the tipÕs apex. A SEM image of the opened layer is given in Figure 22. It clearly resolves

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2.6

90 80 70 60 50 40 30 20 10 0

SiO2

y [μm]

2.8 3.0 Si

3.2 3.4

(a)

3.6

3.8

4.0 4.2 x [μm]

4.4

SiO2

170 nm Si (b)

Figure 21. a) Calculated stress distribution in a thermally grown silicon dioxide layer in a silicon trench with 70.5◦ opening angle. Oxidation was assumed to be conducted at 1000◦ C. It gives rise to maximum intrinsic stress of about 80Ð90 MPa in the vicinity of the concave apex. Calculations were performed with the code FLOOPS (kindly provided by M. Law and adapted by D. Clark). (b) SEM image of a cross section of an oxidized silicon trench (oxidation temperature: 1000◦ C) revealing the two thinnest parts in the apex vicinity separated by 170 nm. (Reproduced with permission from Volkopf, A., Rudow, O., M¬ uller-Weigand, M., Gerogiev, G., and Oesterschultze. E. (2001). Inßuence of the oxidation temperature on the fabrication process of silicon dioxide aperture tips. Submitted).

Al

SiO 2

500 nm

Figure 22. Oxide growth retardation in the vicinity of the apex of pyramidal hollow silicon dioxide tips was proven by cross sectioning tips via FIB milling. Prior to cross sectioning, the tip was metalized from both sides with aluminum (Reproduced with permission from Volkopf, A., Rudow, O., Gerogiev, G., and Oesterschultze, E. Technology to reduce the aperture size of microfabricated aperture SNOM tips. (2001), accepted for publication in the Journal of the Electrochemical Society). (FIB milling was performed by WITec GmbH, Ulm, Germany.)

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167

φ

h

d

(a)

(b)

(c)

(d)

Figure 23. (a) Schematic setup of combined AFM/SNOM cantilever probes with an integrated silicon dioxide aperture tip. The opening angle φ is given as the angle between two (111) silicon crystal planes, that is 70.5◦ . (b) SEM image of the pyramidal hollow silicon dioxide tip with integrated aperture. (c) Close-up of a typical aperture in the silicon dioxide tip revealing dimensions of ca. 170 nm. (d) Aperture of ca. 50 nm size after metallization with an aluminum layer (Vollkopf et al., 2001b). (Aperture tips are commercially available at WITec GmbH, Ulm, Germany.)

the inhomogeneous thickness in the apex area. Based on this tip structure, aperture formation in the hollow silicon dioxide tip is Þnally accomplished by piercing its thinner parts by an additional KOH etching process that is used to remove the substrate, and also to free the silicon dioxide tips, and Þnally to open reproducibly apertures with dimensions of 170 nm. SEM images of typical silicon dioxide aperture tips are shown in Figure 23. In (a) the schematic of the aperture probe is illustrated as a cross section through the symmetry plane of the pyramidal tip. A complete aperture tip is given in (b), whereas (c) is a close-up of a typical aperture of about 170 nm in the oxide layer (see also Fig. 21b). The aperture dimension is further reduced by the deposition of a metal layer Þnally to receive apertures of about 50 nm size. The discussed aperture formation process by thermal oxidation of silicon was experimentally endorsed by the results of Minh and Ono (1999) and Minh et al. (2000a).

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Theoretical as well as experimental results conÞrmed that a further reduction of the aperture dimension is feasible in two different ways. In the Þrst process, already fabricated silicon dioxide tips are subject to thermal annealing to release the intrinsic stress in the tip material. It was observed that this process entails a noticable reduction of the aperture dimensions owing to stress release (Vollkopf et al., 2001b). The second process exploits the particular rheological behavior of silicon dioxide at elevated oxidation temperatures (Vollkopf et al., 2001a). Although the stress relaxation time in the oxide is reduced, that is, the mechanical stress is relaxed, aperture formation is even more effective. Aperture dimensions of only 60 nm in the silicon dioxide layer were successfully obtained at oxidation temperatures of 1100◦ C followed by KOH etching (Vollkopf et al., 2001a). This contradiction to the results discussed earlier is eliminated by taking into account that the strong reduction of the oxidation time at elevated temperatures substantially overcompensates the decrease of the stress relaxtion time owing to a reduced viscosity. In comparison to the solid aperture probes just mentioned silicon dioxide aperturre tips offer some noteworthy advantages. Owing to their hollow geometry, Þlling with materials, such as dielectric, semiconducting, or polymer materials adapted for certain applications is an important way to realize a huge variety of other tips. With respect to near-Þeld optics the aperture tip setup might be improved by Þlling it with a high-index material adapted to the desired wavelength range. In correspondence to the solid aperture tips presented above, this further increases the already remarkably high transmission efÞciency. Figure 24 gives an example of a near-Þeld device of heterogenous structure, consisting of a cleaved Þber with an already-mounted microfabricated silicon dioxide aperture tip. This probe proÞts from the ease of guiding light by means of the optical Þber. A similar approach was discussed by Genolet et al.

Figure 24. SEM images of (a) silicon pads with an integrated aperture tip and (b) a cleaved Þber provided with the aperture tip. (By courtesy of Fumagalli et al., FU Berlin, Institut f¬ ur Experimentalphysik, Berlin, Germany.)

169

RECENT DEVELOPMENTS OF PROBES

(a)

(b)

Figure 25. SEM images of cleaved Þber probes provided with batch-fabricated aperture tips made of SU-8 photoresist. (Reproduced with permission from Genolet, G., Cueni, T., Bernal, M. P., Despont, M., Staufer, U., Noell, W., Vettiger, P., Marquis-Weible, F., and de Rooij, N. F. (2000). In Proceedings of the 14th European Conference on Solid State Transducers. Eurosensors XIV: 641Ð644).

(2000) mounting microfabricated aperture tips on cleaved Þbers as presented in Figure 25. Their aperture tips were molded from SU-8 photoresist material. The silicon dioxide tip conÞguration (Figure 26a) enables coaxical probes (Figure 26b) or electroßuorescent, that is, active light-emitting probes based on polymers or macromolecules (Figure 26c). The application range is widely extended by taking the electronically passivating properties of silicon dioxide

Metal Si

70.5

SiO2

o

Metal 1 Si

SiO2

(a)

(b)

Aperture

Metal 2

Metal1 Polymer Metal 2

Passivation layer Metal Si

Si

SiO2

SiO2

(c)

Aperture

Aperture

(d)

Aperture

Figure 26. The hollow silicon dioxide aperture tip conÞguration is proposed to be proper for the realization of different sensors. (a)Ð(c)show several conÞgurations as near-Þeld optical probes: an aperture tip, a coaxial tip, and an active tip with an electroßuorescent polymer working as an active optical near-Þeld emitter, respectively. (d) shows the conÞguration of an isolated miniaturized electrode used, for examples, in scanning ion conductance microscopy. However, the same setup is also proposed for the batch fabrication of nanotube probes.

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EGBERT OESTERSCHULZE

Figure 27. SEM image of a near-Þeld cantilever probe for combined SNOM/AFM. The aperture in the metallized silicon tip was obtained by head-on FIB milling. (Reproduced with permission from Dziomba, Th., Danzebrink, H. U., Lehrer, Ch., Frey, L., Sulzback, Th., and Ohlsson, O. (2001). High resolution constant-height imaging with apertured silicon cantilever probes. Journal of Microscopy, 202, in press).

into consideration. This beneÞts the realization of any kind of ultrasmall electrodes or electronic devices in the tip (Figure 26d), such as for ion conductance microscopy or the batch fabrication of nanotube probes. Finally, another serial fabrication scheme has gathered attention. It utilizes the serial FIB method to open an aperture in the metallization of completely coated silicon tips of conventional AFM cantilever probes. Dziomba et al. proposed head-on ion beam etching to accomplish apertures of only 50Ð100nm dimensions (Dziomba et al., 1999). A typical example of this kind of tip is shown in Fig. 27. The SEM image of the aperture region reveals a clearly deÞned aperture of ca. 50 nm size. FIB milling is a dedicated process to control the aperture geometry. Its importance inßuence on polarization properties was already studied in the case of silicon dioxide tips (Werner et al., 1998). The high refractive index of silicon inside the tip substantially improves the transmission efÞciency. But of course the tip has to be operated in the infrared, that is, below the gap energy of silicon of ca. 1.1 eV. This gain in transmission is advantageous and necessary, in particular for applications such as near-Þled optical data storage. However, the bottleneck with respect to temporal bandwidth is constituted by serial processing in scanning near-Þeld optics. Yatsui et al. introduced an array of silicon (and also silicon dioxide) aperture tips for parallel data storage (Ohtsu, 1998; Lee et al., 1999b, 1999c). A close-up of a silicon tip array carried on a sliding head is shown in Figure 28. Meanwhile, phase change recording with a bit length of 110 nm at 2.0 MHz transmission rate with part of the tip was performed at a wavelength of 830 nm (Yatsui et al., 2000).

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Figure 28. SEM images of an array of silicon tips used as near-Þeld tips in the infrared for high-density optical data storage. (Reproduced with permission from Lee, M. B., Kourogi, M., Yatsui, T., Tsutsui, K., Atoda, N., and Ohtsu, M. (1999). Silicon planar-apertured probe array for high-density near-Þeld optical data storage. Applied Optics 38(16):3566Ð3571).

In conclusion the aperture tip approach has been proven very successful in near-Þeld optics. Neverthless, there are some important drawbacks that have to be overcome in future. The rather poor transmission efÞciency as well as the complicated saddle-shaped electric Þeld distribution in the aperture plane are the important ones to emphasize. In the two following sections we introduce two concepts that have the potential to solve these problems: the coaxial probe and the how-tie antenna probe. 2. Coaxial Probes In transmission waveguide theory it is derived that propagation of electromagnetic Þeld energy between two separated metal structures is not affected by the cutoff effect (Collin, 1991). The most prominent candidate is the coaxial line consisting of a hollow cylindrical ground electrode that encapsulates a second cylindrical conductor. Coaxial lines are widespread, for example, in communication electronics (Collin, 1992). a. Energy Transport in Coaxial Waveguides. The electric Þeld energy distribution of coaxial probes has been investigated on base of the silicon dioxide aperture tip introduced earlier (see Figure 23). As can be seen in Figure 29a, the setup was modiÞed by metallizing the silicon dioxide aperture tip of Figure 23 from both sides with an aluminum layer to establish a constant gap between the two electrodes that constitute the coaxial line, as already depicted in Figure 26b. Figure 29b shows the respective energy density distribution of the coaxial tip on a logarithmic gray scale with a factor of 10 between successive lines. The tip was powered from the left via an adapted coaxial waveguide not shown here assuming that transport is supported by the lowest TEM mode. Because

172

EGBERT OESTERSCHULZE

E

Al SiO2

x,y

0d

z 100 nm

(a)

(b)

Figure 29. (a) Cross-sectional model of a coaxial tip based on the aperture tip in Figure 23. All geometrical parameters are identical to these mentioned in Figure 23. For the optimization of the geometry the position d of the aperture plane was varied with respect to the terminal of the center electrode. (b) Energy density distribution of the electric Þeld in the symmetry plane of the coaxial aperture tip on a logarithmic gray scale with a factor of 10 between successive lines. The aperture plane coincides with the terminal plane of the inner conductors, that is, d = 0. All material and geometrical parameters used are deÞned in Figures 15 and 23. (Reproduced with permission from Rudow, O., Vollkopf, A., Muller-Weigand, M., Gerogiev, G., and Oesterschultze, E. (2001). Theoretical investigations of a coazial probe concept for scanning near-Þeld optical microscopy. Accepted for publication in Optics Communication).

of the symmetry of the TEM mode the distribution shown is identical to the one in the perpendicular direction. The local electric Þeld density perpendicular to the electrodes is almost constant and resembles that of a plane-parallel capacitor as expected from waveguide theory. Furthermore, the electric Þeld is exponentially damped inside the metal layers owing to the skin effect. The Þeld density inside the dielectric layer establishes a standing wave pattern along the taper with a maximum at the tip apex in the aperture plane (Ward and Pendry et al., 1997). The standing wave is evoked by the impedance mismatch that occurs crossing the aperture plane going from the coaxial tip to free space. The electric Þeld vector originates at the pointed terminal of the inner conductor and ends on the inner rim of the outer conductor and vice versa. It thus shows both a longitudinal and a transverse component. The focusing of the coaxial geometry gives rise to a strong maximum at the center accompanied by low side lobes at the rim of the outer conductor, as is obvious from the proÞle in Figure 30a. In contrast to the aperture tip, it offers a symmetric distribution and furthermore, the electric Þeld energy prevails over the magnetic Þeld energy by one to two orders of magnitude (Rudow et al., 2001). The full width

173

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-10

0,8 0,6

Energy/m [J/m]

Energy density [a.U.]

10

19,6nm

0,4

-11

10

-12

10 10

0,2 10

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0 -200

(a)

-150

0

100

x [nm]

-100

200

(b)

-50

0

50

100

150

200

z [nm]

Figure 30. Electric Þeld energy density (a) in the aperture plane and (b) along the propagation direction of the coaxial tip of Figure 29b. (Reproduced with permission from Rudow, O., Vollkopf, A., Muller-Weigand, M., Gerogiev, G., and Oesterschultze, E. (2001). Theoretical investigations of a coazial probe concept for scanning near-Þeld optical microscopy. Accepted for publication in Optics Communication).

log. Energy/m [J/m]

at half maximum (FWHM) of the energy density in the center is on the order of 2Ð3times the skin depth in the metal of the pointed inner conductor. This roughly determines the lowest attainable focus width of the coaxial line probe. The strong decay of the Þeld energy density along the propagation direction is obvious from Figure 30b. In contrast to the aperture tip the transmission efÞciency is near unity, that is, it is larger by four orders of magnitude than the aperture tip of the same aperture size (Figure 31b) (McCutchen et al., 1995).

FWHM [nm]

120 100 80 60 40 20

-8 -9 -10 -11 -12

-10 0 10 20 30 40 50 60

(a)

d [nm]

-10 0 10 20 30 40 50 60

(b)

d [nm]

Figure 31. (a) Full width at half maximum (FWHM) of the density distribution of the electric Þeld and (b) energy density both evaluated in the aperture plane of the coaxial tip of Figure 29b. (Reproduced with permission from Rudow, O., Vollkopf, A., M¬ uller-Weigand, M., Gerogiev, G., and Oesterschultze, E. (2001). Theoretical investigations of a coazial probe concept for scanning near-Þeld optical microscopy. Accepted for publication in Optics Communication).

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Varying the geometrical parameter d (Figure 29a) resulted in a slightly lower FWHM of the Þeld energy density distribution in the aperture plane of about two times the skin depth with a slight decay of the transmission efÞciency compared to the case d = 0 as indicated in Figure 31b. b. Probes. The Þrst near-Þeld coaxial probe realized was introduced by Fee et al. in 1988 for microwave applications (Fee et al., 1989). A conventional coaxial cable was cut to obtain a very blunt tip. The latter was fed with a sinusoidal microwave of 2.5 GHz frequency. Scanning with the open end across a sample surface allowed a resolution of about λ/4000 utilizing an interferometric detection scheme. A similar microwave arrangement was also discussed by Keilmann et al. (1996). They extended the spectral range into the infrared (Keilmann, 1991, 1988). An analogous scheme for the optical range was presented by Fischer (1989b) and Fischer and Zapletal (1992). Fischer used silver-Þlled pipettes, so-called Taylor wires, that were thermally tapered to improve the attainable resolution. However, tapering led to a noncontinuous silver wire inside the pipette owing to the strong difference in their thermal expansion coefÞcients. The imperfections of the silver Þlling allowed local launch of surface plasmons. However, at the same time they were responsible for their radiative decay, and thus focusing of light to the apex of the tapered tip give only minor success. A Þrst approach to a coaxial probe relying on an AFM cantilever was presented by Leinhos et al. (1999). As indicated in Figure 32a the setup consists of

W

Ti

Ti Si (a)

Ti

W (b)

Figure 32. a) Schematic setup of a coaxial probe consisting of a partly hollow Ti tip residing on a silicon cantilever. A tungsten wire was CVD deposited on the silicon basis by gas-phase deposition with an FIB. (b) SEM image of the Þrst coaxial probe accomplished (fabricated in collaboration with Micrion Munich and M. Stuke). (Reproduced with permission from Leinhos, T., Stopka, M., and Oesterschultze, E. (1998). Micromachined fabrication of Si cantilevers with Schottky diodes integrated in the tip. Applied Physics A 66, 65Ð69).

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Figure 33. SEM image of a batch fabricated coaxial near-Þeld cantilever probe for combined SNOM/AFM based on the silicon dioxide aperture tip. (Reproduced with permission from Minh, P. N., Ono, T., and Esashi, M. (2000). High throughput aperture near-Þeld scanning optical microscopy. Reviews of Science Instruments).

a hollow Ti tip that resides on a stub made of silicon. A Ti pillar only 200 nm in diameter is grown inside the hollow tip to realize the center conductor. Deposition of Ti was performed by ion-assisted CVD deposition from a Ti containing organic precursor utilizing a focused ion beam. An SEM image of the complete coaxial cantilever probe is given in Figure 32b. The low mechanical stability of the thin pillar is the major factor that inhibits the application of such tips. Minh et al. (2000b) presented the Þrst batch-fabricated coaxial near-Þeld probe for the optical range. It is based on the silicon dioxide aperture probe presented in Figure 23 and is identical to the scheme of Figure 29a. The corresponding SEM image in Figure 33 clearly reveals the inner pointed conductor made of Cr whereas the outer electrode is deposited as a thin metal Þlm on the silicon dioxide layer. The complete fabrication scheme with Þrst results obtained with such probes is discussed in detail in. 3. Bow-Tie Antenna Probes Another approach to realize waveguides unaffected by the cutoff effect was introduced by Grober et al. (1997a, 1997b). It relies on a planar bow-tie antenna setup that is shown in Figure 34. The bow-tie antenna proposed features two triangular metal electrodes that face each other separated by a small gap. If light is directed perpendicular to this antenna arrangement with the polarization spanning the gap between the electrodes, then the electromagnetic wave energy is focused into the gap even in the case of gap geometries that are small in comparison to the applied wavelength. Impedance matching with respect to the external power source can be achieved varying the opening angle ψ to

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y Al

Al

Al SiO2

SiO2 Al

Al (a)

z

( b)

(c)

Figure 34. Schematic setup of (a) a planar bow-tie antenna proposed by Grober et al. (Reproduced with permission from Grober, R. D., Rugtherford, T., and Harris. T. D. (1996). Model approximation for the electromagnetic Þeld of a near-Þeld optical probe. Applied Optics 35(19), 3488Ð3495). (b) Three-dimensional view and (c) cross section of an aperture tip (see Fig. 23) with only two side walls coated with aluminum, forming a narrow gap at the apex. The structure constitutes a tapered electric dipole bow-tie antenna. (Reproduced with permission from Oesterschultze, E., Georgiev, G., Vollkopf, A., and Rudow, O. (2001). Transmission line probe on base of a bow-tie antenna. Journal of Microscopy. 202(1): 39Ð44).

obtain a wavelength-independent impedance ZD:   π/2 dϕ 1 μ0 K (cos(ψ/2))  ZD = (28) with K (x) = 2 ǫ0 K (sin(ψ/2)) 0 1 − x 2 sin2 ϕ

◦ K (x) is the elliptic √ integral of the Þrst kind. For ψ = 90 a constant impedance of Z D = 1/2 μ0 /ǫ0 = Z 0 /2 = 189 is obtained where Z 0 is the vacuum impedance. Bow-tie antennas are usually applied as broadband radiofrequency antennas (Collin, 1992).

a. Energy Transport in Bow-Tie Antennas. The application of a bow-tie antenna as near-Þeld probe demands a pointed tip rather than the planar arrangement proposed by Grober et al. Thus we suggest a bow-tie antenna on base of the silicon dioxide tip discussed earlier (see Fig. 23). The schematic setup is depicted in Figures 34b and 34c. The two arms of the electric dipole antenna are deposited on opposing sides of the hollow silicon dioxide pyramid and are separated by a small gap. The calculated electric Þeld energy density distributions in the two planes of symmetry are shown in Figure 35 on a logarithmic gray scale. Obviously the electric Þeld is concentrated in the gap between the antenna arms, whereas in the perpendicular direction the Þeld is less conÞned. Part of the incoming wave is reßected from the antenna structure owing to the mismatch between the wave impedance of the antenna and free space. This in turn gives rise to a standing wave pattern in the negative z-direction. This

RECENT DEVELOPMENTS OF PROBES

E

E

100 nm

(a)

177

100 nm

(b)

Figure 35. Energy density distribution in the symmetry planes of the bow-tie antenna tip of Figure 34 with the polarization orthogonal to the gap between the two electrodes. (Reproduced with permission from Oesterschultze, E., Georgiev, G., Vollkopf, A., and Rudow, O. (2001). Transmission line probe on base of a bow-tie antenna. Journal of Microscopy. 202(1): 39Ð44).

effect is much easier to identify in the proÞle shown in Figure 36b. The electric Þeld peaks out in the plane of the gap and constitutes a strong electrical dipole, which is advantageous for near-Þeld optical applications. Simultaneously, the magnetic Þeld curls arround this electric dipole. ProÞles of the electric Þeld distribution in the plane of the gap are shown in Figure 36a revealing that the Þeld conÞnement in the polarization direction is almost identical with the sum of the skin depth and the gap size. The latter was assumed to be 100 nm. An additional minor static Þeld enhancement at the rim of the gap electrodes occurs. In the perpendicular direction the Þeld conÞnement is less pronounced and exceeds the geometrical width of the tapered antenna arm by ca. 16%. Although the Þeld distribution is more complex in comparison to the coaxial tip, the bow-tie antenna probe offers two important advantages. It reveals an almost unity transmission efÞciency and simultaneously it seems to be much easier to realize. b. Probes. Thus far no approach has been discussed to a bow-tie antenna in the optical range. A very early attempt to check the feasibility of the approach discussed in Figure 34b is shown in Figure 37. A silicon dioxide tip (without aperture) was Þrst completely metallized with an aluminum layer. Subsequently, the aluminum Þlm was sputtered by a focused ion beam to slice

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ε0 |E|

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4 3 2

μ0|H| 2

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x

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Gap plane

-15

10

0 -200

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106nm

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(a)

2

10

5

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Energy density [a.U.]

