Advances in Imaging and Electron Physics

EDITOR-IN-CHIEF

PETER W. HAWKES CEMESILaboratoire d'Optique Electronique du Centre National de la Recherche Scientifique 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 VOLUME 92

Advances in

Imaging and Electron Physics EDITEDBY PETER W. HAWKES CEMESILaboratoire d Optique Electronique du Centre National de la Recherche Scientifique Toulouse, France

VOLUME 92

ACADEMIC PRESS San Diego New York Boston London Sydney Tokyo Toronto

This book is printed on acid-free paper.

@

Copyright 0 1995 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher.

Academic Press, Inc. A Division of Harcourt Brace & Company 525 B Street, Suite 1900, San Diego, California 92101-4495

United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW 1 7DX International Standard Serial Number: 1076-5670 International Standard Book Number: 0- 12-014734-3 PRINTED IN THE UNITED STATES OF AMERICA 95 96 9 7 9 8 99 0 0 B B 9 8 7 6

5

4

3 2 1

CONTENTS

CONTRIBUTORS . PREFACE . . . .

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

vii ix

Image Enhancement PIEROZAMPERONI I. Outlining the Task . . . . . . . . . . . . . . . 11. Enhancement by Means of Gray-Scale Transformations 111. Enhancing the Uniformity of Homogeneous Regions by Smoothing . . . . . . . . . . . . . . . . . IV. Shading Compensation . . . . . . . . . . . . . V. Local Contrast Enhancement . . . . . . . . . . . VI. Detail Enhancement . . . . . . . . . . . . . . VII. Line Patterns Enhancement . . . . . . . . . . . . VIII. Enhancement of Binary Images . . . . . . . . . . References . . . . . . . . . . . . . . . . . .

I. 11. 111. IV. V. VI.

. . . . . . . .

. . . . . . .

75

Electromagnetic Propagation and Field Behavior in Highly Anisotropic Media CLIFFORDM. KROWNE Introduction . . . . . . . . . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . Conductivity Tensor for Carriers and Fields . . . . . . . . . Anisotropic Energy Bands . . . . . . . . . . . . . . . . Quantum Phenomena from Magnetic Fields . . . . . . . . . Band Structure Anisotropy Effect on Conductivity Tensors . . .

80 81 83 85 88 90

V

. . . . . . .

. . . . . . .

. . . . . . .

1

4 9 20 26 38 52

64

vi

CONTENTS

Variational Formula for the Propagation Constant . . . . . . Finite-Element Formulation for Propagating Structures . . . . Planar Guiding Structures . . . . . . . . . . . . . . . . Magnetoplasma Permittivity Tensor . . . . . . . . . . . . Chiral and Chiral-Femte Media . . . . . . . . . . . . . . Vector Variational and Weighted Residual Finite Element Procedures for Highly Anisotropic Media . . . . . . . . . . XI11. Femte Media . . . . . . . . . . . . . . . . . . . . . XIV. Numerically Calculated Results for Guided-Wave Structures . . XV. Overall Conclusion . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

VII . VIII . IX . X. XI . XI1.

INDEX

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

93 94 95 114 117

132 158 183 200 201

215

CONTRIBUTORS Numbers in parentheses indicate the pages on which the authors’ contributionsbegin. CLIFFORDM. KROWNE(79), Microwave Technology Branch, Electronics Science and Technology Division, Naval Research Laboratory, Washington, D.C. 20375

REROZAMPERONI (l), Institut fur Nachrichtentechnik, Technische Universitat Braunschweig, D-38092 Braunschweig, Germany

vii

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PREFACE

The two long contributions in this volume correspond exactly to two of the regular themes of this series, namely, image processing and device physics. In the opening chapter, F? Zamperoni surveys one of the broad areas of activity in digital image processing, image enhancement. This includes many different ways of improving images or rendering them more informative without explicit use of any knowledge we may have about the causes of any deficiencies in image quality. Enhancement is thus different from restoration: for the latter all possible use is made of any information available about the image-forming process; three-dimensional reconstruction, Wiener-like filtering, and attempts to solve the phase problem are typical examples of restoration. The range of techniques for enhancing images is wide, including the many linear convolutional filters, the nonlinear morphological operations, and median and rank-order filters; these and other approaches to enhancement are all described in full in P. Zamperoni’s review. In the other survey, C. M. Krowne presents in great detail the special problems of electromagnetic wave propagation in such highly anisotropic media as chiral materials and anisotropic ferrites as well as other “as yet undiscovered materials which result in additional or induced anisotropy.” This is a very promising research area and it is important to have a good grounding in the physics of carrier transport in these unusual substances and of the laws governing the propagation of electromagnetic waves, since the applications are mainly in the very high frequency domain. In addition, C. M. Krowne provides guidance to the numerical methods that are being employed to study these properties. I conclude by thanking F? Zamperoni and C. M. Krowne for all the trouble they have taken to make their surveys accessible to a wide readership and by listing contributors to forthcoming volumes of these Advances.