6

100

x,y [nm]

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(b)

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100

200

z [nm]

Figure 36. ProÞles of the energy density distribution (a) in the plane of the gap and (b) along the axis of symmetry of the tip discussed in Figure 35. (Reproduced with permission from Oesterschultze, E., Georgiev, G., Vollkopf, A., and Rudow, O. (2001). Transmission line probe on base of a bow-tie antenna. Journal of Microscopy. 202(1): 39Ð44).

the metal electrode on two opposite walls of the pyramid. A bow-tie antenna with a 50 nm gap between the antenna arms could be achieved in a Þrst attempt (Oesterschulze et al., 2001). 4. Solid Immersion Lens Probes Kino et al. MansÞeld and Kino (1990) and Kino and MansÞeld (1991) transferred the principal idea of the well-established immersion microscopy

(a)

(b)

Figure 37. The Þrst miniaturized bow-tie antenna tip was accomplished by slicing the metal layer on two adjacent side walls of the completely coated aperture tip shown in Figure 23 by focused ion beam milling (in collaboration with H. U. Danzebrink, Th. Dziomba, and Ch. Lehrer). (a) shows the opened slit on the side wall of the tip and (b) gives a top view of the tip apex revealing a slit size of approximately 50 nm. (Reproduced with permission from Oesterschultze, E., Georgiev, G., Vollkopf, A., and Rudow, O. (2001). Transmission line probe on base of a bow-tie antenna. Journal of Microscopy. 202(1): 39Ð44).

RECENT DEVELOPMENTS OF PROBES

179

Lens R/nSIL SIL (a)

(b)

Figure 38. Schematic view of (a) the conventional SIL setup consisting of a hemisphere proposed by MansÞeld and Kino (1990) and (b) the supersphere setup with a sphere of radius R cut at a distance of R/n S I L from the center to realize stigmatic imaging. (Reproduced with permission from Terris, B. D., Mamin, H. J., and Rugar, D. (1994). Near-Þeld optical data storage using a solid immersion lens. Applied Physics: Letters. 65(4): 388Ð390).

method to near-Þeld optics. In conventional immersion microscopy the gap between the imaging lens and the sample is Þlled with a ßuid of high refractive index n to reduce the wavelength by a factor 1/n. This is accompanied by an improvement of the lateral resolution by a factor 1/n in correspondence with Eq. (21). Instead of ßuids, they suggested utilizing solids of high refractive index which gave the method its name: solid immersion lens (SIL) (Ichimura et al., 1997). Figure 38a shows the SIL setup proposed by MansÞeld and Kino (1990). A hemisphere is illuminated with a converging light beam that is not refracted at the curved surface of the hemisphere. All rays are focused on the plane glass/air interface. Owing to FresnelÕs formulas, some of the rays are totally reßected at this interface and give rise to evanescent surface waves that couple with a sample surface for distances of some nanometers between them. The refracted part of the beam is superimposed as a far-Þeld contribution (Milster et al., 1999). An improved SIL arrangement proposed by Terris et al. (1994) is based on stigmatic imaging with a supersphere (Kino, 1998). For this purpose a sphere must be cut at a distance of R/n SIL from its center as depicted in Figure 38b (Born and Wolf, 1959). The advantage of this arrangement is the reduction of the wavelength by the focusing of the

TABLE IV Refractive Indices of Some Solids That Are Suitable for the Fabrication of SIL Probes Material

Refractive index n

Wavelength (nm)

PBS CdS ZnS BK7 glass Diamond

3.912 2.529 2.340 1.517 2.417

589.3 589.3 650.0 589.3 589.3

Citation (Naumann and Schr¬ oder, 1992) (Naumann and Schr¬ oder, 1992) (Milster, 1999) (Naumann and Schr¬ oder, 1992) (Naumann and Schr¬ oder, 1992)

180

(a)

EGBERT OESTERSCHULZE

(b)

Figure 39. (a) Close-up of a silicon nitride SIL of a single AFM/SNOM cantilever and (b) SEM image revealing four silicon nitride cantilevers with integrated SIL. The bottom part of the SIL is simultaneously used as a AFM tip. Cantilevers are 92 μm long, 10 μm wide, and 1 μm thick. (Reproduced with permission from Crozier, K. B., Fletcher, D. A., Kino, G. S. and Quate, C. F. (2001). Micromatched silicon nitride solid immersion lenses. Submitted).

stigmatic lens. In Table IV some interesting materials with exceedingly high refractive indices are compiled that might be interesting for future development of SILs. The approach of the rather large SIL to the sample surface in a near-Þeld microscope is substantially facilitated by grinding the SIL at the bottom to form a curved rather than a ßat surface (Crozier et al., 2001; Ghislain and Elings, 1998). Figures 39a and 39b show SEM images of a single silicon nitride SIL integrated into a cantilever beam and an array of SIL cantilever probes, respectively (Crozier et al., 2001). They clearly resolve the SIL on top of the cantilever and its curved surface below the cantilever that establishes the contact with the sample surface during scanning. With similar probes an optical resolution of 65 nm was demonstrated (Kino and MansÞeld, 1991). Novel developments were conjectured applying aspheric lenses (Minyu et al., 2000) and mirrors (Ueyanagi and Tomono, 2000) to optimize the focusing. Although one possibly might not expect that future developments of SILs offer a further dramatic improvement of the lateral resolution in near-Þeld optics, their robust and rigid design is quite advantageous for several applications. A very popular one is near-Þeld optical data storage. SIL arrays residing on a sliding head are proposed for parallel magnetooptical data storage with an anticipated data storage density in the range of 40Ð100Gbit/in.2 (Imanishi et al., 2000; Liu et al., 2000; Otaki et al., 2000).

RECENT DEVELOPMENTS OF PROBES

181

5. Scattering Tip In far-Þeld optics the well-known theorem of Babinet states that complementary apertures show the same far-Þeld diffraction pattern (Born and Wolf, 1959). This effect suggests that an aperture in an opaque screen shows optical behavior similar to that of a circular slab of the same radius. Based on this idea, Bachelot et al. (1995), Kawata and Inouye (1995), Furukawa and Kawata (1998), Zenhausern et al. (1994, 1995), and Martin et al. (1996) proposed use of a pointed tip as-near-Þeld scatterer to transform high-frequency spatial Þeld components in the vicinity of the sample into propagating ones. In view of the availability of tips with a radius of curvature of less than 5 nm (see Section II.C.1). a scattering tip approach is a fascinating alternative to aperture probes, although also in the case of scattering tips, image interpretation is strongly affected by, for example, topographical artifacts. a. Theory of Scattering Probes. In a simplifying theoretical description, a scattering tip is described in terms of a polarizable sphere of radius R made of a material with dielectric function ǫ Sc and immersed in a medium of dielectric function ǫ Med . Its scattering σ Sc and absorption σ Abs cross sections are given as (Zenhausern et al., 1995) σ Sc =

k4 |α|2 6π

and σ Abs = k Im (α)

(29)

with α = 4π R 3

ǫ Sc − ǫ Med ǫ Sc + 2ǫ Med

(30)

Owing to the k4-dependence scatterers experience a much higher efÞciency in the optical in comparison to the infrared spectral range (Knoll, 1999). Metal and semiconductor tips are preferred materials because their polarizability exceeds that of dielectrics (Kawata and Inouye, 1995). Furthermore, it is suggested by virtue of Eqs. (29) and (30) that the absorption cross section of a tip exceeds the scattering one noticeably for radii smaller than the wavelength because of the square dependence on α instead of a linear one. However, Wang et al. showed that this model is not adequate to describe the near-Þeld optical experiment. Taking the interaction with the sample and also the antenna effect with the pronounced Þeld enhancement of the elongated tip into account, they showed that the scattering cross section increases by 5Ð6orders of magnitude in comparison to what is predicted by Eq. (29). In consequence Hagman (1997) and Martin and Girand (1997) conÞrmed that the lateral resolution attainable with scattering tips is governed by the size and shape of the scattering tip apex, whereas the volume of the tip body determines the signal yield.

182

EGBERT OESTERSCHULZE

b. Experimental Results. Wickramasinghe et al. Zenhausern et al. (1994, 1995) and Martin et al. (1996) presented results with a lateral resolution of ca. 1Ð3nm exploiting the tip of a AFM cantilever as near-Þeld scatterer. Similar results were reported by Inouye and Kawata (1997) in the case of a metal tip used in an STM conÞguration with the restraint that measurements are restricted to electrically conducting samples. Knoll and Keilmann (1998, 1999) extended the spectral range into the infrared with the objective of performing chemical analysis of samples. However, the scattering tip approach also has some inherent problems that exacerbate image formation and interpretation. Some of them are brießy addressed in the following. In the scattering tip approach the tip and therefore the sample is continuously illuminated, which might cause damage in case of light-sensitive samples. A complicated interference pattern emerges in the vicinity of the tip generated by the superposition of the incoming light on that reßected and/or scattered from both the sample and the tip. This makes a reproducible illumination of the tip apex in case of nonplanar samples of varying composition rather complicated. The introduction of a reÞned homodyne technique solved the problem of the exceedingly low signal-to-noise ratio (Zenhausern et al., 1994; Kawata and Inouye, 1995). For this purpose, the tip is vibrated perpendicular to the sample surface and the scattered light is detected by means of an optical interferometer. However, the nonlinear dependence of the optical nearÞeld on the distance between tip and sample adds signiÞcant problems to interpretation of images, as was pointed out by Hamann et al. (1998). Thus topography-induced artifacts become an important issue in image interpretation (Hecht et al., 1997, 1998; Carminati et al., 1997; Valle et al., 1999). Furthermore, the homodyne detection scheme prohibits the measurement of incoherent signals in for example, Raman or ßuorescence microscopy. Xiao (1997) emphasized that in the case of a reßection conÞguration the illumination angle of the incoming light has a distinct inßuence on measured images.

D. Active Probes 1. Light-Emitting Active Probes The terminus technicus light-emitting active probe is thought of as a near-Þeld device with an active light emitter in the probe tip powered by an external energy source, that is, an optical light source or an electrical power supply. a. Plasmon Probes. The metal-coated Þber probes mentioned earlier were discussed in view of guiding light from the external light source via the link, the near-Þeld aperture and the sample to the external detector. However, Wessel

RECENT DEVELOPMENTS OF PROBES

183

(1985) anticipated taking advantage of the collective excitation of the free electron gas of metals igniting surface plasmons on the metal surface with the aim of a signiÞcantly increased Þeld enhancement. In a very simple model of a thin metal (ǫ 1) on a dielectric substrate (ǫ 0) immersed in air (ǫ 2) the maximum Þeld enhancement for a proper Þlm thickness is expressed as 

2

r e

ǫ1 (ǫ0 − 1) − ǫ0 1 2 ǫ1r e



(31) Tmax = im ǫ2 ǫ 1 1 + ǫ r e

1

following RaetherÕs (1988) discussion. For Ag and Au the expected Þeld enhancement is quite large owing to their low surface plasmon frequencies of ω S P = 3.5 and 2.5 eV, respectively (Bohren and Huffman, 1983). In the case of Ag the very small imaginary part of the dielectric function gives rise to rather formidable values of the Þeld enhancement of 50Ð250.In contrast, Al with its rather large surface plasmon frequency of 11 eV is rather inapt. Next to the dielectric function, the geometry of the metallic particle also plays a signiÞcant role, as was proven in the case of spheres (Barber et al., 1983; Ruppin, 1983), ellipsoids (Bohren and Huffman, 1983), and tips (Denk and Pohl, 1991). The Þeld distribution of propagating surface plasmons is exponentially damped perpendicular to the metal surface. The respective penetration depth lz is written as  1 ǫ1r e + ǫ2 lz = (32) k0 |ǫ1r e | in terms of the respective dielectric functions and the vacuum wavevector k0 of the surface plasmon. The typical penetration depth of Au and Ag of ca. 25Ð35nm provokes the reduction of the cutoff effect in case of coated waveguides, such as tapered Þbers, and thus an improvement of the transmission efÞciency (Wolff, 1998). However, the price paid for this is the increase of the effective aperture radius, as was experimentally proven (Ruppin, 1983; Ebbesen et al., 1999). Matching of the propagation vector for the resonant excitation of surface plasmons is conventionally achieved by a prism or a grating arrangement. However, the roughness of a thin metal Þlm with its extended spectrum of spatial frequencies is also sufÞcient for this purpose. In near-Þeld optics this was Þrst demonstrated in the optical range by Fischer (1986) and Fischer and Pohl (1989) in case of small silver particles. Similar approaches were discussed later by Silva and Schultz (1992, 1993), Silva et al. (1994), and Ohtsu (1998). For a theoretical description see, for example, Barber et al. (1983) or Ruppin (1983). A more practical approach was introduced by Fischer and Zapletal (1992) utilizing silver-Þlled tapered pipettes. However, their strongly deviating thermal expansion coefÞcients added difÞculties during the thermal

184

(a)

EGBERT OESTERSCHULZE

(b)

Figure 40. (a) Schematic of the tetrahedral tip consisting of a glass fragment that is coated in a two-step deposition process with a 50-nm thin gold layer leaving the edge K1 free of metal. (b) SEM image of a complete tetrahedral tip. (Reproduced with permission from Fischer, U. Ch. (1993). The tetrahedral tip as a probe for scanning near-Þeld optical microscopy. In Near Field Optics. Pohl, D. W., and Conrjon, D. Eds. (pp. 255Ð262)Kluwer Academic).

tapering process in reproducibly fabricating a continuous Ag wire inside the pipette. These irregularities of the silver wire provide the opportunity to fulÞll k-matching and thus to excite surface plasmons that are presumably attributted to TM0 modes (Gurevich and Libenson, 1995). But the same irregularities are also responsible for the irradiation of light, and thus focusing of light to the Þber is almost impossible. Fischer (1993, 1998) developed an improved version of a plasmon-powered near-Þeld probe: the tetrahedral tip. It consists of a tetrahedral glass prism that is coated with a thin gold layer with the exception of one edge (denoted K1 in Figure 40a). This setup resembles the widespread Kretschmer conÞguration (Lipson et al., 1997). Surface plasmons are excited at the uncoated edge, propagate toward the tip, and are presumably scattered from a small gold cluster residing on the apex (Koglin and Fischer, 1995). An SEM image of the gold coated tip is depicted in Figure 40b. An excellent lateral resolution of only a few nanometers was demonstrated (Koglin and Fischer, 1997). In future plasmon probes are expected to be quite important for near-Þeld optical applications. The perspective to concentrate the power of the plasmons on single particles of only nanometer size with the enormous Þeld enhancement might be the key to routinely achieving nanometer lateral resolution in nearÞeld optics with an acceptable signal-to-noise ratio. However, it should be noted that the effective spectral bandwidth of plasmon probes is limited, as is obvious from Figure 41. b. Fluorescent Near-Field Probes. In general any kind of microscopic excitation might be exploited for near-Þeld probes. Liebermann et al. suggested Þlling tapered pipettes with ßuorescent material that is externally driven to

185

RECENT DEVELOPMENTS OF PROBES 40

Penetration depth [nm]

250

200

Ag

Tmax

150

100

Al

50

Au

35 30

Ag

25 20

Al

15

Au 0 400

(a)

450

500

550

600

Wavelength [nm]

650

700

10 400

(b)

450

500

550

600

650

700

Wavelength [nm]

Figure 41. (a) Spectral intensity enhancement Tmax owing to Eq. (31) of planar Ag, Au, and Al Þlms on quartz (ǫ 0 = 2) immersed in air (ǫ 2 = 1,0). (b) Penetration depth of surface plasmons [Eq. (32)]. The corresponding dielectric functions of the materials were taken from Raether (1988) and Palik (1985). However, they might substantially deviate from those of real deposited Þlms.

irradiate light (Liebermann et al., 1990). Emission is obviously restricted to the inside of the pipette and thus ultrasmall light emitters are feasible. In this context both optically (Sturmer et al., 1998) and electrically driven (Kuck et al., 1992) probes were suggested. Another interesting approach utilized nanoporous silicon tips for the same technique (Gottlich and Heckl, 1996). In this case the restriction of the porous area on the tipÕs apex causes technological difÞculties that have not been completely solved to date. Nevertheless, ßuorescent probes offer the capability to accomplish near-Þeld optics with subnanometer resolution involving single molecules as emitters or scatterers. First approaches to Þber probes provided with a single ßuorescent molecule at the apex have been discussed in the literature (Michaelis et al., 1999). However, it should not be concealed that single-molecule light emitters make great demands on powerful and sophisticated or demanding optical detection. c. Laser Near-Field Probes. Recent developments in optoelectronic light emitters also opened up new vistas for near-Þeld optics. The typical dimensions of conventional edge-emitting lasers were drastically reduced, introducing the concept of the vertical cavity surface-emitting laser (VCSEL) diode. Today we are concerend with VCSELs of about 5Ð20 μm diameter and some micrometer height, including already the two distributed Bragg reßectors that establish the high-quality longitudinal resonator. Although the emitting area of the VCSEL is not of sub-wavelength size, the VCSEL was anticipated by Heisig (2000) for near-Þeld optical applications. It was suggested to illuminate a sub-wavelength-sized aperture in a sharpened metallized aperture tip. The underlying concept is shown in detail in Figure 42a. The VCSEL is established

186

EGBERT OESTERSCHULZE QPD

Cantilever AuZn/Au Si3N4 n-GaAs AuGe

VCSEL Sample Al

Lens

(a)

Photodetector

(b)

GaAs cantilever

VCSEL

GaAs tip (c)

Metallization

(d) VCSEL

Figure 42. (a) Schematic drawing of the Þrst active, that is, light-emitting GaAs cantilever probe with integrated surface-emitting laser diode of 977 nm wave-length. (b) Top view revealing the VCSEL with its cylindrical shape of 8 μm diameter on the GaAs cantilever. The AuZn/Au electrode is utilized as ohmic contact. (c) Front view showing the VCSEL centered above the sharp GaAs tip that is shown in (d) in more detail. The shape of the tip was optimized to allow a perpendicular orientation with respect to the sample surface. (Reproduced with permission from Heisig, S. (2000). Multifunktionale Galliomarsenid-Sensoren fur die Rastersondenmikroskopie Ph.D. Thesis, Universit¬ at Gesamthochschule Kassel).

on top of a GaAs cantilever that also carries a AuZn/Au ohmic contact. The epitaxially grown active area of the VCSEL comprises three 8-nm thin undoped In0.17Ga0.83As quantum layers separated by 10-nm thin undoped GaAs Þlms. The emission wavelength of 977 nm is above the wavelength of the corresponding band gap of the GaAs substrate material of 870 nm, and thus laser light is transmitted through the cantilever with negligible absorption. It illuminates an aperture in the metal coating of the GaAs tip that establishes the near-Þeld emitting area. The metal coating serves simultaneously as the second ohmic contact to electrically supply the VCSEL. A detailed description of the setup is given in Heisig et al. (2000a). Figure 42 comprises SEM images of the Þrst VCSEL near-Þeld probes realized (Heisig et al., 2000b). Figure 42b shows the GaAs cantilever with the upper electrode and the VCSEL as a cylindrical feature at the cantilever.

RECENT DEVELOPMENTS OF PROBES

187

The cantilever itself shows an excellent homogeneity of its thickness owing to the spray etching process used for its fabrication (see also Section II.C.2). The closeup shown in (c) reveals details of the VCSEL and also its exact alignment on top of the GaAs tip seen below the GaAs cantilever. Although VCSELs offer a rather high quantum efÞciency, the access heat nevertheless adds some difÞculty to its applications in near-Þeld optics because of the restricted thermal heat abduction over the thin GaAs cantilever. As was demonstrated by Heisig et al. (2000b) thermal damage of both the laser and the sample can be avoided by operating the VCSEL in a pulse mode. 2. Light-Detecting Active Probes In comparison to light-emitting near-Þeld probes, it looks exceedingly easier to realize light-detecting near-Þeld devices. This fact also explains the more extensive literature on this topic. The material choice of photodetectors is not as restricted as that for emitting devices because semiconducting material with an indirect electronic band gap is also of interest. In particular the availability of silicon-based AFM cantilever probes has promoted the development of integrated near-Þeld probes because silicon is a quite common and preferred material used for light-detecting devices (Singh, 1994). Nevertheless, the material class of the IIIÐV semiconductors has already been suggested for this purpose (Heisig and Oesterschulze, 1992; Prins et al., 1994; Kolb et al., 1994, 1995; Danzebrink et al., 1995; Heisig et al., 1998). Probes with integrated near-Þeld detectors presented rely on the well-established concepts known from conventional optoelectronics, that is, the Schottky diode or the p/n junction. In both cases the incoming photons are absorbed to generate electrons and holes which are separated in the depletion layer of the diode and thus give rise to an external photocurrent that is proportional to the light intensity. The width w of the depletion layer depends on the dopand concentration of the acceptor NA and donor ND, on the static dielectric constant ǫ stat of the semiconductor material, and on the applied voltage U

  1 2ǫ0 ǫstat 1 (33) + (U D − U ) w= e NA ND with the diffusion voltage UD UD =

kB T NA ND e n i2

(34)

e denotes the electron charge, kB BoltzmannÕs constant, and ni the intrinsic concentration of charge carriers of the desired materials at a given temperature. Table V provides an overview of the respective material parameters of the most common materials.

188

EGBERT OESTERSCHULZE TABLE V Electronic Parameters of Some Common Semiconductor Materials: Si, Ge, and GaAsa

Material

Band gap (eV) (Wagemann and Schmidt, 1997)

ni (cm−3) (M¬ uller, 1991)

ǫ stat (Wagemann and Schmidt, 1997)

Silicon Germanium Gallium arsenide

1.11 0.67 1.43

1.5 × 1010 2.5 × 1013 1.8 × 106

12 16 11

a

Both the band gap and the intrinsic carrier density are given at room temperature.

As a rule of thumb an optimum detection efÞciency of photons is achieved if the depletion width w exceeds the optical absorption length 1/α of photons in the semiconducting material by a factor of 2 or so. Figure 43 shows the spectral dependence of the absorption coefÞcient α, the absorption length 1/α, and the refractive index n of Si and GaAs. In the optical range a typical absorption length of typically 0.1Ð9.5μm for Si and 0.03Ð0.93μm for GaAs is obtained. In accordance with Eq. (33) a proper doping level has to be chosen to adapt the depletion width to the optical absorption length and thus optimize the sensor efÞciency. Obviously, the absorption length is not compatible with the size of a typical near-Þeld probe of less than 100 nm. Thus as a precaution the sensitive area of the probe has to be restricted, such as in the form of a subwavelength-sized aperture masking the diode. It is worth noting that the rather larger refractive index of Si and GaAs is quite opportune because it helps to substantially reduce the inßuence of the cutoff effect on the transmission efÞciency if tapered tips are fabricated from these materials. 10 6

10 4

10 3 0.4

(a)

10 -1

GaAs

10 0

Si

0.5

0.6

Wavelength [ m]

0.7

10 1

5.6

Si Refractive index n

10 5

Absorption length [ m]

Absorption coef. α [cm -1]

10 -2

5.2

GaAs 4.8 4.4 4.0 3.6 0.4

(b)

0.5

0.6

0.7

Wavelength [ m ]

Figure 43. (a) Absorption coefÞcient α and absorption length 1/α and (b) refractive index n of Si and GaAs as a function of wavelength.

RECENT DEVELOPMENTS OF PROBES

189

a. Schottky Diode Probes. The Þrst integrated near-Þeld detecting probes were realized by coating commercially available Si cantilever probes with a metal layer to obtain the desired Schottky diode (Danzebrink et al., 1995). This technological approach looks very promising at Þrst because the complete mechanical structure of the probe including a sharp tip is already deÞned. However, there is the hitch of realizing the subwavelength-size aperture subsequently. Repeated metal deposition processes under oblique angles were used to obtain the aperture by shadow masking certain parts of the tip. But this process could not be proved successful for reproducible and controlled aperture generation. Additional problems arose from the dopand concentration of the cantilever material of about ND = 1018 cm−3. It gives rise to a depletion layer width w of about 50Ð100nm that substantially deviates from the just-mentioned optical absorption length in silicon. Finally, the Schottky diode was not restricted to the tip but to the whole cantilever surface with the disadvantage of a rather large saturation current. An improved version of an A1/Si Schottky diode was later presented by Davis et al. (1995) and Davis and Williams (1996). Fabrication relied on complete batch fabrication of probes. However, the achieved aperture size of about 0.7Ð1.0μm was in the range of the wavelength, that is, at the crossover from the far-Þeld to the near-Þeld regime. The presented tips were integrated onto a planar surface rather than in cantilever probes, which causes difÞculties with respect to the approach of the tip to the sample surface. A complete fabrication process of Schottky diode tips on base of cantilever probes has been presented by Leinhos et al. (1998). In their case lithography and etching processes were optimized to reproducibly deÞne apertures of about 50Ð100nm dimension in the metal Þlm that coats the tip and establishes the locally conÞned Schottky diode at the very tip. Figure 44 shows SEM images of the AFM with the integrated Ti/p-Si/Al Schottky sensor. b. p/n Junction Probes. Akamine et al. (1995, 1996) and Yamada et al. (1996) modiÞed the conventional fabrication process of planar p/n junctions to establish a rather large photosensitive area on a cantilever probe. An almost transparent Spindt tip (see Section II.C) was deposited as a tapered waveguide on top of the 10 × 10 μm2 sized p/n junction to scatter light of the near Þeld and thus to provide near-Þeld resolution (Tanaka et al., 1998). A similar approach was presented by Castagne et al. (1998) based on InP cantilever probes. In their case an almost transparent tapered EBD tip was deposited as a nearÞeld scatterer on the active area of the p/n junction. The main disadvantage of both approaches is the sensitivity of the rather large detectors to scattered background light. Tanaka et al. (1999) invented a method to substantially reduce this effect attaching a small amount of an infrared excitable phosphor (IEP) to the tip. IEPs are optically nonlinear materials that are capable of up-converting infrared to visible light. In case of TanakaÕs experiments the

190

EGBERT OESTERSCHULZE

(a)

(b)

(c)

(d)

Figure 44. (a) Silicon cantilever with an electrically isolated Schottky diode at the apex of the tip. (b) Close-up of the tip revealing the stacked insulating silicon dioxide layer and thin titanium Þlm. (c) The silicon tip was Þrst oxidized and subsequently the oxide was selectively etched at the very tip to conÞne the active Schottky diode area to the tip apex. (d) The Þnal shape of the completed Schottky diode. (Reproduced with permission from Leinhos, T., Stopka, M., and Oesterschultze, E. (1998). Micromachined fabrication of Si cantilevers with Schottky diodes integrated in the tip. Applied Physics A 66: 65Ð69).

infrared wavelength of 1550 nm used for the excitation of the IEP material differed substantially from its emission wavelength at 550 nm and 670 nm. Owing to the fact that the associated energy of the exciting photons is below the band gap of the substrate material, the photodetector is sensitive only to the emission wavelength. Another promising approach for the batch fabrication of p/n-junction probes was introduced by Sch¬ urmann et al. (2000). Reproducible aperture fabrication is accomplished by exploiting the Þeld enhancement at the metal coating tips during plasma treatment to increase the etch rate locally. This selective etching process is capable of reproducibly providing apertures of 50Ð100nm dimensions as reported.

RECENT DEVELOPMENTS OF PROBES

191

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ADVANCES IN IMAGING AND ELECTRON PHYSICS, VOL. 118

Morphological Image Enhancement and Segmentation IVAN R. TEROL-VILLALOBOS Centro de Investigaci« on y Desarrollo Tecnol« ogico en Electroquimica Parque Tecnol« ogico Quer« etaro S/N Sanfandila-Pedro Escobedo, CP.76700-APDO 064. Quer« etaro, M« exico

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . II. Some Basic Tools in Mathematical Morphology . . . . . . . . . . . . . . A. Basic Morphological Transformations (Dilation, Erosion, Closing, and Opening) . . . . . . . . . . . . . . . . . . . . . . . B. Morphological Image Reconstruction . . . . . . . . . . . . . . . . . C. Contrast Detectors (Morphological Gradients and Top-Hat Transformations) . D. Activity Mappings and Toggle Mappings. . . . . . . . . . . . . . . . III. Morphological Nonincreasing Filters Using Gradient Criteria (Morphological Slope Filters) . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Morphological Slope Filters . . . . . . . . . . . . . . . . . . . . . B. Extrema ModiÞcation with Contrast Enhancement . . . . . . . . . . . . 1. Reconstruction Transformations . . . . . . . . . . . . . . . . . . 2. Morphological Slope Filters . . . . . . . . . . . . . . . . . . . . 3. Histogram ModiÞcation . . . . . . . . . . . . . . . . . . . . . C. Fixed Zone Growth: Stability of MSFs. . . . . . . . . . . . . . . . . D. Properties of Morphological Slope Filters . . . . . . . . . . . . . . . IV. A Sequential Family of MSFs. . . . . . . . . . . . . . . . . . . . . . A. Some Intermediate Results Using a Family of Sequential MSFs . . . . . . B. Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Image Segmentation using MSFs . . . . . . . . . . . . . . . . . . . . A. Homotopy ModiÞcation. . . . . . . . . . . . . . . . . . . . . . . B. Image Segmentation Using the Watershed Transformation. . . . . . . . . C. An Image Segmentation Algorithm Using MSFs. . . . . . . . . . . . . 1. Quadtree Approach. . . . . . . . . . . . . . . . . . . . . . . . 2. Flat Zone Approach . . . . . . . . . . . . . . . . . . . . . . . 3. A Segmentation Algorithm . . . . . . . . . . . . . . . . . . . . VI. Nonlinear Multiscale Approach Using a Sequential Family of MSFs. . . . . . A. A Geodesic Approach . . . . . . . . . . . . . . . . . . . . . . . B. MSF Using Flat Zone Notion: Flat Zone Gradient, Graphs, and Connected Operators . . . . . . . . . . . . . . . . . . . . . . C. Nonlinear Multiscale Representation using MSF. . . . . . . . . . . . . 1. Multiscale Representation. . . . . . . . . . . . . . . . . . . . . 2. Weighted Morphological Slope Filters . . . . . . . . . . . . . . . 3. Some Comments about Invariants . . . . . . . . . . . . . . . . . 4. Kramer and Bruckner ModiÞed Algorithm . . . . . . . . . . . . . . VII. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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C 2001 by Academic Press ADVANCES IN IMAGING AND ELECTRON PHYSICS Copyright  All rights of reproduction in any form reserved. ISSN 1076-5670/01 $35.00

208

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I. Introduction Image segmentation and image contrast enhancement are two useful techniques in image processing. Generally, both techniques use contrast operators to transform an image and to get the Þnal one. In mathematical morphology (MM), the basic contrast operators are the top-hat transformations and the gradient operators. These contrast operators are important tools for the watershed-plusmarker approach that is used to segment images in mathematical morphology (Beucher, 1990; Meyer and Beucher, 1990). In the watershed-plus-marker approach, the homotopy of the gradient image is modiÞed by imposing some region markers as its only minima. Then, the watershed transformation on the modiÞed gradient is applied. However, as expressed by Crespo et al. (Crespo, 1993; Crespo et al., 1993), the need for locating each marker inside an image region poses an important limitation. For example, some problems occur when the features are very small, when the shape is quite elongated, or when there are thin regions. In other words, the watershed-plus-marker approach shows a limited resolution when the purpose is the extraction of features. This was the main reason that originated the proposition of another image segmentation technique, called the ßat zone approach (Crespo, 1993; Crespo and Schafer, 1994), which gives another option for image segmentation. In this technique, the morphological connected Þltering plays a fundamental role. In addition, in the watershed-plus-marker approach, the main tools for detecting a set of markers are morphological Þlters. This is the main reason for studying image segmentation from a morphological Þltering point of view. Morphological Þltering is one of the most interesting subjects of research in MM. Morphological Þlters are nonlinear transformations, locally modifying geometric features of images. The basic morphological Þlters are the morphological opening and the morphological closing with a given structuring element. In general, this element is a set that describes a simple shape that probes the image and provides information about it. By using basic Þlters, we can build others with different properties. Morphological Þltering has been tested in many practical problems with good results. Recently, Þlters by reconstruction (a class of geodesic transformations) have become powerful tools that enable us to eliminate undesirable features without affecting desirable ones. Intensive work has been done on the characterization of these transformations (Serra and Salambier, 1993; Crespo et al., 1995; Serra, 1998, among others). These transformations by reconstruction, which form a class of connected Þlters, involve not only the homotopy modiÞcation but also opening, closing, alternated Þlters, alternating sequential Þlters, and even new transformations called levelings (Meyer, 1998). By deÞning

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monotone planings and ßattenings and by combining both concepts, the levelings notion has been introduced; these deÞnitions enable us to have a framework of algebraic theory for studying connected Þlters. Serra (2000) extends these concepts by means of a marker approach and the activity mapping notion. In the present work, a family of nonincreasing Þlters is presented. The Þlters use a slope criterion to modify images in order to contrast and segment them. Applying these Þlters, the image zones with weak slopes are eliminated and zones with stronger slopes become more contrasted. To build these Þlters, the concept of morphological gradient has been applied. Two versions of the morphological gradient are used: (1) The internal gradient obtained by the arithmetical difference between the original function and the erosion function, and (2) the external gradient given by the arithmetical difference between the dilation function and the original function. These two deÞnitions generate two pairs of Þlters. This class of nonincreasing Þlters was proposed with the name of morphological slope Þlters (MSFs) (TerolVillalobos, 1995, 1996a). These Þlters can be considered toggle mappings. The notion of toggle mappings was proposed by Serra (1988b) in order to build contrast operators. The MSFs have excellent properties and provide essential contrast to the images. For example, the transformed image using an MSF presents a well-deÞned contrast. Thus, we can deÞne a class of invariants (roots) (Terol-Villalobos, 1996a) where an element (image with an essential contrast) of this class is left unchanged by the Þlters. Furthermore, by applying MSF, minima or maxima on the image are modiÞed, allowing the use of the watershed transformation to obtain a good segmentation. The main purpose of the Þlters is to attenuate the zones where the gradient is weak and to leave the rest of the regions unchanged. To discriminate between both zones we use a parameter φ. By attenuating zones of weak contrast, we increase the contrast of other zones that have a strong gradient without changing the gray level of the points belonging to them. Several important contributions are presented in this work. First, we investigate the Þlters when they are sequentially applied (Terol-Villalobos and CruzMandujano, 1998; Terol-Villalobos, 1998). This proposition allows the establishment of some intermediate results of the output image. We show that these sequential transformations allow the selection of features at each level of a family of morphological slope Þlters. Second, the domain of invariance associated to these transformations is studied to better understand the Þltering process (Terol-Villalobos and Cruz-Mandujano, 1998). In addition, the transformations are used as a contrast-oriented segmentation approach: subsequently, this procedure is compared with the ßat zones approach (Terol-Villalobos and Cruz-Mandujano, 1998; Terol-Villalobos et al., 1999).

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To obtain homogeneous zones from the image, we study these Þlters in a geodesic way using a MaxÐMincriterion, and a MaxÐMin-to-Areacriterion. These criteria enable us to select the homogeneous regions on the images. Then, both complementary techniques in image analysis, edge extraction and image partitioning, are studied by means of these Þlters. Finally, a multiscale approach is presented. We discuss the notion of image analysis and diffusion process in order to compare this technique with our approach. In fact, the Gaussian family is an isotropic diffusion process and this family is the multiscale paradigm within the linear Þltering. Principally, we propose other gradient deÞnitions to look for better control of the output image and to avoid splitting the ßat zones. SpeciÞcally, the notion of ßat zones was used to build a gradient that permits construction of MSFs that do not break ßat zones. Also, a weighted gradient criterion using gray-level intensity is proposed. This notion allows attenuating some sensitivity of the MSF to parameter φ. All the work is presented in the discrete case and, more speciÞcally, with real-valued images. The present work is organized as follows. In Section II, some basic concepts of MM are presented. The notion of morphological slope Þlters and their algebraic properties are introduced in Section III and the sequential morphological slope Þlters are described in Section IV. Image segmentation using morphological slope Þlters is studied in Section V. Finally, in Section VI, a multiscale approach is discussed and new results concerning MSF on graphs are proposed.

II. Some Basic Tools in Mathematical Morphology The aim of this section is to deÞne the basic transformations, the geodesic transformations, the traditional contrast operators, and the notion of activity mappings.

A. Basic Morphological Transformations (Dilation, Erosion, Closing, and Opening) The basic morphological transformations in mathematical morphology are the morphological dilation and erosion of f (x) (numerical case) by a structuring element λB. These basic transformations are given by ∨

δλB ( f )(x) = ( f ⊕ λB)(x) = ∨{ f (x − y) : y ∈ λB} ∨

ελB ( f )(x) = ( f  λB)(x) = ∧{ f (x − y) : y ∈ λB}

(1)

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where B is the elementary structuring element (3 × 3 pixels in this work) and λ is an homothetic parameter. Both transformations, the dilation and the erosion are increasing, but they are not idempotent transformations. The dilation transformation is extensive while the erosion is antiextensive. Using these basic transformations, however, we can build the morphological closing and opening that are increasing and idempotent transformations. The morphological closing is an extensive transformation and the morphological opening is an antiextensive one. These Þlters are given by ϕλB ( f )(x) = ε ∨ (δλB ( f ))(x) λB

∨

γλB ( f )(x) = δ ∨ (ελB ( f ))(x) λB

where B is the transposed set of B.

B. Morphological Image Reconstruction The reconstruction transformations are connected Þlters that enable us to modify minima or maxima without considerably changing the structure of the remaining components. Geodesic transformations are used to build the reconstruction transformations (Beucher, 1990). In this case, the geodesic transformations are iterated until idempotence is reached. Consider the functions f and g, with f ≥ g or f ≤ g. The reconstruction transformations by geodesic dilation and geodesic erosion are respectively expressed by R( f,g) and R∗ ( f,g) and they are deÞned by R( f,g) = lim δ nf (g) = δ 1f δ 1f · · · δ 1f (g) with f ≥ g n→∞

∗

R ( f,g) =

lim εn (g) n→∞ f

until stability

=

ε1f ε1f · · · ε1f (g) until stability

(2)

with f ≤ g

where ε1f (g) = f ∨ ε B (g) is the geodesic erosion and δ 1f (g) = f ∧ δ B (g) is the geodesic dilation that are obtained by means of morphological dilation and erosion. When the function g is equal to the dilation or to the erosion of the original function by a given structuring element, we obtain the closing and the opening by reconstruction. ϕ÷λB ( f ) = lim ε nf (δλB ( f )) = ε1f ε1f · · · ε1f (δλB ( f )) n→∞

Until stability

γ÷λB ( f ) = lim δ nf (ελB ( f )) = δ 1f δ 1f · · · δ 1f (ελB ( f )) n→∞

(3)

Until stability

In Figures 1b and 1c the morphological opening γ λB and the opening by reconstruction γ÷λB , computed from the original image in Figure 1a, are illustrated.

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Figure 1. (a) Original image, (b) morphological opening γ λB, (c) opening by reconstruction γ÷λB , (d) top-hat transformation, (e) top-hat by reconstruction.

C. Contrast Detectors (Morphological Gradients and Top-Hat Transformations) Morphological gradient operators are, in fact, contrast detectors (Rivest et al., 1993). Let f be a function deÞned on Z2. The digital version of this algorithm is given by the formula grad B ( f )(x) = δ B ( f )(x) − ε B ( f )(x)

(4)

However, in this work we use more speciÞcally the morphological internal and external gradients deÞned respectively from the following: gradi B ( f )(x) = f (x) − ε B ( f )(x)

grade B ( f )(x) = δ B ( f )(x) − f (x)

(5)

On the other hand, the top-hat transformation is simply the arithmetic pointwise difference of the closed function from the original one (or the difference

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of the original function from the opened one). Generally, these transformations are followed by a threshold operation: ThwλB ( f )(x) = f (x) − γλB ( f )(x)

ThbλB ( f )(x) = ϕλB ( f )(x) − f (x)

Because opening is antiextensive, the function f (x) − γλB ( f )(x) is nonnegative (in the same way as the function ϕλB ( f )(x) − f (x)). This transformation was proposed by Meyer (see Serra, 1982). The top-hat leads to a size distribution involving contrast of the image and also provides one of the best algorithms for segmenting images. Figures 1d and 1c illustrate the top-hat transformations using the morphological opening and the opening by reconstruction, respectively.

D. Activity Mappings and Toggle Mappings To introduce our main results, in this section we present the notion of toggle mappings and some derived non-increasing transformations called morphological slope Þlters. The main idea of toggle mappings (Serra, 1988b) is to compare a real-valued function f with two patterns, and to choose at each point x the closest value between them and the original function. The original idea was suggested by Kramer and Bruckner (1975). The proposed transformation is given by  δ ( f (x)) if δλB ( f (x)) − f (x) < f (x) − ελB ( f (x)) δε (6) W ( f (x)) = λB ελB ( f (x)) otherwise Some problems in Kramer and BrucknerÕs transformation are the oscillations and jumps produced when it is iterated. The notion of toggle mappings progressed in the way suggested by the Kramer and Bruckner algorithm. These developments were initially studied by Meyer and Serra (1989) to develop the theory of contrast mappings in MM. The following deÞnition formalizes the concept of toggle mappings. DeÞnition II.1 Let F be the class of the functions E → R(Z in this work) and F ′ that of the mappings F → F. Given a family {i } of elements of F ′, one calls toggle mapping of primitives  i any mapping W such that: r r

At each point x, Wx equals one of the  i,x. The criterion that affects one  i to Wx at a given point x depends only on the primitives  i, on the value Wx, and on possible constants.

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In particular, if at point x one of the  i coincides with the identity mapping I, then Wx = Ix = x .