FORTHCOMING ARTICLES Group invariant Fourier transform algorithms

Nanofabrication

ix

Y. Abdelatif and colleagues (vol. 93) H. Ahmed

X

PREFACE

Use of the hypermatrix Image processing with signal-dependent noise The Wigner distribution Parallel detection Discontinuities and image restoration

Hexagon-based image processing Microscopic imaging with mass-selected secondary ions Nanoemission Magnetic reconnection Sampling theory ODE methods The artificial visual system concept Projection methods for image processing The development of electron microscopy in Italy Space-time algebra and electron physics The study of dynamic phenomena in solids using field emission Gabor filters and texture analysis Group algebra in image processing Miniaturization in electron optics Crystal aperture STEM The critical-voltage effect Amorphous semiconductors Stack filtering Median filters Bayesian image analysis RF tubes in space Mirror electron microscopy Relativistic microwave electronics Rough sets The quantum flux parametron The de Broglie-Bohm theory Contrast transfer and crystal images

D. Antzoulatos H. H. Arsenault M. J. Bastiaans P. E. Batson L. Redini, E. Salemo and A. Tonazzini S. B. M. Bell M. T. Bemius Vu Thien Binh A. Bratenahl and P. J. Baum J. L. Brown J. C. Butcher J. M. Coggins P. L. Combettes G. Donelli C. Doran and colleagues M. Drechsler J. M. H. Du Buf D. Eberly A. Feinerman J. T. Fourie (vol. 93) A. Fox W. Fuhs M. Gabbouj N. C. Gallagher and E. Coyle S. and D. Geman A. S. Gilmour R. Godehardt V. L. Granatstein J. W. GrzymalaBusse W. Hioe and M. Hosoya P. Holland K. Ishizuka

xi

PREFACE

Seismic and electrical tomographic imaging

€! D. Jackson and

Morphological scale-space operations Algebraic approach to the quantum theory of electron optics Electron holography in conventional and scanning transmission electron microscopy

l? Jackway

colleagues

Quantum neurocomputing Surface relief Spin-polarized SEM Sideband imaging High-definition television Regularization Near-field optical imaging Vector transformation SEM image processing Electronic tools in parapsychology Image formation in STEM The Growth of Electron Microscopy Phase retrieval Phase-space treatment of photon beams Image plate Z-contrast in materials science Electron scattering and nuclear structure Multislice theory of electron lenses The wave-particle dualism Electrostatic lenses Scientific work of Reinhold Riidenberg Electron holography X-ray microscopy Accelerator mass spectroscopy Applications of mathematical morphology Set-theoretic methods in image processing Texture analysis

R. Jagannathan and S. Khan E Kahl and H. Rose (vol. 95) S. Kak J. Koenderink and A. J. van Doom K. Koike W. Krakow M. Kunt A. Lannes A. Lewis W. Li N. C. MacDonald R. L. Morris C. Mory and C. Colliex T. Mulvey (ed.) (vol. 94) N. Nakajima (vol. 93) G. Nemes T. Oikawa and N. Mori S. J. Pennycook G. A. Peterson G. Pozzi (vol. 93) H. Rauch E H. Read and I. W. Drummond H. G. Rudenberg D. Saldin G. Schmahl J. l? E Sellschop J. Serra M. I. Sezan H. C. Shen

xii

PREFACE

Focus-deflection systems and their applications New developments in ferroelectrics Orientation analysis Knowledge-based vision Electron gun optics Very high resolution electron microscopy Spin-polarized SEM Morphology on graphs Cathode-ray tube projection TV systems

Signal description

The Aharonov-Casher effect

T. Soma J. Toulouse K. Tovey (vol. 93) J. K. Tsotsos Y. Uchikawa D. van Dyck T. R. van Zandt and R. Browning L. Vincent L. Vriens, T. G. Spanjer, and R. Raue A. Zayezdny and I. Druckmann (vol. 95) A. Zeilinger, E. Rasel, and H. Weinfurter

ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL. 92

Image Enhancement P I E R 0 ZAMPERONI Institut fur Nachrichtentechnik. Technische Universitat Braunschweig Braunschweig. Germany

I . Outlining the Task . . . . . . . . . . . . . . . . . . . . I1. Enhancement by Means of Gray-Scale Transformations . . . . . . . . A Contrast Stretching and Nonlinear Gray-Value Transformations . . . . B Histogram Equalization and its Variants . . . . . . . . . . . . I11. Enhancing the Uniformity of Homogeneous Regions by Smoothing . . . . A . Uniformity Enhancement in the Presence of Additive Noise or Texture . . B Uniformity Enhancement in the Presence of Multiplicative Noise or Texture C . Uniformity Enhancement Based on Rank-Order Filtering . . . . . . . D . Streak Suppression . . . . . . . . . . . . . . . . . . . IV . Shading Compensation . . . . . . . . . . . . . . . . . . . V. Local Contrast Enhancement . . . . . . . . . . . . . . . . . A Local Range Stretching and Rank Transformation . . . . . . . . . B. Inverse Contrast Ratio Mapping . . . . . . . . . . . . . . . C Extremum Sharpening . . . . . . . . . . . . . . . . . . D . Adaptive Contrast Enhancement in the Neighborhood of Edges . . . . . E Contrast Enhancement Based on a Pyramidal Image Model . . . . . . VI . Detail Enhancement . . . . . . . . . . . . . . . . . . . . A . Detail-Preserving Multistage One-Dimensional Filters . . . . . . . . B An Extended Set of One-Dimensional Subwindows for Multistage Filtering C . Detail Enhancement by Means of Adaptive Extremum Sharpening Filters D Detail Enhancement Methods Derived from the Mathematical Morphology VII . Line Patterns Enhancement . . . . . . . . . . . . . . . . . A . Iconic Maps of a Local Anisotropy Measure . . . . . . . . . . B Line Detection by Aid of Linear Filters . . . . . . . . . . . . C . Topographical and Morphological Approaches to Line Enhancement . . . VIII Enhancement of Binary Images . . . . . . . . . . . . . . . . A Rank-Selection and Majority Filters . . . . . . . . . . . . . B Enhancement of Intrinsically Binary Images by Polynomial Filtering . . . C . Enhancement Methods Based on Contour Chain Processing . . . . . . D Smoothing of Discretization Noise by Aid of the Distance Transform . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . .

. . .

. . .

.

. . .

1 4 4 6 9 10 13 14 19 20 26 21 30 32 34 36 38 39 43 44 48 52 53 55 62 64 65 68 10 12 15

.

I OUTLINING THE TASK Image enhancement is any one of a group of operations which improve the detectability of objects . These operations include. but are not limited to. contrast stretching. edge enhancement. spatial filtering. noise suppression. image smoothing. and image sharpening . 1

Copyright 0 1995 by Academic Press. Inc . All rights of reproduction in any form reserved . ISBN 012-0147343

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PIER0 ZAMPERONI

This definition, given by Haralick and Shapiro (1991) in their “Glossary of Computer Vision Terms,” is very broad. However, if we consider the kind of tasks that in practice are the object of image enhancement methods in most of the scientific or technical applications, it is opportune to introduce some limitations, which will be useful also for tracing a clear outline of the scope of this contribution. In most cases, enhanced images are intended to be visually assessed by a user, who is very often an expert on a specific field, as for instance biology, medical diagnostics, remote sensing, or nondestructive testing. The criteria for evaluating the quality of an enhanced image, and therefore of an enhancement method, are application-specific, since there is nothing like the field-test-based image-quality evaluation procedures used in commerical television. Further, in the definition just quoted, the “detectability” is related to a visual image assessment, and not, for instance, to a subsequent automatized image analysis process. The latter kind of detectability is rather the aim of an early processing for segmentation. It seems that the nature of image enhancement can be better reduced to a common denominator based on the utilization made by the user, rather than by considering the algorithms used for obtaining the enhancement effect. In fact, depending on the values chosen for the parameters (e.g., window size, number of iterations, coefficients, weights, etc.), the same algorithms can produce processed images which deviate more or less strongly from the original image, as shown schematically in Fig. 1. Within the scope of image enhancement, the deviations are mostly of “cosmetic” nature, i.e., they are relatively weak and they are intended only as a help to the visual evaluation. This help consists of attenuating the irrelevant (from the user’s point of view) patterns and in reinforcing the relevant ones. On the other hand, in pre-processing for segmentation, the deviations

Source Image

I

I ”

I I

minor

substantial

Extent of the deviations from the source image

I I I I I

visual evaluation oftheenhanced image, madeby experts

FIOURB1. Schematic representation processing tasks.

’,

I I I

automatic segmentation of the processed image by means of simple edge detectors

of the relationships between some typical image

IMAGE ENHANCEMENT

3

introduced are mostly more substantial; they may be visually striking, but they are specifically directed at an image approximation tailored for the subsequent segmentation process. Several well-established textbooks on digital image processing, as for instance: (a) (b) (c) (d) (e) (f)

Gonzalez and Wintz (1987), Chapter 4; Hall (1979), Chapter 4; Jain (1989), Chapter 7; Jensen (1986), Chapter 7; Rosenfeld and Kak (1982), Chapter 6 of Vol. 1; and Yaroslavsky (1985), Chapter 8,

dedicate a special chapter to the image enhancement topic. In these textbooks image enhancement is more or less unanimously considered as the task of improving the image quality without making use of any knowledge about the degradation process or about the undegraded image, while image restoration takes account of this knowledge. The image may also be not degraded at all, but the user may wish to enhance some features or structures, for getting a falsified, but subjectively preferable image, just as in the reproduction of a faithfully registered music piece the hearer can “falsify” the original, e.g., by enhancing the treble. In the references (a) to (f), one can find a variety of interpretations to the task of image enhancement, depending upon the feature that is the object of the enhancing action. There are some main classes of enhancement operators, emerging more or less neatly from the textbook chapters quoted above, which can be used for classifying the methods described in the very copious image enhancement literature. The contents of this contribution has been subdivided into the following sections, corresponding roughly to these classes: (11) Enhancement by means of gray-scale transformations (111) Enhancement by smoothing

(IV) (V) (VI) (VII) (VIII)

Shading compensation Enhancement of the local contrast Detail enhancement Line enhancement Binary image enhancement

The enhancement techniques to be described in the next sections concern only monochrome single pictures, and they do not take account of relationships between channels or between images of a sequence.