Because toggle mappings generate jumps, the Þrst way, as expressed by Serra (1988b), for keeping down this effect is to look for idempotent toggles. III. Morphological Nonincreasing Filters Using Gradient Criteria (Morphological Slope Filters) The basic problems in image analysis are edge extraction and image partitioning. By solving one of these problems, the other is also solved. For example, by correctly extracting the edges from an image it is possible to deÞne the regions in an image as connected components. Then, edge extraction and image segmentation are complementary problems. While edge extractors search for discontinuities in the image, region segmentation methods locate groups of pixels that share certain common properties. This section describes a family of nonincreasing Þlters that can be used for both basic goals, edge detection and image partitioning. Other approaches in the literature also have been treated the problem of integrating edge-based and region based information as presented by Haddon and Boyce (1990) and by Pavlidis and Liow (1990). The morphological nonincreasing Þlters presented here arise from the notion of the traditional image segmentation algorithm in MM, the so-called watershedplus-marker approach: image homotopy modiÞcation by a set of markers and application of the watershed. Gradient homotopy modiÞcation is the main step to achieve the image segmentation. Here, morphological Þltering is the base for detecting a set of markers that is used to modify the homotopy of the gradient image. This is the main reason why we focus our study in a family of Þlters that modiÞes gradient image. A. Morphological Slope Filters In Terol-Villalobos (1995, 1996a), a new class of morphological nonincreasing Þlters was proposed. The main idea to build these Þlters is the notion of morphological gradient criteria and idempotent mappings. Zones of weak contrast (weak gradient) are attenuated and zones of great contrast remain unchanged. This allows zones of great contrast on the image to be retained (without changing the gray-level values) and enhanced by attenuating the rest of the zones. Consider the deÞnitions of internal and external gradients (Beucher, 1990; Rivest et al., 1993): gradi B ( f )(x) = f (x) − ε B ( f )(x)

grade B ( f )(x) = δ B ( f )(x) − f (x)

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215

From these equations and from the notion of toggle mappings, we deÞne two operators,  ε B ( f )(x) if gradi B ( f )(x) ≤ φ ε1 ξφ ( f )(x) = f (x) if gradi B ( f )(x) > φ (7)  δ B ( f )(x) if grads B ( f )(x) ≤ φ δ1 ξφ ( f )(x) = f (x) if grads B ( f )(x) > φ Here, we will be working with these operators by iterating them. For the second step we have     ξφε2 ( f )(x) = ξφε1 ξφε1 ( f ) (x) and ξφδ2 ( f )(x) = ξφδ1 ξφδ1 ( f ) (x)

At the nth step when stability is reached (n → ∞):    δ  ε ξφεn ( f )(x) = ξφε1 ξφn−1 ( f ) (x) and ξφδn ( f )(x) = ξφδ1 ξφn−1 ( f ) (x)

(8)

For the sake of clarity, we will present results for the ξφεn ( f ) Þlter case, but similar comments and results can be expressed for the other Þlter ξφδn ( f ). In addition, we will invariably write ξφε∞ or ξφεn at the nth step when the idempotence or stability is reached. To understand these Þlters from a geometrical point of view, consider a function such as that illustrated in Figure 2a. Although the function used in this example is 1-D (one-dimensional case), it enables us to understand the 2-D case. We use a structuring element composed by three connected points and centered at the middle point, as shown in Figure 2, and the φ value is equal to one (φ = 1). We analyze the case of ξφεn Þlter. First, the eroded of the function f is computed to determine the gradient transformation (Figure 2b). Next, the new function ξφε1 , shown in Figure 2c, is calculated by applying Eq. (7). A similar procedure is applied until idempotence is reached, as illustrated by Figure 2e. These Þlters, ξφε∞ and ξφδ∞ , are antiextensive and extensive transformations respectively, they are idempotent transformations at the nth step, but they are not increasing. These Þlters have other interesting properties. In Section III.C, a complete study is presented. To test these Þlters, several experiments have been performed. Figure 3 shows one example for the Þlter ξφε∞ . The original image is composed of white objects with a black background. In fact, this is a very simple image where the only problems are the intensity changes on the image. However, this example correctly illustrates the behavior of these Þlters. Filters enhance not only the edges of the image but also the objects. In Figure 4, other characteristics of the Þlters are illustrated with a different example. Figure 4b shows the morphological slope Þlter ξφε∞ with φ = 20, while Figures 4c and 4d show the binary images obtained by threshold (between

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ε

Figure 2. (a) Original function, (b) gradient of the original function, (c) ξφ11 ( f ), ε (d) gradi B (ξφ11 ( f )), (e) ξφε1n until stability.

174 and 255 gray levels) from the images in Figures 4a and 4b. The image in Figure 4b illustrates the elimination of the zones with weak slopes, even if the gray level of these pixels is higher than that of others, but with a stronger slope. The φ parameter is selected by studying the gradient values and the φ value is selected in such a way that the zones of interest are detected.

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Figure 3. (a) Original image; (b) output image ξφε∞ with φ = 20.

B. Extrema ModiÞcation with Contrast Enhancement Image extrema (regional minima and maxima) play a fundamental role in MM. We remember that a regional minimum is a plateau of uniform altitude without lower neighbors and a regional maximum is a plateau of uniform altitude without upper neighbors. Regional extrema are particularly important to solve practical problems using the morphological segmentation approach by applying the watershed transformation. However, an inherent problem in such procedure is the large amount of noise in the original image or in the gradient image (the gradient transformation is very sensitive to noise), producing a lot of minima and maxima in the image. In this case, the watershed transformation produces many catchment basins, which are associated with the minima of the gradient. Figures 5a and 5b illustrate the original image and the watershed of the gradient image. In fact, this is the main drawback when the watershed is directly applied, without modifying the minima of the gradient image or without Þltering the original image. A solution for preventing this oversegmentation consists of a prior selection of the objects or regions to be extracted from the image. This selection gives a collection of markers that is used to apply the traditional image segmentation approach, the so-called watershed-plus-marker approach. Locating each marker inside a region is a complication that poses many problems. Generally, it is necessary to know the correct Þltering tools to obtain a good set of markers to solve the segmentation problems. Here, we show that our Þlters enable us to modify the minima (or maxima) to obtain good

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IVAN R. TEROL-VILLALOBOS

(a)

(b)

(c)

(d)

Figure 4. (a) Original image, (b) morphological slope Þlter ξφε∞ with φ = 20, (c) threshold of image (a), and (d) threshold of image (b).

results when the watershed transformation is applied. Equally, we illustrate that MSF allows the modiÞcation of maxima to extract the different regions of the image. 1. Reconstruction Transformations The Þltering step is aimed at the simpliÞcation of the image. The reconstruction transformations are powerful tools that enable us to modify minima

MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION

(a)

219

(b)

Figure 5. (a) Original image; (b) watershed of the gradient image.

and maxima to decrease the oversegmentation without considerably changing the structure of the remaining components. These transformations are very useful in MM, especially in image Þltering. Geodesic transformations are used to build reconstruction transformations. In this case, the geodesic transformations are iterated until idempotence is reached. This process propagates the minima (using geodesic erosions) or the maxima (using geodesic dilations) and merges them. These transformations have been designed to identify characteristic topographic features on images, such as large and deep valleys, sharp crests, and high summits. These features are then used to localize objects. Figure 6 illustrates (1-D example) the case of the reconstruction transformation using geodesic erosions [gray-level case; see Eq. (2)]. To apply this transformation we need a marker function g, which is obtained in this case by arithmetical addition between the original function f and one constant h. The function f is used as the mask function. Next, the following equation is applied [see Eq. (2)]: R ∗ ( f, f + h) = R ∗ ( f, g) = lim εnf (g) = ε1f ε1f · · · ε1f (g) n→∞

(9)

until stability

The marker function g becomes closer to the original function f after reconstruction. This algorithm, also called wrap-up in MM, enables us to detect an overset of the regional minima called cuvettes. Observe the merging process of the minima, which reduces their number. However, the contrast is attenuated, because vertical movements of the reconstruction transformation are prohibited. Images in Figures 7d, 7e, and 7f show, respectively, the

Figure 6. ModiÞcation of minima using reconstruction transformation by geodesic erosion (1-D case).

(a)

(d)

(b)

(c)

(e)

(f)

Figure 7. (a) Original image, (b) Þltered image using reconstruction transformation applying geodesic erosions, (c) Þltered image using ξφε∞ with φ = 6. (d) Minima of the original image, (e) minima of the Þltered image using reconstruction transformation, (f) minima of the Þltered image using MSF.

MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION

221

minima of the original and Þltered images showed in Figures 7a, 7b, and 7c. The Þltered image in Figure 7b was obtained by a reconstruction transformation (h = 25, applying geodesic erosions); the Þltered image in Figure 7c was computed using the Þlter ξφε∞ with φ = 6. Observe the minimum reduction process. 2. Morphological Slope Filters Filters by reconstruction reduce the number of minima, but attenuate the contrast of the remaining structures. Here, using MSF, we combine the minima (or maxima) modiÞcation and the contrast enhancement of remaining structures. As expressed earlier, with our Þlters we try to attenuate the zones with weak contrast without affecting the other ones, which means that incorrect minima (maxima) created by the noise and inhomogeneities are merged to form good minima (maxima). However, minima or maxima created by impulsional noise of great magnitude remain. To test the Þlters, we applied the Þlter ξφε∞ on the image shown in Figure 5a, using different φ values. In Figures 8 and 9, it is possible to observe a reduction of the oversegmentation for φ = 6 and φ = 8 (the structuring element is given by a square having 3 × 3 pixels). With increasing φ value, the minima of the gradient image merge and the oversegmentation is reduced when the watershed transformation is applied. In fact, two minima (or more) will merge if there exists a path C joining them with a slope weaker than the slope criterion for all points belonging to

(a)

(b)

Figure 8. (a) MSF ξφε∞ with φ = 6; (b) watershed of the gradient image.

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IVAN R. TEROL-VILLALOBOS

(a)

(b)

Figure 9. (a) MSF ξφε∞ with φ = 8; (b) watershed of the gradient image.

C. In Figure 2 (1-D case) the minima m1 and m2 are not merged because the point p shows greater contrast than the slope criterion. Then, at the nth step (stability), the minima m1 and m2 are modiÞed but they are not merged. Figures 8a and 9a show the Þltered images with φ = 6 and φ = 8, respectively, while Figures 8b and 9b show their watershed images. This is a most interesting image. The scanning electron microscope (SEM) images present well-deÞned contours that can be considered as an inherent gradient on the image. By observing the Þltered images (φ = 6 and φ = 8), we note that the contours are preserved. With this type of image, the morphological slope Þlter does not considerably affect the contours and it merges minima by looking for a path between the minima. We will show below that there are inclusion relations of the transformed images depending on the ordering relations of the φ parameter. Finally, maxima modiÞcation is illustrated by applying the operator ξφδ∞ . Figures 10a, 10c, 10e, and 10g show the original image and the Þltered images ξφδ∞ ( f ) for φ = 7, 8, 9, respectively. The different regions in the Þltered images are more deÞned than the original image. Furthermore, the contours of different regions are well deÞned and imposed to the gray level image. To verify the results, the maxima of the original and Þltered image are computed (white regions). Maxima images are illustrated in Figures 10b, 10d, 10f and 10h. The morphological Þlter ξφδ∞ behaves as a connected operator, by modifying the maxima of the image (in the same way as ξφε∞ works with minima) and deÞnes the transitions between the zones that compose the image. In Section VI, the

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 10. (a) Original image; (b) original image maxima; (c) and (d) ξφδ∞ with φ = 7 and its maxima; (e) and (f) ξφδ∞ with φ = 8 and its maxima; (g) and (h) ξφδ∞ with φ = 9 and its maxima.

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IVAN R. TEROL-VILLALOBOS

notion of connected Þlters is analyzed. We will show that MSFs are not strictly connected but it is possible to build connected MSFs by modifying the gradient criterion. 3. Histogram ModiÞcation We know that the histogram of an image represents the relative frequency of occurrence of the various gray levels in the image, and it has been used as a powerful tool for image enhancement. Because our Þlters enable us to modify minima and maxima by combining contrast enhancement, it is interesting to know how the gray-level histogram is affected. If we consider that our Þlters merge the different minima (or maxima), it is possible to suppose that the gray-level histogram of the Þltered images will present well-deÞned histogram points. Observe this behavior by analyzing the images using the gray-level histogram. In Figure 11a the gray-level histogram of the original image (Figure 7a) is shown, while Figures 11b and 11c illustrate the gray-level histograms of the Þltered images (using ξφε∞ ) for φ = 5 and φ = 11, respectively. Note that the different ßat zones of the images are detected in the histogram (see the peaks). Each peak represents a well-deÞned zone in the image, as shown in Figures 12a to 12c (φ = 7, 9 and 12) for a threshold of 137 to 137. C. Fixed Zone Growth: Stability of MSFs When morphological erosion and dilation are used as primitives to build toggle mappings, the risk of degenerating the image by iteration of these operators can appear. In our case, both basic transformations are used to build toggle mappings but in a separated way. This option enables us to have some control of the output image. Let us analyze the stability problem, as presented in Terol-Villalobos (1998), by studying the case of Þxed zone growth. DeÞnition III.1 (Serra, 1988). The activity of a mapping  is said to exhibit a Þxed zone growth when  ◦ ≻

and x = Ix ⇒ ( ◦ )x = Ix

This deÞnition means that the successive iterated values at point x, namely x1 , x2 , . . . , xn , may only increase (or decrease), or stop. If  does not modify the function value at point x, it remains unchanged as long as the operator is iterated. An example, which shows that nonidempotent toggles generate jumps and oscillations, is the Kramer and Bruckner transformation (Kramer and Bruckner, 1975) (Fig. 13).

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MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION 1800

Histogram

1200

600

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243

155

177

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111

89

67

45

23

1

0

Intensity (a) (a) 4000

Histogram

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111

89

67

45

1

0

23

2000

Intensity (b) (b) Figure 11. (a) Gray-level histogram of the original image (Figure 7a); (b) and (c) gray-level histograms of the Þltered images (using ξφε∞ ) for φ = 5 and φ = 11.

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40000

Histogram

20000

241

217

193

169

145

121

97

73

49

25

1

0

Intensity (c)(c) Figure 11. (continued )

This transformation is given by the following relationship:  ε B ( f )(x) if gradi B ( f )(x) < grade B ( f )(x) W δε ( f )(x) = δ B ( f )(x) otherwise When expressing it by gradient criteria the interpretation becomes more complex. At each point x, the erosion transformation is selected if gradi B

(a)

(b)

(c)

Figure 12. Threshold at 137 to 137 gray level of images: (a) Þltered image φ = 7, (b) Þltered image φ = 9 , (c) Þltered image φ = 12.

MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION

(a)

227

(b)

Figure 13. (a) Kramer and Bruckner transformation using a structuring element λB with λ = 3 at the Þrst iteration; (b) same transformation after 5 iterations.

( f )(x) < grade B ( f )(x); otherwise the dilation is choosen. In other words, we compare two gradient deÞnitions to select the smallest gradient value between both deÞnitions. The interpretation of this contrast operator is problematic, because the supports of two gradient operators (i.e., the two sets of points of the domain of deÞnition where these functions are strictly positive) are relatively disjoint. In contrast to this transformation, which includes both gradient deÞnitions, more consistent interpretations are obtained when working separately with the internal and external gradients. In this case, the successive iterated εn ε1 ε2 ε1 , ξφ,x , . . . , ξφ,x , decrease or stop. If ξφ,x does not values at point x, namely ξφ,x modify the function value at point x, it remains unchanged as long as the operaε1 ( f )(x) = f (x) (gradi B ( f )(x) > φ). tor is iterated. Suppose that at point x, ξφ,x Then, in a neighborhood Bx and using Eq. (7), the points y ∈ Bx can only take ε1 ( f ))(x) ≤ ε B ( f )(x) and the eroded or the original function value; thus ε B (ξφ,x ε1 gradi B (ξφ,x ( f ))(x) ≥ gradi B ( f )(x) ≥ φ. The function value f (x) remains unε1 ε1 ε2 (ξφ,x ( f ))(x) = ξφ,x = f (x) if gradi B ( f )(x) ≥ φ. Therefore, we changed; ξφ,x can deÞne a set Sφ (great contrast points), associated to this operator, where the function value f(x) remains unchanged. Let Df be the domain of deÞnition of the function f . Then, for all function f we have the following property: Property III.1 Let Sφ ⊂ D f be the set of points x such that ξφε1 ( f (x)) = f (x) with gradi B ( f (x)) > φ. Then ξφε∞ ( f (x)) = f (x) ∀x ∈ Sφ . In the following sections, the set Sφ deÞnes one support of the points of great contrast. In addition, a set of weak contrast points can be deÞned. These

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sets enable us to ensure that at each iteration, the function value of a weak contrast point will decrease or stop. ε

Property III.2 ∀x ∈ SφC ⇒ ξφk+1 ( f (x)) = ε B (ξφεk ( f (x))), where SφC is the complement of the set Sφ . SφC is the set of weak contrast points. Then, at the Þrst iteration of this operator, all points are classiÞed in two categories and remain in them at each iteration. Similar comments can be expressed for ξφδ1 . Then, by iterating nonidempotent mappings (as the reconstruction transformations case), two idempotent mappings are built. The behavior under iteration of the Þxed zones growth enables us to control the stability.

D. Properties of Morphological Slope Filters In this section, we illustrate other properties of MSFs. For a better understanding of the Þltering characteristics, it is interesting to study the properties depending on the φ parameter. Here, we mainly present properties for ξφεn case. However, similar results can be expressed for the ξφδn case. In fact, because by complementation (in the digital case), f C (x) = gl max − f (x) where gl max = 255, we obtain [ε B ( f (x))]C = δ B (gl max − f (x))

ε B ( f (x)) = gl max −δ B (gl max − f (x)) then gradi B ( f )(x) = grade B ( f C )(x)

= δ B (gl max − f (x)) − f C (x)

= δ B (gl max − f (x)) − gl max + f (x)

= −[gl max −δ B (gl max − f (x))] − f C (x)

= f (x) − ε B ( f )(x)

As expressed by properties III.1 and III.2, since at the Þrst step the points of image are classiÞed (Sφ and SφC ), it is easy to show that for two given parameters φ2 > φ1 , there exists an inclusion relation. Property III.3 For φ2 > φ1 ⇒ Sφ2 ⊂ Sφ1 . For two given parameters φ 1 and φ 2 such that φ1 < φ2 , the support Sφ2 is included in the support Sφ1 . Then, by thresholding the gradient image between φ 1 + 1 and 255 to obtain Sφ1 , and between φ 2 + 1 and 255 to obtain Sφ2 , it is possible to observe this

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relation. In fact, because the pixels of the image can be classiÞed at the Þrst iteration, it is possible to obtain a marker in order to use the fast algorithms that are used for building the reconstruction transformations. This will be used later to propose general MSFs based on a geodesic approach. On the other hand we can observe that there are inclusion relations between the MSFs, depending on the ordering relation ≤ of the φ value. That means: Property III.4

For φ 1 and φ 2 such that φ 1 < φ 2, we have that  ε∞ ξφ1 ( f (x)) ≥ ξφε2∞ ( f (x)) ξφδ1∞ ( f (x)) ≤ ξφδ2∞ ( f (x))

By choosing the maximum value (or greater than this value) of the gradient image as the φ parameter, the Þnal image at the nth step (when the idempotence is reached) will be the eroded image by λB with λ → ∞. This is expressed by the following property:  Property III.5  Let f (x) be a function deÞned on Df and φ = max gradi B ( f (x)); x ∈ D f . Then, we have that ∀x ∈ D f ,

ξφε1 ( f (x)) = ε B ( f (x))

(Sφ = ∅)

and   ξφε∞ ( f (x)) = ε∞ B ( f (x)) = ∧ f (z); z ∈ D f

Finally, the next property shows that the transformed image by MSF will have a well-deÞned contrast. For any point in the domain of deÞnition, the contrast with regard to a neighborhood Bx will be zero or greater than φ. Property III.6 Let φ be a given parameter and B the structuring element. We suppose that the idempotence is reached at the nth step. Then,     ∀x ∈ Sφ , gradi B ξφεn ( f (x)) > φ and ∀x ∈ SφC , gradi B ξφεn ( f (x)) = 0 IV. A Sequential Family of MSFs In mathematical morphology it is common to employ by composition a family of Þlters depending on some particular parameter. This notion, frequently used for morphological Þltering, motivates us to study the MSF from this point of view. We will show that sequential MSF retain contrast features at different levels of the family (Terol-Villalobos and Cruz-Mandujano, 1998).

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A. Some Intermediate Results using a Family of Sequential MSFs For the Þltering of the input image, we employ a family of MSFs in a sequential way. This family of Þlters depends on a family of parameters {φ i} with i ∈ S = {1,2, . . . , m} and φ j < φk ; j < k. Let us consider a family composed by two elements with associated parameters φ 1, φ 2. For a better understanding of our propositions we study them from a geometrical point of view. Let us consider a 1-D example. The structuring element is composed of three connected points centered at the middle point. Figures 2a to 2e show the different steps (until stability is reached at the nth step) used to obtain the Þltered function ξφε1n ( f )(x) (Figure 2e). We study this example in a subset DR included in the domain of deÞnition Df. First, we calculate the eroded of the function f(x) to determine the gradient transformation at Figure 2b. Then, applying Eq. (7), the new function ξφε11 ( f )(x) is obtained (see Figure 2c). For the second step, in the same way as for step 1, we obtain the gradient shown in Figure 2d. A similar procedure is applied until idempotence (Eq. (8)) is reached, as shown in Figure 2e. We observe that some points remain with the similar function values in the original and Þltered functions (i.e., ξφε1n ( f )(x) = f (x)). Now, let us perform a similar processing on the original function f (x), but using a φ 2 parameter with φ 1 < φ 2. Figure 14c shows that several points (or regions) which remain unchanged after applying ξφε1n ( f )(x) are removed using the Þlter ξφε2n ( f )(x). Filtering with parameter φ 2 is more discriminative than that with parameter φ 1 (see property 4). Finally, let us transform the function f (x) in a sequential way using both Þlters. In other words, we apply by composition ξφε1n ( f )(x) and ξφε2n ( f )(x). Figure 14d illustrates this transformation. We observe in this Þgure that more points (regions) of the original function remain unchanged, with similar function values, in the Þltered function ξφε2n (ξφε1n ( f )(x)) than in the Þltered function ξφε2n ( f )(x). This means that to obtain intermediate results we can apply the Þlters sequentially. For the sake of simplicity, we use the notation ξφε2n (ξφε1n ( f ))(x) = ξφε1n,φ2 ( f )(x). From Figure 14c, we note that the strong contrast points x ′ and x ′′ in f (x), with regard to the φ 1 parameter (see Figure 2e), are weak contrast points when we apply ξφε2n . However, the strong contrast points x ′ and x ′′ in ξφε1n are also strong contrast points using ξφε1n,φ2 ( f )(x) (i.e., ξφε2n (ξφε1n ( f ))(x ′ ) = f (x ′ )). Similar comments hold for x ′′ (see Figure 14d). This is not true for the strong contrast point p in ξφε1n , because it is a weak contrast point with regard to the sequential family ξφε1n,φ2 ( f )(x). Figure 15d illustrates this sequential transformation applied on a real image (Figure 15a). The processing stage used on the image has a Þrst Þltering step using ξφε1n , next a second Þltering step ξφε2n (with φ 2 = 16 greater than φ 1 = 6), to obtain ξφε1n,φ2 . Then, by applying the Þrst Þlter ξφε1n , with parameter φ 1, before

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DR

x'

f x" y'

εB(f) p

m1

(a)

m2

φ1 = 2

(b)

(c)

x' x"

p

(d) Figure 14. (a) Original function, (b) gradient of the original function, (c) ξφε2n ( f ) until stability, (d) ξφε1n ,φ2 until stability.

applying ξφε2n , we can change the contrast in some points of the original function εn . Compare this Þltered image with in order to keep these when we apply ξφ2 those in Figures 15b and 15c corresponding to Þlters ξφε1n and ξφε2n , respectively. n Another example of sequential MSF ξφε1,..., φ6 using a family of MSF Þlters with parameters φ 1 = 6, φ 2 = 8, . . . , φ 6 = 16 is illustrated in Figure 15e. Now, using properties III.1 and III.2, we can express the next relations between the supports of the strong contrast points.

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(a)

(b)

(d)

(c)

(e)

Figure 15. (a) Original image, (b) Þltered image ξφε1n with φ 1 = 6, (c) Þltered image ξφε2n with φ 2 = 16, (d) Þltered image ξφε1n ,φ2 with φ 1 = 6, φ 2 = 16, (e) Þltered image ξφε1n ,...,φ6 with φ 1 = 6, φ 2 = 8, . . . , φ 6 = 16.

For φ 1 < φ 2 we have that ξφε2n ( f )(x) = ξφε1n ( f )(x) = f (x)   ε ξφε1n ( f )(x) = f (x) > ξφε2n ( f )(x) = ε1B ξφ2n−1 ( f ) (x)

∀x ∈ Sφ2 ⊂ Sφ1 ,

∀x ∈ Sφ1 ∩ SφC2 , and

∀x ∈ Sφ2 ⊂ Sφ1 ,φ2 ⊂ Sφ1 ,

ξφε2n ( f )(x) = ξφε1n,φ2 ( f )(x) = ξφε1n ( f )(x) = f (x)

∀x ∈ Sφ1 ∩ SφC1 ,φ2 , ∀x ∈ Sφ1 ,φ2 ∩ SφC2 ,

ξφε1n ( f )(x) = f (x) > ξφε1n,φ2 ( f )(x)

ξφε1n,φ2 ( f )(x) = f (x) > ξφε2n ( f )(x)

Now, let us analyze the case of weak contrast points. Point y ′ in Figure 2e is a point of weak contrast using ξφε1n . At the nth step (stability), we have that ξφε1n ( f )(y ′ ) = f (m1). The function value ξφε1n at point y ′ cannot have the function value f (m2) ( f (m2) < f (m1)), because there is a strong contrast point p belonging to a path linking f (y ′ ) and f (m2). The function value f (m2)

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cannot propagate to the point y ′ . This is not the case for point m1. However, for ξφε1n,φ2 ( f )(y ′ ), the point p in Figure 14d is no more a point of strong contrast and we have that ξφε1n,φ2 ( f )(y ′ ) = f (m2) < ξφε1n ( f )(y ′ ) = f (m1). Finally, we observe that the strong contrast points x ′ and x ′′ with regard to ξφε1n and ξφε1n,φ2 (Figures 2e and 14d, respectively) are not strong contrast points for the Þlter ξφε2n in Figure 14c. In this case, the function value for any weak contrast point belonging to DR is smaller or equal to min{ f (x); x ∈ D R }. In fact, because there are no more strong contrast points x ′ and x ′′ , there will be a propagation of function values coming from points that do not belong to DR. Then, the choice of a sequential family of Þlters leaves us a certain range of freedom, as expressed later. From the previous geometrical analysis and from property III.4, we have that, for φ 1 < φ 2: ξφε2n ≤ ξφε1n,φ2 ≤ ξφε1n

and ξφε2n,φ1 = ξφε1n ξφε2n = ξφε2n

(10)

In a family of MSFs, the strong contrast zones, according to a φ 1 parameter are passed to a greater subindex φ 2 (φ 2 > φ 1) level or eliminated. Moreover, let φ i be a family of parameters with i ∈ S = {1,2, . . . m} and such that φ j ≤ φ k for j < k,  n   εn  (11) = ξφε1n,...,φm ≤ ξφε1n . . . ξφ1 ξφεmn ≤ ξφεmn ξφεm−1

Then, if we apply a family of Þlters with parameters φ i between φ 1 and φ m, the output image obtained by ξφε1n,...,φm contains some strong contrast regions of the levels φ 1, φ 2, . . . , φ m employed for its computation, but with a greater contrast than φ m. This can be observed from the following: εn εn εn εn n ξφεmn ≤ ξφεm−1 ,φm ≤ ξφm−1 ≤ . . . ≤ ξφ2 ≤ ξφ1 ,φ2 ≤ ξφ1

(12)

n ξφεi−1 ,φi ( f ) preserves some strong contrast regions from f that are eliminated by ξφεin ( f ). On the other hand, if ξφε1n,φ2 ,...,φm contains some strong contrast regions of the levels φ 1, φ 2, . . . , φ m employed for the sequential family, then ξφε2n,...,φm does not contain details from ξφε1n , similarly, ξφεkn,...,φm does not contain details from all ξφεnj with φ j < φ k. εn n (13) ξφε1n,φ2 ,...,φm ≥ ξφε2n,...,φm ≥ . . . ≥ ξφεm−1 ,φm ≥ ξφm

Concerning the supports of the gradients, from property III.6, we know that in a Þltered image ξφε∞ , the support of the gradient (the set of points of the domain of deÞnition where this function is strictly positive) is the same than that of points of strong contrast Sφ . From the analysis presented in this section, we observe that there is an inclusion relation between the supports of the gradients of Þltered images. We will describe this inclusion relation in Section V.

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B. Invariants The invariants set (or roots) is a useful notion in mathematical morphology to characterize an idempotent transformation. In fact, for all idempotent transformations  there is an associated domain of invariance. We can see the invariants from a contrast point of view (Terol-Villalobos and Cruz-Mandujano, 1998; Terol-Villalobos, 1998). In our case, the class of invariants is given by the set of function βφε (for the Þrst Þlter) such that, for all function f ∈ βφε , we have ξφεn ( f ) = f . According to property III.6, this means that f has a well-deÞned contrast: that is, for each point x of the domain of deÞnition, gradi B ( f (x)) > φ or gradi B ( f (x)) = 0. However, another interesting interpretation can be done by means of results proposed in Section IV.A. Frequently, morphological Þlters posses a leftward absorption property  j i = i  j = i if i ≥ j (Serra, 1988). However, let us explain why our Þlters do not verify this notion and can be of great interest for increasing the contrast of images. First, we analyze the traditional opening in mathematical morphology given by γ B ( f ) = δ ∨ (ε B ( f )) B

which is an idempotent, increasing, and antiextensive transformation. Let λ and μ be two homothetic parameters with λ > μ. Then, γλB ( f ) ≤ γμB ( f ) and γλB (γμB ( f )) = γλB . Thus, we obtain similar results by transforming the function f by γλB or by applying in a sequential way γλB γμB . In our case, for φ1 < φ2 we have ξφε2n ( f ) ≤ ξφε1n ( f ) and   ξφε2n ( f ) ≤ ξφε2n ξφε1n ( f ) ≤ f

This means that in general, we cannot obtain similar results as for the opening by transforming f by ξφε2n or by applying it in a sequential way, ξφε2n ξφε1n . In fact, ξφε1n contrasts some regions of f in such a way that they remain unchanged εn directly on f. when ξφε2n is applied, but they are eliminated when we apply ξφ2 εn ε Then, the domain of invariance βφm associated to ξφm becomes a more interesting class. For a family of parameters φ i with i ∈ S = {1,2, . . . , m} such that φ j ≤ φk for j < k, we have:    n  ξφεmn , ξφεmn ξφεm−1 . . . . ξφε1n ( f ) ∈ βφεm εn εn εn ε n ξφεm−1 ,φm , ξφm−2 ,φm , . . . . , ξφ2 ,φm , ξφ1 ,φm ∈ βφm

ε n ξφε1n,φ2 ,...,φm , ξφε2n,... ,φm , . . . . , ξφεm−1 ,φm ∈ βφm

Similar comments can be made for Þlter ξφδn .

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V. Image Segmentation using MSFs In this section we will use the morphological slope Þlters as a tool for segmenting images. A. Homotopy ModiÞcation Eroding or dilating a region where the contrast is low and leaving it unchanged where contrast is high will ßatten the low-contrast region, thereby enhancing contrast and modifying the gradient homotopy. Property 6 shows that the output image will have a well-deÞned contrast,   ∀x ∈ Sφ ⇒ gradi B ξφεn ( f (x)) > φ

and

  ∀x ∈ SφC ⇒ gradi B ξφεn ( f (x)) = 0

Moreover, using sequential MSF, the output image obtained by ξφε1n,...,φm contains some contrast regions of levels φ1 , φ2 , . . . , φm employed for its computation. A low-contrast region at level φk of the family can be transformed in a highcontrast region with regard to φm . Then, the gray-level intensity at level φk of the gradient of the output image can be ampliÞed. In a Þltered image ξφεn , the set deÞned by the support of the gradient is the same than that of the points of strong contrast Sφ . This can be seen from property III.6. Using properties III.1, III.2, and III.3, we can express the next relations between the supports of the strong and weak contrast points. For a sequential family of parameters φi with i ∈ S = {1,2, . . . , m} and such that φ j ≤ φk for j ≤ k we have that ∀x ∈ Sφm ⊂ Sφ1 ,...,φm ⊂ Sφ1 ,

ξφεmn ( f )(x) = ξφε1n,...,φm ( f )(x) = ξφε1n ( f )(x) = f (x)

∀x ∈ Sφ1 ⊂ SφC1 ,...,φm ,

ξφε1n ( f )(x) = f (x) > ξφε1n,...,φm ( f )(x)

∀x ∈ Sφ1 ,...,φm ⊂ SφCm ,

 ε  ξφε1n,...,φm ( f )(x) = f (x) > ξφεmn ( f )(x) = ε B ξφmn−1 ( f ) (x)

In Figure 16a, the gradient support of the original image is shown. Figures 16b to 16e show the gradient supports of the Þltered images in Figures 15b to 15e (the image in Figure 15e has been obtained using {φ1 = 6, φ2 = 8, φ3 = 10, . . . , φ6 = 16}). We have that Sφ1 ,φ2 ,...,φm ⊇ Sφ2 ,...,φm ⊇ . . . ⊇ Sφm−1 ,φm ⊇ Sφm

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(a)

(b)

(d)

(c)

(e)

Figure 16. (a) Gradient support of original image in Figure 15a. (b) Gradient support of Þltered image in Figure 15b. (c) Gradient support of Þltered image in Figure 15c. (d) Gradient support of Þltered image in Figure 15d. (e) Gradient support of Þltered image in Figure 15e.

where Sφk ,φk+1 ,...,φm is the support of strong contrast points of ξφεkn,φk+1 ,...,φm (also the gradient support of ξφεkn,φk+1 ,...,φm ). Similar relations can be expressed from Eqs. (10)Ð(12). By observing the gradient supports of Þltered images in Figures 16b and 16c (ξφε1n and ξφε6n , with φ1 = 6, φ6 = 16), notice that some contours are preserved, especially for the Þrst Þltered image. The image in Figure 16d clearly shows why sequential Þlters depending of a family φi with i ∈ S = {1,2, . . . , m} give more interesting results. Some contours from level φ1 = 6 of the family have been preserved after applying the Þlter ξφε6n in a sequential way (ξφε1n,φ2 ,...,φ6 ). This inclusion relation between gradient supports can be used in order to minimize the effect of degradations for great values of the φ parameter. For example, using the gradient support image in Figure 16c as contour marks allows the recovery of some contours from the gradient support image in Figure 16e. Now, let us study the gray-level histogram of the gradient images computed only in the gradient support. Property III.6 shows that the gray level histogram of the gradient of Þltered images presents well-deÞned zones. Observe the histograms in Figure 17, obtained from the gradients of the original and Þltered images (Figures 15b, 15d, and 15e).

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Histogram 1200 1000 800 600 400 200 0 1

9

17

25

33

41

49

57

65 73

81

89

97 105 113 121 129 137

Intensity Figure 17. Histograms of gradient images computed from images in Figures 15b (black), 15d (gray), and 15e (white).

B. Image Segmentation Using the Watershed Transformation In the watershed-plus-marker approach, the computation of markers plays a fundamental role in solving the oversegmentation problem that occurs when a watershed is computed directly on an original image gradient. Because our Þlters have shown some properties in modifying image extrema (TerolVillalobos, 1996), we apply the watershed transformation directly on the Þltered image gradient. The new results obtained by a composition of MSFs, enable us to obtain intermediate results between both Þlters ξφε1n and ξφε2n (with φ1 < φ2 ). Then, it is interesting to apply the watershed transformation to these Þltered images (Terol-Villalobos and Cruz-Mandujano, 1998). We will not describe the watershed transformation. Some references about this subject can be seen in Beucher (1990) and Meyer and Beucher (1990). As expressed in Section III, our Þlters attenuate zones with weak contrast without affecting other regions, which means that incorrect minima (maxima), created by noise, and inhomogeneities are merged to form good minima (maxima). Moreover, we show in Section IV that it is possible to obtain intermediate results by applying MSF ξφεn (or ξφδn ) sequentially, using a family of parameters {φi }. In Figure 15d we show the case ξφε2n (ξφε1n ( f )) using two given parameters φ1 and φ2 with φ1 < φ2 (i.e., ξφε1n,φ2 ( f )). The original image (in Figure 15a) is an interesting scanning electron microscope image (SEM), with well-deÞned contours. From the Þltered images shown in Figures 15b and

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15c (using parameters φ = 6 and φ = 16, respectively), we note that some contours are preserved, especially for the Þrst Þltered image (see Figure 15b). The watershed computation successfully partitions the gradient image of the input image along its crests. In Figures 18a and 18b we illustrate the contours obtained by applying the watershed transformation to the images of Figures 15b and 15c. Now, let us use the intermediate images ξφε1n,φ2 that preserve more features than ξφε2n , but eliminate more features than ξφε1n . Figures 18c and

(a)

(b)

(c)

(d)

Figure 18. (a) Watershed of Þltered image in Figure 15b; (b) watershed of Þltered image in Figure 15c; (c) watershed of Þltered image in Figure 15d; (d) watershed of Þltered image in Figure 15e.

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18d show the watershed images computed from the gradient of two Þltered images shown in Figures 15d and 15e. From a segmentation point of view, using the Þlter ξφε1n,φ2 {φ1 = 6, φ2 = 16}, we observe intermediate results when we compare it with the watershed for ξφε1n and ξφε2n Þltered images in Figures 18a and 18b. Note the difference between the partitions obtained in Figures 18c and 18d. Nothing about inclusion relations between the different partitions, obtained by means of watershed, can be expressed. Clearly, the output partition obtained by the watershed transformation depends on the choice of φi values. A sequential family of MSFs ßattens low-contrast regions at each level of the family, enhancing some contrast regions which remain in the output image. This procedure enables us to obtain better results when we apply the watershed directly on the Þltered image.

C. An Image Segmentation Algorithm Using MSFs Another technique in MM for segmenting an image is the so called ßat zone approach (Crespo, 1993; Crespo et al., 1997). This technique provides a solution to the resolution problem that occurs under the traditional watershedplus-marker approach. In the watershed-plus-marker approach, some markers signal the location of the signiÞcant regions in the image. Locating each marker inside an image region poses a great problem when the features are small. Thus, the loss of small features using the watershed-plus-marker approach was the origin of the ßat zone approach (Crespo et al., 1997). In this section we will show a simple algorithm to extract homogeneous zones. Let us Þrst describe some methods in image processing, the quadtree method and the ßat zone approach, in order to compare our algorithm. 1. Quadtree Approach The quadtree approach has been a powerful tool in image processing for coding. This term is used to describe a class of hierarchical data structures whose common property is that they are based on the principle of recursive space decomposition. A complete study on this subject has been presented in a tutorial survey by Samet (1984). In the quadtree approach, the coding by regions is carried out by following a homogeneity criterion (or criteria) that enables us to discriminate whether a square region can be considered a connected component. Here we consider the geometrical construction of a quadtree in a square lattice. We start with a square frame of 2n pixels that is devised in four square zones as shown in Figure 19a. Each square zone is studied on the original image using one or several homogeneity criteria (variance, maxÐmin,. . .). If the homogeneity criterion (or criteria) is veriÞed, a function value is given at

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(a)

(b)

(c)

Figure 19. Image representation by quadtree structure.

all points on the square region (for example, the average of the intensity values in the square). For any square that does not verify the homogeneity criterion a similar procedure is performed in a recursive way by devising each square region by four. This procedure is illustrated in Figures 19b and 19c. The idea of coding a real-valued function (or binary) is linking to a recursivity property of a square lattice (see Figure 20 for a quadtree representation). The quadtree is a tool for representing images that is useful for description and processing images because of the hierarchical procedure. The output image is deÞned by a partition that will have well-deÞned zones corresponding to homogeneous zones in the input image.

Figure 20. Tree representation of different steps in Figure 19.

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2. Flat Zone Approach Let us write the ßat zones deÞnition (Serra and Salambier, 1993; Crespo, 1993). DeÞnition V.1 (Flat Zones) The ßat zones of a gray-level function f: E → T are deÞned as the (largest) connected components of pixels with the same function value. This is called the partition of ßat zones of a function. This ßat zone approach is based on the notion of connectivity as proposed by Salambier and Serra (1993). Note two important remarks: 1. The set of ßat zones of a function constitutes a partition of the space. Let Im be a ßat zone image of a gray-level function. We will denote by the set of ßat zones {Im}. 2. There is no restriction on the size of the ßat zones and they can be reduced to a single point. In the ßat zone approach, if a region (ßat zone) is to be preserved, then all its component pixels will be preserved. Otherwise, it is merged into another one in its entirety. In this approach, the deÞnition does not say how we process the ßat zones and does not state the property of transformations to be used (increasing, idempotent, . . .). The basic idea in this approach is to merge ßat zones according to several criteria looking for a good region-number reduction. In other words, the main objective of this approach is the computation of a good segmentation with a relatively small number of regions. The original images have a great number of ßat zones (connected components with the same function value). Then, the Þrst stage in this approach is to reduce the number of ßat zones by means of a Þltering stage. Connected Þlters (Þlters by reconstruction) are successful at simplifying features that are brighter or darker than their neighboring regions. A complete study about these connected Þlters is presented by Crespo et al. (1995). Because there exist transition regions after the Þltering stage, an intermediate stage that assigns these transition regions to one of their neighboring regions is performed. Finally, an image with a given number of zones is computed by merging different regions. The merging procedure is computed by using several criteria (area, region gradient, . . .) in order to decide which regions are merged to their neighboring regions. By comparing this segmentation approach with the quadtree representation, we can see that the goal is similar: both techniques look for homogeneous zones, reducing their number without changing the main features of the image in any considerable way. However, several differences exist between both techniques. The quadtree approach algorithm splits a nonhomogeneous region in four. The size and shape of the region in the quadtree

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approach are stated exactly. In the ßat zone approach, the main idea is the use of a merging process of regions where the size and shape are not strictly established. 3. A Segmentation Algorithm In this section we propose an algorithm for segmenting images that can be related to the two techniques described above for reducing the number of regions (Terol-Villalobos and Cruz-Mandujano, 1998). Our goal is to use MSFs to compute a relatively small number of homogeneous zones from the images. As for the ßat zone approach, the Þrst step is to reduce the number of regions using the following procedure. Although we use a one-dimensional function example as illustrated in Figure 21a, it allows us to understand the two-dimensional case. Initially, we look for the max and min values. Next, a thresholding operation is carried out at the middle value (see Figure 21b). Consider n connected components with max and min values. These are ßat zones

Max

Min

Max

(a)

Min

(b) Max

Min

(c) Figure 21. (a) Original function f and the maxÐminvalues; (b) thresholding operation at the middle value between max and min ones; (c) max and min approximation of f according to each ßat zone.