4

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11. ENHANCEMENT BY MEANS OF GRAY-SCALE TRANSFORMATIONS

Most of the enhancement operators presented in this section can be described by so-called point operations (Jain, 1989). These operations are transformations of the gray scale, in which the current pixel’s new gray value is a function only of the original gray value, but not of the gray values of some neighboring pixels. The operator is completely specified by the input/output gray-value characteristic, also called the gray-value transformation function u = g(v), where u and v are the input and the output gray value, respectively. In digital image processing, the gray values are integers in a range determined by the system gray-scale resolution, which is usually equal to G = 256 gray levels. Therefore, u and o are between 0 and G - 1. A. Contrast Stretching and Nonlinear Gray-Value Transformations

Poor scene illumination or the automatic gain control characteristics of the acquisition device can cause the input image’s gray-value histogram to occupy only a portion, between the gray values u1 and u 2 , with u1 Iu 2 , of the full available gray scale. Linearly stretching the interval u l , ...,u2 into the full range 0, ..., G - 1 improves not only the visibility of low-contrast details, but also the apparent sharpness, which is influenced by the contrast (Schreiber, 1978). The input/output gray-value characteristic v(u) used for contrast stretching is shown in Fig. 2a; it corresponds to the analytical expression

fo \G-

for u

1

c ul,

for u > u2.

The values of u1 and u2 can be determined automatically before running the contrast stretch routine by recording the global histogram. Then, scanning the histogram from 0 and from G - 1, one determines the first gray values for which the histogram entries exceed a given threshold. For many pictures, the values of u1 and u 2 , as defined earlier, are equal to 0 and to G - 1. In these cases, soft input/output characteristics, like those displayed in Fig. 2b, avoid the clipping of image contents at very low and at very high brightness levels. Among the numerous possibilities for establishing a nonlinear relationship between input and output gray value, the two variants described next allow a broad variety of enhancement effects to be obtained by varying the parameters s (location of the inflection point) and r (exponent), both to be specified by the user.

5

IMAGE ENHANCEMENT

b v

a v

4

U

0

P

G-1

FIOURB2. Input/output characteristicsused for gray-scale transformations. (a) Stretching of the gray-value range between u, and u2. (b) “Soft” nonlinear characteristics for contrast stretching in the bright zones ( A and C) or in the dark zones (B and D ) . (From Klette and Zamperoni, 1992.)

Variant I (see Fig. 2b, curves A and B).

Here u(u) is a power function of the normalized input gray value. Depending upon the value of r, one obtains a convex curve like A for r > 1, and a concave curve like B for r < 1. In the first case the dynamic range of the bright gray values is enhanced, and that of the dark ones is reduced; for r < 1, the converse is true.

Variant 2 (see Fig. 2b, curves C and D ) . Let us consider the normalized gray values U = u /( G - 1) and S = s/(G - 1). The enhanced output gray value u is then

This variant aims at obtaining combinations of contrast expansion and compression in different gray value ranges. Taking r > 1 (curve C), the contrast is expanded around the gray values, and it is compressed elsewhere; for r < 1, the converse is true. the value of r determines the slope of u(u) at points s, and therefore the expansion or compression ratio at the inflection point of the input/output characteristic curve. The preceding expression of v(u) results in a continuous characteristic curve, since it fulfills the following conditions: ul_(s) = u;(s) = r(G - 1) and u-(s) = u+(s) = s.

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B. Histogram Equalization and its Variants The aim of a histogram equalization is that of obtaining an equal gray-value distribution. The resulting image features the highest possible contrast enhancement, regularly distributed over the whole gray-value dynamic range. Yet this does not insure the best visual quality for an arbitrary input image. For instance, the equalization of a sharp bimodal histogram, associated with a nearly binary image, spreads the gray values over a wide range, thereby blurring the object-to-background transitions. The gray-value transformation function u(u), performing the histogram equalization, can be determined on the basis of some simple considerations, illustrated with the aid of Fig. 3. Here, u(u) is considered as a continuous function, while p,(u) and Pu(u) represent, respectively, the gray value density function and the distribution function of the image before equalization. Let p,(u) be the gray-value density function in the equalized image. Under the constraint that u(u) should be monotonically growing, i.e., that the equalization does not cause any “negative image” effect on the output image (Gonzalez and Wintz, 1987; Jain, 1989), we obtain the condition p,(u) * du

=p

du , ( ~* )dv = G - 1’

i.e., u(u) = (G - 1)

s:

p,(t)dt.

It follows that the required transformation function u(u) is given by the distribution function P,(u). For discrete gray values (u and u are integers), the G values of Pu(u)= C:,,,p,(t), determined in a preliminary image scan, can be stored in a look-up table, addressed by the current pixel’s gray value u for calculating the equalized gray value u(u) = G * P,(u). Figure 4 shows an example of an image before and after equalization, together with the respective histograms. Notice that, if the original histogram spans only a subset of G’ gray values, with G’ c G, the equalization process can introduce gaps and holes into the “stretched” zones of the

G-1

FIOURE3. Histogram equalization. The input/output gray-scale characteristics u(u) are obtained from the global gray-value distribution function P,,(u).

IMAGE ENHANCEMENT

7

a

b

FIGURE4. Effects of gray-value equalization upon a weakly contrasted natural image. Beside the original (a) and the equalized image (b), the respective gray-value histograms are shown. (From Zamperoni, 1991.)

histogram, as shown in Fig. 4b. The height of the isolated histogram peaks and the spacing between peaks compensate each other. What the histogram equalization ensures is a constant average density over the whole gray value range.