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Figure 22. (a) Original function and the maxÐmin approximation at step 1. (b) Original function and the maxÐminapproximation at step 2.

in accordance with deÞnition III.1 having a domain of deÞnition Di,max,min i = 1, . . . , n. Now, each connected component that belongs to a max or min region is approximated by the max and min values obtained from each region as illustrated in Figure 21c. The average intensity of pixels can also be used. We perform a similar procedure until step k is reached. Figure 22 illustrates the procedure for k = 1 and k = 2 for another function. To express this transformation with the morphological slope Þlters and the toggle mappings we perform the next procedure. Using property III.5 we have that   and φ1 = Sup gradi B ( f (x)) : ∀x ∈ D f   φ2 = Sup grads B ( f (x)) : ∀x ∈ D f

Thus,

ξφε1n ( f (x)) = εn B ( f (x))

and ξφδ2n ( f (x)) = δn B ( f (x))

Now, consider the case when idempotence is reached (nth step). By working in a similar way as the toggle mappings, we deÞne the next transformation: ⎧   ⎨δn ( f (x)) = max f (x) : x ∈ Dgeo if   [δn − f ](x) <[ f − εn ](x) W δnεn f (x)) = (14)  ⎩ εn ( f (x)) = min f (x) : x ∈ Dgeo otherwise

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(a)

(b)

(d)

(c)

(e)

Figure 23. (a) Original image, (b) Þrst step, (c) second step, (d) third step, (e) fourth step.

Note in Eq. (14) that we have replaced Df with Dgeo where Dgeo is a subset of Df. We observe that this is an idempotent mapping as the thresholding operation. Figure 23b shows the transformed image obtained from the original in Figure 23a using Dgeo = Df. This is a traditional operator in mathematical morphology called the morphological contrast operator (Kramer and Bruckner transformation). However, we do not simply use it for a given number of iterations, but we rather look for idempotence (nth step). Now, we iterate this transformation [Eq. (14)] in a geodesic way. Here, we work with gray-level images, using a binary geodesic mask Dgeo, where a transformation will be applied. In other words, Dgeo is the domain of deÞnition of the transformation [see Eq. (14)]. The elementary geodesic dilation and erosion are given by:    Sup f (y) : y ∈ Bx ∩ Dgeo if x ∈ Dgeo 1 δ Dgeo ( f (x)) = f (x) otherwise    Inf f (y) : y ∈ Bx ∩ Dgeo if x ∈ Dgeo ε 1Dgeo ( f (x)) = f (x) otherwise

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where Bx is the elementary ball centered at point x. The gray-level geodesic dilation and erosion of size n are given by (1) (1) δ (n) Dgeo ( f ) = δ Dgeo • · · · • δ Dgeo ( f ) n times

(1) (1) ε(n) Dgeo ( f ) = ε Dgeo • · · · • ε Dgeo ( f ) n times

These transformations are applied not only in a simple region Dgeo ⊂ D f , but in a partition of Df. Let Im be an image composed by different ßat zones and {Im} its associated partition. Each ßat zone corresponds to a given geodesic mask Dgeo. The function Im is called a geodesic mask image. We shall work using two main remarks: 1. The function Im is composed of a set of ßat zones obtained from the original image. 2. Each element of {Im} is used as an independent geodesic mask, where Eq. (14) is applied. Observe the image for the Þrst step in Figure 23b with {Im} = Df. Note the thresholding operation using the maxÐmin criterion. In the second step, the transformation at Eq. (14) is applied conditionally to each white and black region in Figure 23b. Thus, this geodesic function Im will be obtained by applying Eq. (14). The original function in Figure 23a is approximated by a geodesic transformation as illustrated in Figure 23c in the second step. This image is later used as the image Im in order to perform the next step for obtaining the image illustrated in Figure 23d. Figure 23e shows the image obtained for four steps. This procedure recalls the quadtree approach. However, there is an important difference because the partition zones in our procedure are not square regions with a given size 2k, 2k+1, . . . , but geodesic regions. Scanning of square regions is simple, but for geodesic regions it is necessary to use a special data structure. We employ a single queue that is a frequently used algorithm in mathematical morphology (Vincent, 1993). Note that we abandon the region marker concept and the computation of the gradient operator. Note also that the transformed image will be used as the mask function for the next iteration. We perform a similar procedure until step k or using a stop criterion (in this work maxÐmin< Valthresh). This procedure is illustrated in Figure 23 where four segmenting results corresponding to four steps are shown (using Valthresh = 20). At each step we obtain a Þner partition. Let us remember that a partition {Ai} is said to be Þner than another partition {Bi} if any pair of points belonging to the same class Ai also belong to a unique partition class Bj. By assigning a zero value to Valthresh and by iterating this procedure until stability is reached, we will Þnd the original image.

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(a)

(b)

(c)

(d)

Figure 24. (a) Threshold of the gradient of image 23a (4 to 255 gray levels); (b) threshold of the gradient of image 23b; (c) threshold of the gradient of image 23c; (d) threshold of the gradient of image 23d.

In order to show the boundary of each ßat zone, we apply an internal gradient on each image in Figures 23a to 23d. Next a thesholding operation between 4 and 255 gray levels is applied on the gradient images. In Figure 24 we observe that the ßat zone contours clearly show the partition of the ßat zone images. A second step is performed in a similar way as the ßat zone approach. An image with a relative number of regions is computed using a merging procedure. An useful data structure to perform this procedure is a graph whose vertices are linked to the homogeneous zones and whose edges describe adjacency between those zones. Such a representation was used for

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Figure 25. Image approximation using minÐmax and area criteria. (a) Original image; (b) image approximation using a split procedure for step 3; (c) merge procedure using an area criteron of 30 (using image in Figure 25b); (d) area criterion of 40 (using image Figure in 25b); (e) area criterion of 50 (using image in Figure 25b).

connected morphological operators (Potjer, 1996). The merging procedure is computed by using an area criterion in order to decide which regions are merging to their neighboring regions. In Figures 25c to 25e we present a set of images obtained by our procedure using maxÐmin and area criteria. The original image shown in Figure 25a has 1810 ßat zones. The image in Figure 25b has 110 ßat zones obtained after applying the split procedure (three steps). Finally, Figures 25c, 25d, and 25e have been obtained using an area criterion of 30, 40, and 50 pixels, respectively. The number of regions in these images are 23, 19, and 17, respectively. This allows a selection of the ßat zones by both criteria: contrast and size. The maxÐmin criterion splits a nonhomogeneous region and the size criterion merges the small ßat zone to another one in its entirety as in the ßat zone algorithm. In this case we combine both notions, split and merge, to obtain the output image.

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VI. Nonlinear Multiscale Approach Using a Sequential Family of MSFs A. A Geodesic Approach From the stability study made in Section III.C, a geodesic approach can be deÞned by means of a set of strong contrast points Sφ (or its complement composed of the weak contrast points). Then, by thresholding the gradient image, we obtain the geodesic mask SφC, where the original image will be transformed by erosion or dilation transformations. Moreover, for sequential MSF with parameters φ i with i ∈ S = {1, 2, . . . , m}, a similar threshold procedure can be performed at each level of the family. This process enables us to use a geodesic mask to generalize Eqs. (7) and (8) and to use an efÞcient algorithm to process the image using MSF. In fact, other gradient deÞnitions or Þltered gradient images can be used for obtaining the geodesic mask to build other Þlters. This is not directly possible from the following relationship:  δλB ( f (x)) if gradeλB ( f (x)) ≤ φ δλ (15) ξφ ( f (x)) = f (x) otherwise This transformation has a different behavior than Eq. (7), which uses an elementary structuring element. For λ > 1, Eq. (15) deÞnes a conditional transformation and not a geodesic one. Consider two points y and y ′ on an image such that f (y ′ ) ≤ f (y). The function value f (y ′ ) cannot propagate to point y if there is not a path of weak contrast points. When λ > 1, a function value can propagate even if a path of weak contrast points is not present. Figures 26a, 26b, and 26c show the original image and the Þltered images using ξφδn ( f ) with φ = 8 and φ = 6, respectively. Figure 26d illustrates the Þltered image using ξφδλn ( f ) with λ = 2 and φ = 12. By comparing the Þltered images using ξφδn ( f ) in Figures 26b and 26c to the Þltered image ξφδλn ( f ) in Figure 26d, we observe a greater contrast enhancement in ξφδλn ( f ) than in ξφδn ( f ). Now, let us brießy study other gradient deÞnitions. First, consider the directional gradient case where the structuring element B is given by a segment of length l in a given direction α. In this case, the Þlters given by Eqs. (7) and (8) are built using linear dilations (or erosions) in order to calculate directional gradients. In this example, zones of weak contrast are attenuated using an elementary structuring element. To protect the edges from the noise, a Þltering processing in a perpendicular direction for each directional gradient is applied. This enables us to retain the most signiÞcant contours at each direction. Finally, the supremum of directional gradients is computed. The image in Figure 26e has been Þltered using this procedure. A sequential family of these directional transformations with parameters φ i = {9, 10, 11, 12, 13} was used. When

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Figure 26. (a) Original image, (b) MSF ξφδn ( f ) with φ = 8, (c) MSF ξφδn ( f ) with φ = 6, δ (d) ξφλn ( f ) with λ = 2 and φ = 12, (e) MSF using directional gradients with φ i {9,10,11,12,13}, size 2, (f) MSF using Sobel gradient with φ = 4.

comparing the quality of Þltered images in Figures 26b and 26c with that in Figure 26e, we observe that the Þltered images in Figures 26b and 26c are better contrasted. However, some zones are merged. This fact leads to a loss of quality. In contrast, the Þltered image in Figure 26e presents well-deÞned contrast and well-deÞned zones. Finally, in Figure 26f the Sobel gradient was used as a contrast criterion to build the MSF. It is well known that the Sobel gradient is one of the most interesting operators for detecting contours. Consequently, this is the reason why the output image has an excellent quality. This operator is given by the masks ⎡ ⎡ ⎤ ⎤ −1 0 1 1 2 1 ⎣−2 0 2⎦ ⎣0 0 0⎦ −1 0 1 −1 −2 1 These detect vertical edges and horizontal edges, respectively. However, this operator is an empirical gradient. This is a major drawback and it is the reason

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(a)

(b)

Figure 27. (a) Original function; (b) MSF output.

why it is not possible to ensure that properties III.1 to III.6 will be veriÞed when this gradient is used as a criterion. Next, a ßat zone approach will permit the proposal of other criteria that will allow us to obtain better results by preserving theoretical properties.

B. MSF Using Flat Zone Notion: Flat Zone Gradient, Graphs, and Connected Operators In general, our Þlters are not connected operators. However, the algorithm for segmenting images proposed in Section V.C is connected. By deÞnition an operator is connected if and only if it extends the input image ßat zones. In other words, connected operators do not break ßat zones. The morphological slope Þlters are not connected Þlters. In Figure 27 we observe this behavior. Figure 27a illustrates the input function, while in Figure 27b, the output function of the MSF (using internal gradient deÞnition) is shown. Point p is a strong contrast point. The contour of strong contrast is preserved, but the ßat zone is broken. Then, in order to create a connected operator and to preserve a well-deÞned contrast (contrast invariants set), different gradient deÞnitions must be used. Since a connected operator does not split components of the level sets, connected operators must act on the level of ßat zones rather than on the pixel level. Thus, it is clear that the ßat zone notion, and not the pixel concept, will be used to deÞne a gradient. In order to use a gradient deÞnition using the notion of ßat zone a study made by Vincent (1989), concerning MM and graphs, will be used. This author deÞnes the main morphological operators on graphs (on partitions). In his work many morphological transformations such as dilation, erosion, opening, closing, and also reconstruction transformations are deÞned on graphs. He also introduced the notion of connected components, or more strictly speaking the notion of a connectivity class, directly into graphs.

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On the other hand, work by Potjer (1996) describes region adjacency graphs and connected operators. Potjer shows that the action of connected operators can be given in terms of region adjacency graphs. These studies enable us to deal with ßat zone operators in the gray-level case. In fact, since the ßat zone notion allows us to partition an image, dilation and erosion on graphs can be established for gray-level images. In other words, a partition is a subdivision of the underlying space into disjoint zones. DeÞnition VI.1 (Partition) Given a space E, a function P : E → ℘ (E) is called a partition if: (a) x ∈ P(x), x ∈ E

(b) P(x) = P(y) or P(x) ∩ P(y) = ∅, for x, y ∈ E where ℘(E) denotes the collection of subsets of E and P(x) is the zone of P that contains x. Therefore, it is possible to use both concepts on graphs: the ßat zone notion and morphological transformations. We will use ßat zone notation to describe the morphological operator for graphs, and the notation of Vincent will be avoided. However, all morphological transformations on graphs deÞned in this work are the same as those proposed by Vincent, but on the gray-level case. To the authorÕs knowledge, this topic (the gray-level case) has not been previously treated. Let us express some useful deÞnitions to introduce the MSFs on ßat zones. The following deÞnition of connectivity is due to Serra (1988): DeÞnition VI.2 (Connectivity Class) on the subsets of a set E when:

A connectivity class C is deÞned

(a) ∅ ∈ C and for all x ∈ E, {x} ∈ C.

(b) For all families Ci in C, ∧ Ci != φ ⇒ ∨ Ci ∈ C. i

i

This deÞnition is equivalent to the deÞnition of a family of connected pointwise openings {γx , x ∈ E} associated to each point of E: Theorem VI.1 (Connectivity Characterized by Openings) The deÞnition of a connectivity class C is equivalent to the deÞnition of a family of openings {γx , x ∈ E} such that: (a) ∀x ∈ E, γx ({x}) = {x}

(b) ∀x, y ∈ E and A ⊂ E, γx (A) = γ y (A) or γx (A) I γ y (A) = ∅

(c) ∀x ∈ E and A ⊂ E, ∀x ∈ / A ⇒ γx (A) = ∅

When the operation γ x is associated with the usual connectivity in Z2, the opening γ x(A) can be deÞned as the union of all paths containing x that are

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included in A. Thus, when a space is equipped with the opening γx, connectivity issues in E can be expressed using γ x. A set A ⊂ Z 2 is connected if and only if γx (A) = A. DeÞnition VI.3 Let f be a function f : Z 2 → Z . The set of ßat zones at gray level t is given by: Z t ( f ) = {x : f (x) = t} All propositions in this study are based upon this threshold deÞnition. As previously expressed, the ßat zones of a numerical function are deÞned as the (largest) connected components with the same gray level. Let us give another deÞnition. DeÞnition VI.4 (Flat Zone) A ßat zone Fx at the gray level t of a function f is a connected component of Zt( f ), i.e., Fx = γx (Z t ( f )). We observe that (a) x ∈ Fx , and (b) for x, y ∈ D f , Fx = Fy or Fx ∩ Fy = ∅. Then, the ßat zone notion partitions the image. DeÞnition VI.5 Let x be a point of E equipped with γ x. The set of adjacent ßat zones Ax to Fx is given by A x = {Fx ′ : x ′ ∈ Z 2 , Fx ∨ Fx ′ = γx (Fx ∨ Fx ′ )} Now, let Pf be the partition of the domain of deÞnition Df induced by f by means of the ßat zone concept. Since the gray-level image is now formed by the function f and the partition Pf induced by f, the morphological operators must work on pairs ( f, Pf). We will deÞne the element ( f, Pf)(x) as the gray-level value of the connected element Fx = γx (Z t ( f )). The morphological dilation and erosion applied on ßat zones are given by: δ(( f,P f ))(x) = max{( f,P f )(y), Fy ∈ A x ∪ Fx }

ε(( f,P f ))(x) = min{( f,P f )(y), Fy ∈ A x ∪ Fx }

(16)

Thus, the dilation or erosion value on the ßat zone Fx will be given by the respective maximum or minimum gray-level value of the gray-level values of components formed by the ßat zones adjacent to Fx. Because the notion of a structuring element for the morphological operators does not exist for graphs, this element is eliminated from the dilation and erosion deÞnitions. Observe that the partition of the original image Pf is always used for computing the transformations. For example, to calculate the opening on the ßat zone (the erosion following by a dilation), this transformation will be given by γ (( f,P f ))(x) = δ((ε( f,P f ), P f ))(x) The dilation is computed on the pair (ε( f,P f ), P f ) . Images in Figures 28a and

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Figure 28. (a) Morphological erosion, (b) morphological dilation, (c) ßat zone erosion, (b) ßat zone dilation.

28b illustrate the morphological erosion and dilation; images in Figures 28c and 28d show the ßat zone erosion and dilation, respectively. Since we work with the pair ( f, Pf), one could write the output image by the pair (ε(( f , P f )), P f ) or (ε, Pf) instead of ε(( f,P f )). However, for the sake of simplicity, the output image is given by ε(( f,P f )). In a similar way, this convention is used for any transformation proposed in this section. The duality between the dilation and erosion is preserved. It is only necessary to specify the complement of the pair ( f, Pf). The complement of a function f was previously deÞned by f C (x) = gl max − f (x), where gl max = 255.

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Figure 29. (a) Flat zone at gray level 40 and four adjacent ßat zones; (b) erosion value; (c) internal gradient value.

Thus, the complement of the pair ( f, Pf) is given by ( f c , P f c ). However, the partition induced by f using the ßat zone notion is the same that the partition induced by f c : P f = P f c . Therefore, we have that  c δ(( f,P f ))(x) = ε(( f c ,P f ))(x) By means of the erosion and the dilation on the ßat zone, the internal and external gradients for the ßat zone image can be deÞned by grade(( f,P f ))(x) = δ(( f,P f ))(x) − ( f,P f )(x) gradi(( f,P f ))(x) = ( f,P f )(x) − ε(( f,P f ))(x)

(17)

Figure 29 illustrates the internal gradient for ßat zones (graphs). Consider the ßat zone at gray level 40 adjacent to four ßat zones (Figure 29a). The erosion value is shown in Figure 29b; Figure 29c illustrates the internal gradient value of the ßat zone. Now, by using the deÞnitions given by Eqs. (7) and (8), we obtain the following deÞnitions:  ε(( f,P f ))(x) if gradi(( f,P f ))(x) ≤ φ ε1 ξφ (( f,P f ))(x) = ( f,P f ) (x) if gradi(( f,P f ))(x) > φ  δ(( f,P f ))(x) if grade(( f,P f ))(x) ≤ φ ξφδ1 (( f,P f ))(x) = ( f,P f )(x) if grade(( f,P f ))(x) > φ At the nth step, when stability is reached (n → ∞):  ε  ξφεn (( f,P f ))(x) = ξφε1 ξφn−1 (( f,P f ))(x)   δ ξφδn (( f,P f ))(x) = ξφδ1 ξφn−1 (( f,P f ))(x)

(18)

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Figure 30. (b) Filtered image ξφε1n with φ1 = 6; (b) Þltered image ξφε2n with φ1 = 16; (c) Þltered image ξφε1n ,φ2 with φ1 = 6, φ2 = 16; (d) Þltered image ξφε1n ,...,φ6 with φ1 = 6, φ2 = 8, . . . , φ6 = 16; (e), (f), (g), and (h) ßat zone gradients of Þltered images in Figures 30a, 30b, 30c, 30d, respectively.