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In the classical histogram equalization technique just outlined, given the global histogram, the transformation function u(u) is determined automatically, without any degree of freedom for obviating some secondary effects, which may arise. One of them is the noise enhancement occurring when large and noisy gray-value plateaus undergo a contrast spread. The amount of this spread at gray value u depends on the slope of the distribution function Pu(u), which in its turn is nothing else than the associated value of the density function p,(u). Since p&) can be particularly high in large quasi-uniform areas, this can turn out to enhance the noise just in the regions of the image where it is most perceptible. Pizer et al. (1987) developed an improved histogram equalization procedure to compensate for this situation. They proposed obtaining the transformation function v(u) from a modified density function p$(u), in which the peaks have been clipped off. The measurements corresponding to the pixels of the clipped-off peaks are equally redistributed over the whole gray-scale range, as shown schematically in Fig. 5 . The integration of p$(u) results in a distribution function Pz(u)= u(u) with reduced maximum slopes. # of pixels

#of pixels

t

t

level P

Recorded intensity Mapped intensity

Redistributed clipped pixels

Recorded intensity

Mapped intensity

t

t Mapping function

L I

intensity

Recorded Recorded intensity

FIQW 5. Modified histogram equalization method without noise enhancement in the uniform regions. The modified input/output gray-scale characteristic is obtained from a density function with clipped peaks. (From Pizer et a/., 1987.)

IMAGE ENHANCEMENT

9

In the previous approach, a pixel is estimated to belong to a large plateau only by checking whether its gray value is that of one of the main histogram peaks. This criterion, not involving any neighboring pixel, may not be reliable enough, and it can be substituted by a better one, based on the evaluation of the edge intensity. This means that the operations to be carried out are of the local type rather than of the point type, as postulated at the beginning of this section. Also, the method proposed by Leu (1992) aims at a histogram equalization which enhances the contrast, but not the noise visibility. This is obtained by defining the input/output characteristic as the distribution function P$(u)of a modified density function p:(u) (in practice approached by the normalized global histogram). The modification consists in taking account, in the histogram computation, only of the pixels in which the value of the Sobel gradient exceeds a given threshold. In this way, a gray-value spread occurs only for the gray values associated with the image’s edges. On the other side, it can happen that a gray value which is weakly represented (say, less than 1/G) in the modified histogram p:(u) has a high score in the original histogram p,(u), constructed on all the image’s pixels. It follows that the corresponding regions will be merged together with other regions by the equalization process, because the function Pz(u)is flat in correspondence with the gray values of these regions. However, this will not happen if a second modification is performed. Having determined the gray values for which the preceding conditions are met, the score of these gray values is enforced to exceed 1/G, and all the histogram’s entries are then updated for normalization. After integration of the resulting histogram, one obtains the desired input/output characteristics for the normalization. The histogram equalization techniques described in this subsection are all particular cases of a more general type of operation, called “histogram specification” (Gonzalez and Wintz, 1987; Jain, 1989; Kautsky et al., 1984), consisting of the determination of the input/output characteristic apt to convert a given global histogram into another, arbitrarily specified histogram.

111. ENHANCING THE UNIFORMITY OF HOMOGENEOUS REGIONS BY SMOOTHING One of the most common problems encountered in low-level image processing is that of converting an intrinsically homogeneous region into a constant gray-value plateau (“flat model”). This is equivalent to tagging each pixel of such a region with a unique label, for instance with a gray

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value. The key requirement for this label is that it be, as far as possible, strictly the same over the whole region; that it represent a good estimate of a possibly original gray level, bare of all random deviations, is mostly of secondary importance. The aim of fitting gray value functions into such a flat model is that of pre-processing the image in such a way that it can be subsequently segmented by means of very simple edge-detection algorithms. Then, for the edge-detection stage, it makes no difference whether a region is homogeneous because it has a constant gray value, or because it consists of a complex and yet homogeneous texture. With regard to the task just outlined, it does not matter whether the gray value variations inside an intrinsically homogeneous region are due to random noise, to special types of noise originated by the sensor or by the imaging process (e.g., radar images, scintigrams, ultrasonic images), or whether they faithfully reproduce a real-world texture. If the texture does not feature any form of spatial order, i.e., if it does not contain relevant patterns that it could be desirable to preserve or to enhance, the task is the same as for noise; to smooth the local gray-value variations, and therefore to enhance the perception of the region as something uniform. In the following subsections, some approaches to image smoothing will be presented, and each approach will be illustrated by making reference to one or more representative works. For the reasons explained earlier, it is desirable to consider methods which do not distinguish between noise and texture. Therefore, preference has been given to approaches which do not need a priori knowledges about the noise (or texture) statistics, but which make use only of very general statistical parameters, such as mean and variance, or which are based on a distribution-free description by means of rank-order statistics.

A . Uniformity Enhancement in the Presence of Additive Noise or Texture If the gray-value deviations with respect to the flat model inside a homogeneous region are of an additive nature, smoothing can be accomplished by means of a linear minimum square error (LMMSE) technique. This simple and efficient technique lends itself to local adaptation, and it has been used successfully in several applications (Lee, 1980, 1981; Kuan et al., 1985; Chehdi, 1991; Wu and Maitre, 1992). The LMMSE technique determines the least-mean-square estimation f ( x ,y ) at a point (x, y ) under the constraint of being a linear combination of the current gray value f ( x , y ) and of the local expected value, approximated by the local mean p(x, y ) (Melsa and Cohn, 1978; Haralick and Shapiro, 1992). If the noise is

11

IMAGE ENHANCEMENT

uncorrelated, one obtains (Kuan et al., 1985)

f ( x ,y )

=K

- f ( x ,y )

+ (1 - K) - p(x, y )

with K =

OZ(XY Y )

aZ(x, y )

+ a:

where a 2 ( x , y ) and a : are the variances of the noiseless image (approximated by the measured variance) and of the noise, respectively. The quantity a : , supposed to be constant, should be known from a priori considerations, but its exact value is not critical. As a2(x,y) is measured in the current observation window, it results that the smoothed filter output is equal to the current gray value f ( x , y ) where a ( x , y ) a,, i.e., in correspondence to step edges and to other high-contrast zones, in which the whole local variance can be ascribed to the uncorrupted image. Inside noisy plateaus, the variance can be attributed t o the noise (or texture); here the strongest smoothing takes place [ f ( x ,y ) = p(x, y ) ] . For 0 I K I 1, all the intermediate situations can occur. The upper half of Fig. 6 shows at left a natural scene with additive, normally distributed noise (a, = 20 gray levels) and at right the enhanced image. In this example the filter’s performance is particularly good, since Q, was known from the routine used for generating the test image. Lee’s adaptive smoothing filter described earlier has been improved by Wang (1992a) in two directions:

+

0

Smoothing by unweighted averaging has been substituted by smoothing in a selected neighborhood (Wang et al., 1981). Introducing weights w(i,j ) into the smoothing operations aims at averaging the current pixelf(x, y ) only with those neighborsf(x + i, y + j ) that presumably belong to the same region, thus avoiding smoothing over region edges. For this aim, the weights are taken to be inversely proportional to the absolute gray-value differences If(x + i, y + j ) - f ( x , y)l , implicitly assuming the gray-value difference as a valid criterion for estimating whether two pixels belong to the same region or not. There is no need to establish empirically the value of a, , necessary for running the filter. Instead, this value is estimated adaptively for each pixel on the basis of the local data.

As for the first point, the weights w ( i , j ) depend inversely on the gray-value differences according to the following law:

where c = Cij w ( i , j ) represents the normalization coefficient.

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FIGURE 6. Upper half: Example of additive noise suppression by the method of Lee (1980). The original scene at left has been impaired with normally distributed noise (a, = 20 gray levels). Lower half: Example of smoothing by means of an adaptive quantile filter (Alparone et ul., 1992) of an SAR remote sensing view affected by speckle noise (at left). In both examples, the enhanced images at right have been obtained with a window size of 7 x 7. (From Klette and Zamperoni, 1992.)

The adaptive estimation of D,, has been performed by Wang on the basis of the following approximated model. The filter output .f(x,y) results from the same expression as before, where the unweighted average p(x,y) has been substituted by the average g(x,y), weighted in a selected neighborhood: A x , Y ) = K -.f(x, u) + (1 - K ) * g(x, u) with g(x,y) =

C W,j)

+ i, Y + j ) .

u

The smoothed signal g(x, y) is assumed to have been made noise-free; thus, its variance y) is identified with the local variance of the input image signal without noise. The noise variance is identified with the variance a2 before smoothing. From these considerations follows the expression of the

IMAGE ENHANCEMENT

13

weight K

because a; is the variance of X u w ( i , j ) * f ( x+ i, y + j ) , which is equal to a2Cri w 2 ( i , j )if a2 is assumed to be locally approximately constant. This value of K , which does not need to be estimated by the user, is used in the expression that gives the filter output f ( x , y ) as a weighted sum of f ( x , y ) and g(x,Y ) . Other iterative restoration methods, based on Wiener filtering and utilizing apriori knowledge of the noise statistics, can be found in Cheong and Morgera (1989).

B. Uniformity Enhancement in the Presence of Multiplicative Noise or Texture With some imaging devices, the noise is signal-dependent and it has a multiplicative nature. This is the case of speckle noise, arising in radar images, and especially in SAR (synthetic aperture radar) remote sensing images (Jain, 1989; Pitas and Venetsanopuolos, 1990; Haralick and Shapiro, 1992). In order to exploit the advantages offered by SAR imaging in terrain monitoring (independence of cloud coverage, fog, etc.), it is useful to reduce the typical interference patterns caused by the multiplicative speckle noise, which sensibly impair the quality of remote sensing views. A classical method for attaining this aim has been developed and improved by several authors: Lee (1980), Kuan et al. (1985), and Chehdi et al. (1991). Its main idea is that of converting the multiplicative model into an equivalent additive one, thus reducing the problem to a known case. The equivalence of the two models is imposed by the following relation: f ( X , Y ) = g(x,y) + h(X,Y) = g(x,y) * n(x, v),

h(x, Y ) = g(x, Y M X , Y ) - 11, wheref ( x ,y ) and g(x,y ) are the observed and the uncorrupted gray value in (x,y ) , h(x,y ) is the equivalent additive zero-mean noise, and n(x, y ) is the multiplicative noise factor. If E [ . ] is the expected value in the observation window, we have E[h] = 0 and E[n] = 1. Therefore, if g(x,y ) and n(x,y ) are mutually independent, it follows that

u) = E ~ ~ Y)I ( x - ,E[(n(x,Y ) - 1121 = E ~ ~ (Y)Ix - ,a,,.