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In Figure 30 the MSF using ßat zone notion is illustrated. Figures 30a and 30b show the output images using φ = 6, φ = 16, respectively; Figures 30c and 30d illustrate the sequential MSF. Compare these images with those illustrated in Figure 15. Observe that more contours are preserved in the output images when the ßat zone gradient is used as a criterion. Figures 30e to 30h show the ßat zone gradients of the Þltered images 30a to 30d, respectively. As a result, all the propositions presented in Sections III, IV, and V can be expressed for these gradient deÞnitions. Using the ßat zone notion to build the MSF enables us to obtain intermediate results with regard to the MSF using morphological internal and external gradients [Eqs. (7) and (8)]. For example, the output image of ξφεn using a morphological internal gradient is illustrated in Figure 27b, while the output image of ξφεn using the ßat zone gradient would be the same input function shown in Figure 27a. In other words, in this example the input function is an invariant of the MSF using ßat zone gradient as a criterion. Properties III.1 to III.6 are the same, but the notion of weak contrast point and high contrast point is changed by the ßat zone notion. We only express properties III.1 and III.2 for the MSF using a ßat zone gradient. Property VI.1 Let Sφ ⊂ P f be the set of ßat zones such that ξφε1 (( f,P f )) (x) = ( f,P f )(x) with gradi(( f,P f ))(x) > φ. Then ξφε∞ (( f,P f ))(x) = ( f,P f )(x) ∀Fx ∈ Sφ

The set Sφ deÞnes one support of the ßat zones of strong contrast. The set of weak-contrast ßat zones will be characterized by the following property. ε

Property VI.2 ∀Fx ∈ SφC ⇒ ξφk+1 (( f,P f ))(x) = ε(ξφεk (( f,P f )))(x) where SφC is the complement of the set Sφ . SφC is the set of weak contrast ßat zones. Then, at the Þrst iteration of this operator, all ßat zones are classiÞed in two categories and remain in them at each iteration. Similarly, a sequential MSF using ßat zone gradients can also be built. C. Nonlinear Multiscale Representation using MSF 1. Multiscale Representation A multiscale representation will be completely speciÞed, if one has deÞned the transformations going from a Þner scale to a coarser scale. Basically, the goal of multiscale analysis is the computation of a family of descriptions depending on a parameter, called the scale-space parameter. In general, the objects which have to be detected or recognized in an image belong to one scale, and all remaining objects, to be discarded, belong to another scale. Frequently,

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however, such a separation of scales is not possible, and the information is presented at several scales. As a consequence, multiscale approaches have been developed where a series of coarser and coarser representations of the same image are derived. Several important properties are taken into consideration: (a) invariance by translation, (b) invariance by rotation, and (c) invariance under illumination change. Other requirements on the effect of the transformation itself must be added: (d) The transformation should really be a simpliÞcation of the image: some information has to be lost from one scale to the next. (e) It should not create new structures at coarser scales; that is, it should not create new extrema (minima or maxima). (f) Causality: coarser scales can only be caused by what happened at Þner scales. Some publications that present ideas more or less similar to those proposed here are concerned with diffusion processes. The link between diffusion processes and image analysis began with the multiscale description of images. The Gaussian family is the multiscale paradigm within linear Þltering. The Gaussian family of an input image f is obtained by means of its convolution with Gaussian kernels of different variance σ , symbolized by fσ = f ∗ G σ

(19)

where ∗ represents the convolution operator. The variance σ is the scale space parameter. The larger the σ , the coarser the scale. The equivalence between Eq. (19) and the isotropic diffusion was established by Koenderink (1984) and Hummel et al. (1987) using the following equation: ∂f = c f, ∂t where the boundary value f t=0 is equal to the input image f, and  is the Laplacian operator. Perona and Malik (1987, 1989) consider the anisotropic diffusion equation given by ∂f = ∇(c∇ f ) ∂t in which c is not a constant, but a function of position, and the parameter t. ∇ is the gradient operator. The goal of Perona and Malik was to perform smoothing within the image regions and to prevent blurring of the image edges. They used the gradient ∇ f in order to estimate the edge condition of an image pixel: c = g(∇ f ). The expression g(∇ f ) = exp(−(|∇ f |/k)2 )

(20)

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Figure 31. (a) Blurred step edge; (b) asymptotic result for stable edge.

was used as an adaptive smoothing algorithm. The adaptive smoothing algorithm proposed by Saint-Marc et al. (1991) implements an anisotropic diffusion process by means of Eq. (20). The formula employed is a weighted average. In each iteration, a local 3 × 3 pixel weighted average is computed at each image pixel. The weights in the averaging mask are inversely proportional to the likelihood of the pixels under the mask being edge pixels. Saint-Marc et al. (1991) showed the following result when the averaging coefÞcients are computed using the exponential function: let f 0 (x) be a one-dimensional blurred continuous step edge and let x0 be the zero of its second derivative; when |∇ f 0 (x0 )| < k, then |∇ f 0 (x0 )| decreases as t increases; whereas if |∇ f 0 (x0 )| > k, then |∇ f 0 (x0 )| increases as t increases (the edge is preserved). This is illustrated in Figure 31. Crespo (1993) presents a complete study of diffusion processes and introduces a modiÞcation that treats edge pixels differently from the rest of the pixels. The morphological gradients were used to decide which pixels were treated differently. After applying Saint-MarcÐChenÐMedioniÕ s averaging step, pixels that belong to a set of edge pixels that have been extracted from the input image are strengthened by the Crespo algorithm. In mathematical morphology, the basic ingredients of all multiscale morphological operators are the dilations and erosions of increasing size. However, dilations and erosions by themselves cannot be used for representing the successive scales, because they displace the contours. A powerful class of morphological Þlters that can preserve contours are the openings and closings by reconstruction [see Eq. (3)]. They can reconstruct whole objects with exact preservation of their boundaries and edges. In this reconstruction process, the original image is simpliÞed by completely eliminating smaller objects inside which an increasing criterion (erosion or dilation criteria) cannot Þt. However, Þlters by reconstruction treat the image foreground and background

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asymmetrically. A solution to eliminate this problem was proposed by Meyer and Maragos (2000). Based on the notion of levelings (Meyer, 1998), a new general nonlinear scale-space representation was proposed with several interesting features. The main features of this multiscale approach take into consideration contour preservation and no spurious extrema generation. Several criteria are proposed in the word of Meyer and Maragos. Among them, a slope criterion to build multiscale levelings is mentioned. The output images of these transformations using a slope criterion will have well-deÞned contrast according to the gradient notion. The morphological slope Þlters treat the image foreground and background asymmetrically, as is the case with the Þlters by reconstruction. It is possible to combine both Þlters ξφεn and ξφδn (applied on pixels or on ßat zones), but we will work the foreground and the background in separate ways. Properties required for a multiscale approach are satisÞed by the MSF. In particular, the requirements on the effect of the transformation itself given by (d), (e), and (f) are veriÞed. Consider the requirement given by (d). A particular form of simpliÞcation concerning our transformations is the contour elimination: at any scale change, the edge information (gradient support) at the coarser scale given by the φ parameter is always lower than the edge information at the Þner scale. This has been shown earlier (see Figures 15 and 16 for {φ1 = 6, φ2 = 8, φ3 = 10, . . . , φ6 = 16}). The following inclusion relations between the gradient supports express this characteristic of multiscale processing: Sφ1 ⊇ Sφ2 ⊇ · · · ⊇ Sφm−1 ⊇ Sφm Another form of simpliÞcation is expressed by the luminance. The luminance of the coarser scale is always lower than the luminance at the Þner scale. Using property III.4 we have ∞ ( f (x)) ≥ ξφεm∞ ( f (x)) ξφε1∞ ( f (x)) ≥ ξφε2∞ ( f (x)) ≥ . . . ≥ ξφεm−1

However, image simpliÞcation from a contrast (gradient) point of view is not respected. Concerning requirement (e) on the effect of transformations, the MSF ξφεn does not create new minima and the MSF ξφδn does not create new maxima. Furthermore, if the goal is image segmentation, one may require that contours remain sharp and not displaced. This goal is veriÞed by the MSF: if the point x of the output image ξφεn is an edge point, then x is also an edge point of the input image. In other words, morphological slope Þlters preserve contours, and more speciÞcally, MSFs preserve well-deÞned vertical edges (well-deÞned contrast) as well as horizontal contours. Figure 32 illustrates luminance simpliÞcation using the MSF, with the ßat zone gradient criterion, by preserving contours. Now, concerning requirement (f), the coarser scales in the sequential MSF are caused by what happened at Þner scales.

(a)

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(f)

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Figure 32. (a), (b), (c), and (d) MSF ξφεn using ßat zone gradient as a criterion for φ = 6, φ = 8, φ = 10, φ = 12, respectively. (e), (f), (g), and (h) the watershed of the ßat zone gradient of images (a), (b), (c), and (d), respectively.

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One has to be careful with the relations between the various scales. Many scale representations in the literature verify a semigroup property. That means, if ψs is the representation at scale s of an image, then the representation at scale t of ψs should be the same as the representation at scale s + t of the input image: ψs+t = ψt (ψs ). In mathematical morphology another structure is used to introduce order relations. Frequently, morphological Þlters have the absorption property and the following semigroup property is satisÞed: ψt (ψs ) = ψs (ψt ) = ψmax(m,n) . A morphological slope Þlter does not satisfy the absorption property, as seen in Section IV.B. For φ1 < φ2 we have that ξφε2n ξφε1n ( f ) != ξφε1n ξφε2n ( f ) = ξφε2n ( f )

and ξφε2n ( f ) ≤ ξφε2n (ξφε1n ( f )) ≤ f

This means that in general, we cannot obtain similar results by transforming f by ξφε2n , or by applying it in a sequential way, ξφε2n ξφε1n . However, if one chooses to use sequential MSF, the requirements listed previously can be satisÞed. Furthermore, for a given set of invariants, different approaches for segmenting images can be employed. For example, the family of Þlters εn εn ε n ξφεmn , ξφεm−1 ,φm , . . . . , ξφ2 ,...,φm , ξφ1 ,φ2 ,...,φm ∈ βφm

enables us to go from a coarser segmentation to a Þner one, whereas the family of Þlters ε n ξφε1n,φm , ξφε2n,φm , . . . . , ξφεm−1 ,φm ∈ βφm

takes us from a Þner segmentation to a coarser one as illustrated in Figure 33. Next, the form of simpliÞcation expressed by the luminance is treated and introduced to build other families of MSF. SpeciÞcally, morphological gradient (using the pixel notion) and ßat zone gradient are studied. 2. Weighted Morphological Slope Filters Sequential MSF allows the possibility of looking for intermediate results between two given parameters φ1 and φm . Thus, the sensibility of the MSF to the parameter φ is attenuated and it can be controlled. However, in some cases it is impossible to attenuate this drawback by means of sequential MSF, as is illustrated in Figure 34. Figure 34b shows the output image of Þlter ξφεn with φ = 4; Figure 34c illustrates the output image of the Þlter ξφεn with φ = 5. Notice the great difference between these output images. Whereas image in Figure 34b is practically similar to the original one, the image features of the original image have been completely changed in the output image in Figure 34c. Therefore, in this case the sequential MSF cannot be used in order to obtain intermediate results. This is shown in Figure 34d.

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(a)

(b)

(c)

(d)

εn with parameters φ1 = 6, φ2 = 12 using ßat zone Figure 33. (a) Sequential MSF ξφ1,φ2 εn gradient; (b) sequential MSF ξφ1,φ2 with parameters φ1 = 10, φ2 = 12 using ßat zone gradient; (c) and (d) watersheds of the ßat zone gradient from images (a) and (b), respectively.

This sensibility is created by some conÞgurations of the blurred edge, as shown in Figure 35. When applying an MSF ξφεn , the region of the blurred edge at higher gray level is attenuated by the erosion before the propagation of the slopes, coming from lower gray levels, hits this edge region at higher gray level (see Figure 35b). Then, even if a high-contrast region is created, a better contrast region in this example would be given by that illustrated in Figure 35c. A Þrst solution is the use of the ßat zone gradient.

MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION

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(b)

(c)

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Figure 34. (a) Original image; (b) Þltered image ξφεn with φ = 4; (c) Þltered image ξφεn with φ = 5; (d) Þltered image ξφε1n ,...,φ5 with φ1 = 1, φ2 = 2, . . . , φ6 = 5.

In Figure 36, the output images of Þlters ξφεn using a ßat zone gradient as criterion are illustrated. Images in Figures 36a, 36b, and 36c were obtained from the input image in Figure 34a. The output image in Figure 36a was obtained by the Þlter ξφεn with parameter φ = 8; images in Figures 36b and 36c were computed by sequential MSF. Even if it is possible to obtain intermediate results, we observe that there exists a severe degradation of the images. It could be interesting to weight the gradient with respect to some behavior of the edge. We look for a structural approach and not for an adaptive smoothing

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(a)

(b)

(c)

Figure 35. (a) Blurred edge; (b) output edge using MSF; (c) output edge using weighted MSF.

algorithm as that proposed by Saint-Marc et al. (1991). We know that luminance of an object is independent of the luminance of the surrounding objects, and contrast of an object depends on the luminance of the surrounding. In fact, according to WeberÕs law, if the luminance of an object f o is just noticeably different from the luminance of its surrounding f s , then their ratio (| f o − f s |/ f o ) = constant. Thus, it seems that a gradient weighted with respect to the gray-level intensity is in agreement with the notion of contrast in an image. The idea is that visible edges (good gradient) at lower gray levels on the original image are not detected at higher gray levels. Therefore, they must be treated differently: not only according to the gradient image, but also taking into consideration the gray level in the image. Since the edges that are not

(a)

(b) ξφεn

with φ = 8, (b) Þltered image Figure 36. (a) Filtered image φ2 = 8, (c) Þltered image ξφε1n ,...,φ7 with φ1 = 7, φ2 = 8, . . . , φ7 = 13.

(c) ξφε1n ,φ2

with φ1 = 7 and

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visible on the original image can be visible in the image complement (the image negative), they can be treated with the dual transformation. On the other hand, by looking at Figure 35a, it will be possible to weight the lower and higher gray levels when applying ξφεn in such a way that an output edge like that illustrated in Figure 35c is obtained. Consider the following operators:  ε B ( f (x)) if f (x)[gradi B ( f (x))] ≤ φ ε1 ξφ ( f (x)) = f (x) if f (x)[gradi B ( f (x))] > φ  δ B ( f (x)) if f c (x)[grads B ( f (x))] ≤ φ ξφδ1 ( f (x)) = f (x) if f c (x)[grads B ( f (x))] > φ At the nth step, when stability is reached (n → ∞):    δ  ε and ξφδn ( f (x)) = ξφδ1 ξφn−1 ( f (x)) (21) ξφεn ( f (x)) = ξφε1 ξφn−1 ( f (x))

We will avoid a change in notation of MSFs, even if a weighted gradient is used as criterion. We will only specify it in the examples where a gradient criterion is used. Then, when applying the ξφεn operator, using a weighted gradient, the slopes at higher gray levels are weighted in such a manner that they will remain unchanged, while slopes at lower gray levels are contrasted. Inversely, when using the operator ξφδn , the slopes at lower gray levels are weighted in such a way that they remain unchanged, while slopes at higher gray levels are contrasted. In Figure 37, the behavior of a weighted gradient used as a criterion is illustrated. In Figures 37a, 37b, and 37c, Þltered images using a gradient weighted with respect to the gray-level intensity are illustrated for φ = 400, φ = 600, and φ = 1000, respectively; Figures 37d, 37e and 37f illustrate sequential MSF using a gradient weighted with respect to gray-level intensity. The output images show that better results can be obtained when sequential MSFs are used. Mainly, compare the output image in Figure 37f computed by a family of Þlters with parameters between φ1 = 200, φ2 = 210, . . . , φ81 = 1000 with that in Figure 37d using a family of Þlters with parameters φ1 = 200, φ2 = 300, . . . , φ9 = 1000. Now, consider MSFs using the notion of ßat zone gradients that have a better Þltering characteristic than MSFs using the morphological gradient. We can also weight these Þlters with respect to gray-level intensity in order to obtain a better control of the output images. Figure 38 illustrates the output images of the MSF using a ßat zone gradient for the same parameters as in Figure 37, where a morphological gradient was used. Observe the sequential Þltered images in Figures 38d, 38e and 38f. When comparing these images with those in Figures 37d, 37e and 37f we notice the generation isolated edges by MSF. In fact, this behavior was observed above in Figure 27b and it was the main reason for introducing the ßat zone notion to MSF.

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 37. (a) Filtered image using a gradient-intensity criterion (φ = 400); (b) Þltered image using a gradient-intensity criterion (φ = 600); (c) Þltered image using a gradient-intensity criterion (φ = 1000); (d) sequential Þltered image using a gradient-intensity criterion (φ1 = 200, φ2 = 300, . . . ,φ9 = 1000); (e) sequential Þltered image using a gradient-intensity criterion (φ1 = 200, φ2 = 250, . . . , φ17 = 1000); (f) sequential Þltered image using a gradient-intensity criterion (φ1 = 200, φ2 = 210, . . . ,φ81 = 1000).

Finally, we also stated earlier that morphological slope Þlters treat the image foreground and background asymmetrically like the Þlters by reconstruction. However, it is possible to combine both Þlters ξφδn (( f,P f )) and ξφεn (( f,P f )) using a family (similar for the MSF on pixels) ξφδn ξφεn of alternating sequential MSFs of parameters {φi }. This approach enables us to treat the foreground and background of the image, although the approach is not asymmetrical. Images in Figures 39a, 39b, and 39c illustrate these alternating sequential MSF using the same parameters as those in Figures 38d, 38e, and 38f. The dual morphological slope Þlter ξφδn was applied to perform the segmentation of magnetic resonance imaging (MRI) of brain. MRI is characterized for its high soft-tissue contrast and high spatial resolution These two properties make MRI one of the most important and useful imaging modalities in diagnosis of brain-related pathologies. These transformations are applied in a

MORPHOLOGICAL IMAGE ENHANCEMENT AND SEGMENTATION

(a)

(b)

(c)

(d)

(e)

(f)

267

Figure 38. (a) Filtered image using a ßat zone gradient-intensity criterion (φ = 400); (b) Þltered image using a ßat zone gradient-intensity criterion (φ = 600); (c) Þltered image using a ßat zone gradient-intensity criterion (φ = 1000); (d) sequential Þltered image using a ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 300, . . . , φ9 = 1000); (e) sequential Þltered image using a ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 250, . . . , φ17 = 1000); (f) sequential Þltered image using a ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 210, . . . , φ81 = 1000).

two-dimensional case (2-D). The purpose of this procedure was to segment, as accurately as possible, the gray matter and white matter. The Þlters were applied on approximately 50 2-D images with good results. Figure 40 illustrates the Þltered images computed by the MSF using the morphological gradient and the ßat zone gradient as criteria. Both gradient criteria were weighted with respect to the gray-level intensity. Figures 40a and 40b show the original image and the original image without skull, respectively. The output images in Figures 40c and 40d correspond to the Þltered images using a gradient-intensity criterion with parameter φ = 1200 and using a ßat zone gradient-intensity criterion with parameter φ = 1200, respectively. Observe the quality of the Þltered image using a ßat zone gradient with the one using a morphological gradient. Better control of the output image is obtained by means of the sequential MSF as illustrated in Figures 40e and 40f.

268

IVAN R. TEROL-VILLALOBOS

(a)

(b)

(c)

Figure 39. (a) Alternating sequential Þltered image using ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 300, . . . ,φ9 = 1000), (b) alternating sequential Þltered image using ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 250, . . . ,φ17 = 1000), (c) alternating sequential Þltered image using ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 210, . . . , φ81 = 1000).

Figure 40. (a) Original image; (b) original image without skull; (c) Þltered image using a gradient-intensity criterion (φ = 1200); (d) Þltered image using a ßat zone gradient-intensity criterion (φ = 1200); (e) sequential Þltered image using a ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 600, . . . ,φ4 = 2400); (f) alternating sequential Þltered image using a ßat zone gradient-intensity criterion (φ1 = 200, φ2 = 600, . . . , φ4 = 2400).

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The image in Figure 40e was computed by a sequential MSF using a ßat zone gradient-intensity criterion with parameters φ1 = 200, φ2 = 600, . . . , φ4 = 2400; the image in Figure 40f was obtained by means of an alternating sequential MSF using a ßat zone gradient-intensity criterion with parameters φ1 = 200, φ2 = 600, . . . , φ4 = 2400. 3. Some Comments about Invariants Since new gradient criteria have been proposed for constructing idempotent slope Þlters, it is interesting to comment on the notion of the invariant set. Concerning MSFs obtained by the notion of a ßat zone gradient (using morphological gradients for graphs), the concept is strictly the same as that expressed in Section IV.B. Using a ßat zone gradient (gradient deÞnition given by Eq. (17)), the output image ξφεn (( f,P f )) will have a well-deÞned contrast:     gradi ξφεn , P f (x) > φ or gradi ξφεn , P f (x) = φ

Then, all input images ( f ′ , P f ′ ) such that ξφεn (( f ′ , P f ′ )) = ( f ′ , P f ′ ) belong to the invariants set βφεn . However, in relation to MSFs obtained by means of a weighted gradient, an element of the invariants set depends on the gradient and also on the pixel or ßat zone gray level. For example, for the MSF given by Eq. (21) we have     f (x)gradi ξφεn (x) > φ or gradi ξφεn (x) = φ

This means that, at a given point x, the gradient of the output image is equal to zero for a weak contrast point or different from zero for a great contrast point. It is not possible to specify the gradient value of the output image. Thus, in this case, a contrast invariant is given by the output image and its gradient. 4. Kramer and Bruckner ModiÞed Algorithm Now, consider the Kramer and Bruckner algorithm that was analyzed in Section III.C. As expressed earlier, this algorithm has several problems concerning stability. In each iteration, every pixel is updated with the maximum/minimum value of the 3 × 3 neighboring pixels depending on whether the external gradient is lower/greater than the internal gradient. This inconvenience was initially observed by Serra (1988b). Since vertical cliffs and holes that appear on the transformed images can be too strong, they may also degenerate by iteration. An enhancement process was well controlled by using each gradient deÞnition in a separated way. However, it is possible to attenuate instabilities in the

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Kramer and Bruckner algorithm by using a weighted version by using gray level as the weight. The following equation is proposed:  ε B ( f )(x) if f (x) ∗ gradi B ( f )(x) ≤ grade B ( f )(x) ∗ [ f (x)]C δε W f ( f )(x) = (22) δ B ( f )(x) otherwise In Figure 41 we compare the behavior of the Kramer and Bruckner algorithm with that given by Eq. (22). Observe the image degradation when the Kramer

(a)

(b)

(c)

(d)

Figure 41. (a) Kramer and Bruckner algorithm after Þve iterations; (b) weighted Kramer and Bruckner algorithm after Þve iterations; (c) Kramer and Bruckner algorithm after 20 iterations; (d) weighted Kramer and Bruckner algorithm after 20 iterations.