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For the same reason,

E[f2(X,Y)l= Ek2(X,Y)l* E [ n 2 ( x , y ) l , i.e.,

Ek2(X,Y)l = E [ f 2 ( X , Y ) l / ( d+ 1). Applying to f(x, y ) a general relation valid for the variance, E [ f 2 ( x y, ) ] = af + ( E [ f ( x Y, ) ] ) ~we , can write the equivalent of the expression used in the additive noise case, giving the coefficient K of the linear mean-square error estimator as a function of the variances of the two additive terms off(x, y ) :

K=1--ah'<x,Y ) -I--af(x, Y )

0 :

Y)

af(x,Y) + (E[f(X,Y)lY a, + 1

Here, as in the additive noise case, 0, is assumed to be constant and known from a priori measures [therefore it is no function of (x,y ) ] ,and f ( x , y ) is estimated by means of the same expression given earlier for the additive noise case, using the new value of K. This classical approach was first introduced by Lee (1980) and Kuan et al. (1985), and it has been used since then by several authors, especially for the enhancement of remote sensing images and for smoothing of SAR speckle noise, as for instance by Lee (1981), by Chehdi et al. (1991), and by Maitre and Wu (1992)in combination with an adaptation of the window size. Other approaches to multiplicative noise suppression can be found in Moloney and Jernigan (1989).

C. Uniformity Enhancement Based on Rank-Order Filtering Rank-order filters provide robust estimators of location and scale with respect to outliers and to statistically distributed noise. Their properties are described in textbooks (see Pitas and Venetsanopoulos, 1990), and their applications have been the object of innumerable investigations. Median filters, alpha-trimmed-mean filters, L-filters (weighted averages of the rank-ordered gray values of the observation window), rank-selection filters, and their adaptive versions are all special rank-order filter structures which have been taken as a basis for developing a wide variety of image enhancement methods. Some typical examples of this kind will be examined in this subsection.

a. Smoothing by Means of an Adaptive Quantile Filter. The aim of the method developed by Alparone et al. (1992)for the enhancement of SAR images by smoothing of the speckle noise is similar to that of Lee (1980 and 1981, see earlier discussion), but it is based upon a data-dependent

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rank-order selection instead of a weighted average between the current gray value and the result of a location estimation. The value of the local variation coefficient L = a(x, y)/p(x, y ) (local mean square deviation/local average) controls the ranked gray value which is selected as the filter output. The performed smoothing is weak for high values of L, where the variance is supposed to be due to the presence of high-contrast edges. A strong smoothing takes place for low values of L, where a ( x , y ) is attributed substantially to the noise variance. Moreover, the multiplicative nature of the noise requires a stronger smoothing in brighter regions, i.e., for high values of p(x, y). For describing the filter function, let us introduce the thresholds T, and T,, with 0 ITO IT, IG - 1 (usually G - 1 = 255). If T, and 7i are considered as the Ao- and A,-quantiles of the local gray-value distribution function S(u), the following relations hold: A , = S(T,),

A , = S(T,),

with 0

I

A,

I

Al

I

1.

Considering the observation window U,centered on the current pixel, and T = S-’(A), the inverse of the local distribution function S(u) shown in Fig. 7a, a value of A, determines the threshold T,. A , results from the current value of L and from a given monotonically decreasing function A,(L), two examples of whose shape are shown in Fig. 7b. In practice, A , = 1 - A , is chosen symmetrically to A,. Then the filter output g(x, y) is determined by f(r)

=

(T,)

$(,)

=

(T)

i f $ ( x , ~ )< T,, if If(x, y) I T, , iff(x,u) > T,,

where$(,) is the rth rank-ordered gray value in U,and ( ) means “rankordered gray value nearest to.”

F r o m 7. Smoothing by means of an adaptive quantile filter. (a) Distribution function S(u) of the local gray values u. (b) Functional dependence between the lower threshold A, and the control parameter L = a(x,y)/p(x,y). (From Klette and Zamperoni, 1992.)

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It is evident that, for high values of L, T, = 0 and = 1, so that f ( x , y ) and g(x, y ) mostly coincide. On the other side, for L = 0, A, = 0.5, and the filter performs as a median filter. The actual value A,: = Ao(l - C ) used by the authors differs from A , by a correction factor C = ( p - fmed)/(G - l), where fmed is the local median, introduced for compensating the experimentally observed dissymmetry of the density function dS(u)/du, which, for speckle noise, mostly assumes the shape of a Rayleigh distribution. The corrected value for A l is A l c = (1 - A,)(1 - C ) . The lower half of Fig. 6 shows at left a SAR remote sensing view affected by speckle noise, and at right the image enhanced by smoothing.

b. Composite Enhancement Filter of Mallikarjuna and Chaparro. The main idea of this filter is that of considering two additive components of the unimpaired image signal g(x, y ) , a structural, “low-pass” component p(x,y ) , and a detail, “high-pass” component s(x, y ) , and then of performing a separate enhancement of both components by means of a rankorder-based estimation. The observed signalf ( x , y) is the sum of g(x, y ) and a noise signal h(x,y ) , which may include impulsive noise v(x,y ) as well as random noise w(x,y ) . The following model is assumed:

f ( x ,Y ) = g(x, u) + h(x,u) =

[Lc(x,Y ) + s(x,u)l + [ d x ,Y ) + w(x,u)l

= P(X,Y)

+ r(X,Y),

where r(x, y ) = s(x,y ) + h(x,y ) is a zero-mean residual signal consisting of details and noise. The structural component p(x,y ) can be efficiently estimated by means of an L-filter of the type N