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and Bruckner algorithm is iterated. These instabilities are especially clear in the image in Figure 41c. This problem is attenuated when using the modiÞed algorithm as illustrated in Figure 41d, but stability cannot be ensured. VII. Conclusion In this paper we have investigated image enhancement and segmentation using a class of morphological nonincreasing Þlters called morphological slope Þlters. The notion of morphological gradients was used to build this class of MSF. The idea of retaining the zones of the image with a strong gradient and attenuating the other ones gives good contrast enhancement. Although image-enhancement techniques are generally empirical, the imageenhancement technique presented in this work has a well-deÞned theoretical framework, expressed by a set of properties that permits a better understanding of the technique. By applying the MSF in a sequential way, new properties are found that enable us to retain more features from the original image. By increasing the contrast at each step of the sequence of Þlters, the Þltering process is better controlled. A family of sequential MSFs enables us to obtain better results. On the other hand, by working on the extremities of Þlters, and by combining their results with the maxÐmincriterion, we propose an algorithm for segmenting images. We do not use the watershed-plus-markers approach, but we look for a ßat zone one. We use a max-min criterion to split regions and an area criterion to decide which regions are merging to their neighboring regions. Even if this algorithm splits regions, the output image is a simpliÞed version composed of the fusion of the original image ßat zones. In other words, the algorithm does not break ßat zones. However, the morphological slope Þlters in general break ßat zones. Therefore, other gradient deÞnitions were used to look for a better control of the output image and to avoid splitting the ßat zones. SpeciÞcally, the notion of a ßat zone was used to build a gradient that permits the construction of MSFs that do not break ßat zones. Also, a weighted gradient criterion based on gray-level intensity was proposed. This notion allows attenuation in sensitivity of the MSF regarding parameter φ. This was correctly observed when a modiÞed version of the Kramer and Bruckner algorithm was tested. Finally, a multiscale approach was presented. In this last section the notion of using a diffusion process for image analysis was discussed in order to compare this technique with our approach.

Acknowledgments I thank Dr. Rogelio Arellano for several useful suggestions. Also, I am grateful to Marcela Sanchez Alvarez for her careful revision of the English version. Finally,

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the author thanks Diego Rodrigo and Dario T. G. for their great encouragement. This work was funded by the government agency CONACyT (Mexico).

References Beucher, S. (1990). Ph.D. Thesis, Centre de Morphologie Math« ematique, ENSMP, Fontainebleau, France. Crespo, J. (1993). Ph.D. Thesis, Georgia Institute of Technology, USA. Crespo, J., Serra, J., and Schafer, R. W. (1993). Proc. Workshop on Mathematical Morphology, 52Ð57,Barcelona, Spain. Crespo, J., and Schafer, R. W. (1994). In Mathematical Morphology and Its Applications to Image Processing, J. Serra and P. Soille, eds. Kluwer Academic Publishers, pp. 85Ð92. Crespo, J., Serra, J., and Schafer, R. W. (1995). Signal Processing 47, 201Ð225. Crespo et al., (1997). Signal Processing 62, 37Ð60. Haddon, J., and Boyce, J. (1990). IEEE Trans. Pattern Anal. Machine Intell. 12, 929Ð948. Hummel, A., Kimia, B., and Zucker, S. (1987). Comp. Vision, Graphics Image Processing 38, 66Ð80. Koenderink, J. (1984). Biol. Cybern. 50, 363Ð370. Kramer, H. P., and Bruckner, J. B. (1975). Pattern Recognition 7, 53Ð58. Meyer, F. (1998). In Mathematical Morphology and Its Applications to Image and Signal Processing, H. J. A. M. Heijmans and J. B. T. M. Roerdink, eds. Kluwer Academic Publishers, The Netherlands. pp. 199Ð206. Meyer, F., and Beucher, S. (1990). J. Visual Comm. Image Represent. 1, 21Ð46. Meyer, F., and Serra, J. (1989). Signal Processing 16, 303Ð317. Meyer, F., and Maragos, P. (2000). J. Visual Comm. Image Represent. 11, 245Ð265. Pavlidis, T., and Liow, Y. (1990). IEEE Trans. Pattern Anal. Machine Intell. 12, 225Ð233. Perona, P., and Malik, J. (1987). Proc. IEEE Workshop Computer Vision, Miami. Perona, P., and Malik, J. (1989). IEEE Trans. Pattern Anal. Machine Intell. 629Ð639. Potjer, F. K. (1996). In Mathematical Morphology and Its Applications to Image and Signal Processing, P. Maragos, R. W. Schafer, M. A. Butt, eds. Kluwer Academic Publishers, Atlanta pp. 111Ð118. Rivest, J. F., Soille, P., and Beucher, S. (1993). J. Electron. Imaging Eng. 2, 326Ð336. Saint-Marc, P., Chen, J., and Medioni, G. (1991). IEEE Trans. Pattern Anal. Machine Intell. 12, 514Ð519. Salambier, P., and Serra, J. (1995). IEEE Trans. Image Processing 4, 1153Ð1160. Samet, H. (1984). Computing Surveys 16(2), 187Ð259. Serra, J. (1982). Image Analysis and Mathematical Morphology, Vol. I. Academic Press, London. Serra, J. (1988a). Image Analysis and Mathematical Morphology Vol. II. Academic Press, London. Serra, J. (1988b). Technical report N-18/88/MM. Centre de Morphologie Mathematique, ENSMP, Fontainebleau, France. Serra, J. (1998). J. Math. Imaging Vision 9, 231Ð251. Serra, J. (2000). Fundamenta Informaticae 41, 147Ð186. Serra, J., and Salambier, Ph. (1993). Proc. SPIE Image Algebra Math. Morphology, San Diego, CA, SPIE 2030, 65Ð76. Terol-Villalobos, I. R. (1995). Proc. SPIE Intelligent Robots and Computer Vision XIV: Algorithms, Techniques, Active Vision, and Materials Handling 2588, 712Ð722. Terol-Villalobos, I. R. (1996a). Optical Eng. 35, 3172Ð3182.

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Terol-Villalobos, I. R. (1996b). Proc. SPIE Intelligent Robots and Computer Vision XIV: Algorithms, Techniques, Active Vision, and Materials Handling 2904, 557Ð566. Terol-Villalobos, I. R. (1998). In Mathematical Morphology and Its Applications to Image and Signal Processing, H. J. A. M. Heijmans and J. B. T. M. Roerdink, eds. Kluwer Academic Publishers, The Netherlands, pp. 11Ð18. Terol-Villalobos, I. R., and Cruz-Mandujano, J. A. (1998). J. Electron. Imaging 7, 641Ð654. Terol-Villalobos, I. R., Rodr«õguez-Garc«õa, F., and Morales-Aguill« on, C. (1999). In Recent Research Developments in Optical Engineering, S. G. Pandalai, ed., Vol. 2. Research Signpost, India. pp. 87Ð112. Vincent, L. (1989). Signal Processing 16, 365Ð388. Vincent, L. (1993). IEEE Trans. Image Processing 2, 176Ð201.

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Index

Active probes. See Near-Þeld probes, active Adaptive smoothing algorithm, 258, 263Ð264 Agarose gel phantoms, 61Ð64 Airy function, 152 Algorithms adaptive smoothing, 258, 263Ð264 Crespo, 258 full multigrid, 103Ð104,114Ð115 Kramer and Bruckner modiÞed algorithm, 224, 269Ð271 for the segmentation of images, 242Ð247 Ambulation index (AI), 74, 75 Angular momentum, 6 Aperture probes, passive batch fabrication, 165 description of probes, 160Ð171 energy transport in hollow-pipe waveguides, 156Ð158 etching methods, 161Ð162 Þber, 161Ð164 Þeld distribution of tips, 158Ð160 head-on ion beam etching, 170 MEMS, 164Ð171 Atomic force microscopy (AFM) basic principles, 130Ð131 cantilever probes, mechanics of, 131Ð133 carbon and, 143Ð150 conclusions, 150Ð151 frequency modulated, 131 gallium arsenide and, 141Ð143 high-speed and parallel conÞgurations, 138Ð141 introduction of, 129

materials available for probe fabrication, 133Ð150 microactuator, 138 piezoresistive probes, 137 silicon and, 135Ð141 Bessel function, 152 Bloch equations, 10, 21, 22Ð23 comparison of predicted Z-spectra from complete and simpliÞed, 31Ð33 solutions of coupled, 29Ð30, 78Ð79 solutions of simpliÞed, 30Ð31 Bow-tie antenna probes, 175 description of probes, 177Ð178 energy transport, 176 Brain disorders, diffuse, 69Ð77 Cantilever probes high-speed and parallel conÞgurations, 138Ð141 mechanics of, 131Ð133 nano-, 138 Carbon atomic force microscopy and, 143Ð150 diamond, 145Ð150 electron beam deposited tips, 143 nanotubes, 143Ð145 Chemical exchange model, 23Ð24 Closing transformations, 211 Coaxial probes, passive description of probes, 173Ð175 energy transport, 171Ð173 Contamination lithography process, 143

275

276

INDEX

Contour mapping techniques, 75Ð77 Contrast, in magnetic resonance imaging, 18Ð19 Contrast detectors, 212Ð213 Coulomb force, 131 Crespo algorithm, 258 Curvature sensing, 91 Dephasing, 9 Diamond conventional molding, 145Ð147 projection mask technique, 147Ð150 Differential interference contrast (DIC), 89Ð90 Dilation transformations, 210Ð211 Echoes fast spin, 19, 21 gradient, 20Ð21 planar imaging, 21 spin-echo techniques, 9Ð10, 11Ð12,15Ð18 Echo time (TE), 16, 17 Edge extraction, 214 Electron beam deposited tips, 143 Electrons, phase retrieval with, 118Ð119 Ernst angle, 20 Erosion transformations, 210Ð211 Euler buckling force, 144 Expanded disability status scale (EDSS), 74, 75 Faraday induction, 7 Far-Þeld optics, 151Ð153 Fast Fourier transform, 103 Fast spin echo, 19, 21 Field distribution of tips, 158Ð160 Field gradients, 14Ð15 Fixed zone growth and stability, 224Ð228

Flat zone approach, 241Ð242, 250Ð256 Flip angle, 10 Fluorescent probes, 184Ð185 Focused ion beam (FIB) milling, 162 ForsenÐHoffman reaction rate, 55 Foucault mode of electron microscopy, 89 Fourier series expansions, 102Ð103 Fourier-transformed FID, 10 Fourier-transform method, two-dimensional, 18 Free induction decay (FID), 10Ð11 Frequency modulated (FM) AFM, 131 Fresnel diffraction, 92 FresnelÐKirchhoff diffraction theory, 151Ð152 Gallium arsenide, atomic force microscopy and, 141Ð143 Gaussian family, multiscale paradigm, 257 Gaussian lineshape, 64 Generalized radiance, 93Ð94 Geodesic approach, 248Ð250 GerchbergÐSaxtonphase-retrieval algorithm, 113 Gradient echoes, 20Ð21 Gradient operators, 208, 212 Sobel, 249Ð250 GreenÕs functions, 104, 152 Gyromagnetic ratio, 6 Half-Fourier image, 21 HartmanÐShacksensor, 90Ð91 Head-on ion beam etching, 170 Helmholtz decomposition theorem, 104 Helmholtz equation, 151 Histogram analysis, 73Ð75 Histogram modiÞcation, 224

INDEX

Hoffman phase contrast, 88Ð89 Holography in-line, 112Ð114 Holotomography, 92 Homotopy modiÞcation, 235Ð237 HuygensÕprinciple, 152 Image extrema modiÞcation with contrast enhancement, 217Ð224 Image partitioning, 214 Image segmentation. See Morphological slope Þlters, image segmentation using Infrared excitable phosphor (IEP), 189Ð190 In-line holography, 112Ð114 Interferometry, 92Ð93 Invariants, 269 Iterative multiple defocus technique, 115Ð116 Kramer and Bruckner modiÞed algorithm, 224, 269Ð271 Laplacian operator, 257 Larmor relation, 6, 9 Laser probes, 185Ð187 Lauterbur, Paul, 2 Levelings, 208Ð209 Light-detecting active probes, 187 p/n junction, 189Ð190 Schottky diode, 188Ð189 Light-emitting active probes ßuorescent, 184Ð185 laser, 185Ð187 plasmon, 182Ð184 Longitudinal direction, 6 Longitudinal magnetization, transient solution for, 39Ð40 Lorentzian line, 11 Luminance, 259, 264

277

Magnetic force, 131 Magnetic moment, 5Ð6 Magnetic resonance angiography (MRA), 66 Magnetic resonance imaging (MRI) applications, 2 contrast in, 18Ð19 current research, 2Ð3 development of, 2 Þeld gradients and slice selection, 14Ð15 fundamental signals in, 10Ð12 fundamentals of, 4Ð8 gradient echoes and rapid imaging techniques, 19Ð21 methods for obtaining, 12Ð14 spin-echo techniques, 9Ð10, 11Ð12,15Ð18 spin ßips and relaxation, 9Ð10 two-dimensional, 15 Magnetic resonance spectroscopy (MRS), 2 Magnetization transfer (MT), 3 analytical models for, 27Ð29 applications, 65Ð77 Bloch equations, 10, 21, 22Ð23 Bloch equations, comparison of predicted Z-spectra from complete and simpliÞed, 31Ð33 Bloch equations, solutions of coupled, 29Ð30,78Ð79 Bloch equations, solutions of simpliÞed, 30Ð31 chemical exchange model, 23Ð24 effect of exchange on relaxation times, 38 longitudinal magnetization, transient solution for, 39Ð40 nuclear magnetic double resonance technique, 24Ð26

278

INDEX

Magnetization transfer (MT), (Cont.) saturation, 45Ð53 selective hydration inversion technique, 26Ð27 three-site cyclic exchange model, 33Ð37 T1, approximate solution for, 40 T1, effect of exchange on, 42Ð43 T2, approximate solution for, 41Ð42 T2, effects of exchange on, 43Ð45 T2, exact solution for, 40Ð41 Magnetization transfer contrast (MTC), 3 compared to T2 weighted images, 60 Magnetization transfer imaging (MTI), 3, 42 applications, 53Ð55,65Ð77 correlation in images of agarose gel phantoms, 61Ð64 correlation in images of biological tissue, 60Ð61 fundamental model parameters from Z-spectrum, 64Ð65 images compared to T2 weighted images, 60 on-resonance pulsed, 58Ð60 pulsed off-resonance irradiation, 55Ð58 Magnetization transfer ratio (MTR), 62 Magnetogyric ratio, 6 Mathematical morphology (MM) basic tools in, 210Ð214 dilation, erosion, closing, and opening transformation, 210Ð211 ßat zone approach, 208 gradient operators, 208, 212 histogram modiÞcation, 224

image extrema modiÞcation with contrast enhancement, 217Ð224 reconstruction transformations, 211Ð212,218Ð221 toggle mappings, 209, 213Ð214 top-hat transformation, 208, 212Ð213 watershed-plus-marker approach, 208, 214, 217, 237Ð239 McConnellÕs equations, 23Ð24 Microelectromechanical system (MEMS), 134 Morphological Þlters, 208 Morphological slope Þlters (MSFs), 208, 214Ð217 conclusions, 271 Þxed zone growth and stability, 224Ð228 image extrema modiÞcation with contrast enhancement, 217Ð224 invariants, 234 properties of, 228Ð229 results using a family of sequential, 229Ð233 sequential family of, 229Ð234 weighted, 261Ð269 Morphological slope Þlters, image segmentation using algorithm, 239Ð247 ßat zone approach, 241Ð242 homotopy modiÞcation, 235Ð237 quadtree approach, 239Ð240 segmentation algorithm, 242Ð247 watershed-plus-marker approach, 237Ð239 Morphological slope Þlters (MSFs), nonlinear multiscale representation diffusion processes, 257 invariants, 269

INDEX

Kramer and Bruckner modifed algorithm, 224, 269Ð271 multiscale representation, 256Ð261 weighted, 261Ð269 Morphological slope Þlters (MSFs), nonlinear multiscale using sequential family of connected operators, 250Ð251 connectivity characterized by openings, 251Ð252 connectivity class, 251 dilation or erosion value, 252Ð254 ßat zone notion, 250Ð256 geodesic approach, 248Ð250 partition, 251 Multiscale representation, 256Ð261 Multiple sclerosis (MS), 69Ð77 Nanocantilevers, 138 Nanotubes, 143Ð145 Near-Þeld scanning optical microscopy (NSOM), 151 See also Near-Þeld optics Near-Þeld optics classiÞcation of probes, 156 development of, 151 introduction to, 154Ð155 rules of, 155 Near-Þeld probes, active, 156 light-detecting, 187Ð190 light-emitting, 182Ð187 Near-Þeld probes, passive aperture, 156Ð171 bow-tie antenna, 175Ð178 coaxial, 171Ð175 scattering tip, 180Ð182 solid immersion lens, 178Ð180 Neutrons, phase retrieval with, 119Ð122 Nuclear magnetic double resonance technique, 24Ð26

279

Nuclear magnetic resonance (NMR), 4, 21 On-resonance pulsed MT, 58Ð60 Opening transformations, 211 Optical microscopy, 109Ð111 Optical phase tomography, 111Ð112 Paraxial approximation, 100 Passive probes. See Near-Þeld probes, passive Phase deÞned, 95Ð97 generalized radiance, 93Ð94 interaction of generalized phase with a potential, 97Ð99 introduction and overview, 86Ð87 Phase measurement, methods of curvature sensing, 91 HartmanÐShacksensor, 90Ð91 interferometry, 92Ð93 through-focal series, 91Ð92 Phase recovery, propagation-based general case, 99 requirements for, 107Ð108 transport-of-intensity equation, 100Ð107 uniqueness of, 100Ð102 Phase retrieval electron, 118Ð119 neutron, 119Ð122 x-ray, 114Ð118 Phase retrieval, visible light in-line holography, 112Ð114 optical microscopy, 109Ð111 optical phase tomography, 111Ð112 Phase-sensitive imaging, methods of, 87 differential interference contrast, 89Ð90 Hoffman phase contrast, 88Ð89

280

INDEX

Phase-sensitive imaging, methods of, (Cont.) propagation-based phase visualization, 90 Schlieren phase contrast, 89 Zernike phase contrast, 88 Piezoresistive AFM probes, 137 PlanckÕs constant, 4 PlanckÕs law, 4 Plasmon probes, 182Ð184 p/n junction probes, 189Ð190 Point spread function (PSF), 152 Poisson-type differential equations, 96 Probability current, 95Ð96 Probability density, 99 Projection mask technique, 147Ð150 Propagation-based phase visualization, 90 Proton density weighting, 18 images, 19 Pulse, 10 sequences, 15 Pulsed off-resonance irradiation, 55Ð58 Quadtree approach, 239Ð240 Rapid imaging techniques, 19Ð21 Rayleigh criterion, 153 Reconstruction transformations, 211Ð212,218Ð221 Region-of-interest (ROI) analysis, 71Ð73 Relaxation times, effect of exchange on, 38 Repetition time (TR), 16, 18 Resonance phenomenon, 7Ð8 Saturation, 45 dependence on external B1 Þeld, 46Ð47

in two-spin exchanging system, 51Ð53 in two-spin system, 47Ð51 Scale-space parameter, 256 Scanning electron microscope (SEM), 143 Scanning near-Þeld optical microscopy (SNOM), 151 See also Near-Þeld optics Scanning probe microscopy (SPM) atomic force microscopy, 129, 130Ð151 far-Þeld optics, 151Ð153 near-Þeld optics, 154Ð190 recent developments, 129Ð130 Scanning tunneling microscopy (STM), 129, 130 Scattering tip probes, 180Ð182 Schlieren phase contrast, 89 Schottky diode probes, 188Ð189 Segmentation algorithm, 242Ð247 Selective hydration inversion technique, 26Ð27 Sensitive Point Method, 14, 15 Shear-force detection, 163 Signals, in magnetic resonance, 10Ð12 Silicon atomic force microscopy and, 135Ð141 focused ion beam (FIB) method, 136 high-speed and parallel conÞgurations, 138Ð141 microactuator, 138 -on-insulator (SOI) substrates, 136 piezoresistive probes, 137 total thickness variation (TTV), 136 Single-shot fast spin echo, 21 Slice selection, 15

INDEX

281

Sobel gradient, 249Ð250 Solid immersion lens (SIL), 178Ð180 Solomon equations, 25 Spin diffusion, 24 Spin-echo techniques, 9Ð10,11Ð12, 15Ð18 fast, 19, 21 Spin ßips and relaxation, 9Ð10 Spin-lattice relaxation, 10, 21, 23 Spin-spin relaxation, 9, 21, 23 Spin states, 4Ð5 Spin-warp imaging, 15 Stimulated echo, 12

effects of exchange on, 43Ð45 exact solution for, 40Ð41 relationship between magnetization transfer contrast and, 60 Twin images, 112Ð114 Two-dimensional Fourier transform method, 18 Two-dimensional MRI, 15 Two-spin exchanging system, saturation in, 51Ð53 Two-spin system, saturation in, 47Ð51

Three-site cyclic exchange model, 33Ð37 of biological tissue, 33Ð35 general, 36Ð37 solutions of, 36 through an intermediate site, 37 Through-focal series, 91Ð92 Toggle mappings, 209, 213Ð214 T1 approximate solution for, 40 effect of exchange on, 42Ð43 Top-hat transformation, 208, 212Ð213 Transport-of-intensity equation, solution of, 100 algorithm for nonuniform intensity, 103Ð104 numerical stability of reconstruction, 104Ð105 simulated example, 105Ð107 uniform intensity, 102Ð103 uniqueness of phase recovery, 100Ð102 well-posedness of, 102 Transverse magnetization, 7 T2 approximate solution for, 41Ð42

van der Waals force, 131 Vertical cavity surface-emitting laser (VCSEL) diode, 185Ð187 Watershed-plus-marker approach, 208, 214, 217, 237Ð239 WeberÕs law, 264 Weighted morphological slope Þlters, 261Ð269 Wigner function, 93Ð94,98Ð99 WuÕs equations, 29 X-rays, phase retrieval with, 114Ð118 YoungÕs modulus, 132, 133, 144 Zernike phase contrast, 88 Zernike polynomials, orthogonal, 102 Zero direct saturation, 28 Z-spectrum, 28Ð29 comparison of predicted, from complete and simpliÞed solutions, 31Ð33 fundamental model parameters, 64Ð65

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ISBN 0-12014760-2

90018 >

9 780120 147601