N

b(x,y ) =

C aif C i ) i= 1

with

C ai = 1, i= 1

where are the N rank-ordered gray values in the observation window U. It is well-known (see, e.g., Pitas and Venetsanopoulos, 1990) that, for some typical gray-value distributions in U,the optimum estimation of p(x, y ) can be obtained by the preceding formula, for proper values assigned to the weights a i . For instance, for a normal distribution, B(x,y) is the local average (ai = 1/N v i ) ; for a long-tailed exponential distribution, &x, y) is the median [ai = 1 for i = (N + 1)/2 and ai = 0 otherwise]; for an equal distribution, i ( x ,y ) is the midrange (al = aN = 0.5 and ai = 0 otherwise). After having extracted the structural component, the spike noise (outliers) is eliminated from the zero-mean residual image r(x, y ) by means

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IMAGE ENHANCEMENT

of a median filter, which is optimal for this purpose, thereby obtaining the filtered residual image rl(x,y), which still contains random noise. For reducing the latter, i.e., for obtaining the final estimate $ x , y ) of s(x, y), rl(x,y) undergoes a further filtering stage of the type described in Section II1,l (Lee, 1980): i(x, y)

=K

- r l ( x , y )+ (1 - K )

*

rl(x,y)

with K =

Y) Y) +

dl(X,

&x,

4’

where r l ( x , y ) , ~ $ ~ ( x , yare ) the mean and the variance of r:(x,y) in U, and 0; is the constant noise variance, estimated on the basis of a priori information. The final filtered image signal in (x, y) is then represented by the sum @(x,y) + 2. The experimental results, reported in Mallikarjuna and Chaparro (1992), show a good enhancement of the region uniformity, without edge blur, even under heavy noise impairment.

c. Center-Weighted Median Filtering. Center-weighted median filtering can be used for enhancing images affected by a mixture of spike noise and of random noise with parametrized distribution. A center-weighted median filter (see KO and Lee, 1991) is a particular form of the weighted median filter (see Pitas and Venetsanopoulos, 1990), whose output u(x,y) is the median of the set of gray values obtained by taking qi (i = 1, ...,N ) times each ranked gray value hi)of the observation window U. In a centerweighted median filter, all the values of qi are equal to 1, except the value Q = 2k + 1 (odd), associated with the current pixel in the position ( x , y ) ; therefore, u(x, y) is the median of N + 2k gray values. The deterministic and statistical properties of the center-weighted median filter have been investigated by KO and Lee (1991); within the scope of this work, it will suffice to point out some qualitative aspects of this filter. It is clear that, if k is equal to 0, the filter will perform as a conventional median filter, while for k 2 (N - 1)/2, one obtains u(x,y) = f ( x , y ) . Choosing k between these extreme values, the filter will feature the whole range of intermediate properties between those of the median filter (edge preservation, spike noise suppression) and those of the identity filter (detail preservation). This consideration leads to the idea of adaptive center-weighted median filters, in which the value of k is chosen depending on the local data: If the presence of outliers is estimated [e.g., by scale estimation, by evaluating the quantity f&-c+l) - f(c), with 1 s c 5 (N - 1)/2], k should be low; otherwise, k should assume a high value. d. Iterative Noise Peak Elimination Filter of Imme. This filter (Imme, 1991) can be used in a single iteration or in several iterations, depending upon the main aim pursued: spike noise elimination or smoothing. Its central idea is that of replacing the gray value f ( x , y) of a current pixel, which has

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been recognized to be an outlier [ f ( x ,y) = fa)or f ( x , y) = f(,,], with its nearest ranked gray value in the direction of the local median [f(N-l) or&,)]. If u(x,y) is the filter output, the first iteration, which eliminates the outliers, can be described as follows: A N - 1)

if f ( X , Y ) = f ( N ) if f ( x , Y ) = f ( 1 ) otherwise.

9

9

f ( x , y)

This filtering criterion can be applied iteratively, whereby the number of iterations places the filter function nearer to the spike noise elimination or nearest to smoothing, with several intermediate stages. In case of iteration, the filter function must include a stopping criterion enforcing the convergence to a stable state:

U(X,Y) =

i"' &m)

f ( x , y)

i f f @ ,Y ) = f(N)and &M)

f &l)

iff(x, Y ) = 41) and A m ) f f otherwise,

3

( ~9)

where M is the rank of the largest rank-ordered gray value j&,with

hi)< f c N ) ;similarly, m is the rank of the smallest rank-ordered gray value &i), withf(i) > & I ) * Also, the nonlinear mean filters, described by Pitas and Venetsanopoulos (1990) and realized in an adaptive variant by Kundu et a1 (1984), feature good spike noise rejection properties. The general expression of the output u(x,y) of a nonlinear mean filter is

where aijare weights, U is the observation window, a n d p is a real number. For p = 1, one obtains the conventional mean filter. With p > 1 (p < l), the filter performs a nonlinear maximum (minimum) estimation and can therefore be used for suppressing negative (positive) spike noise, whenever the current pixel is judged to be affected by this kind of noise. This aim can be attained by computing the filter output u(x, y) by means of the following filter function, for which it is necessary to introduce the thresholds tl and tz and to determine the local mean C :

up(x,y) uq(x,y )

f ( x , y)

with p > 1 if C - f ( x , y ) > tl , with q < 1 if f ( x , y) - C > t z , otherwise.

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19

FIGURB8. Example of streak elimination from the watermark of a manuscript, with a 5 x 7 window. (From Klette and Zamperoni, 1992.)

D. Streak Suppression In some image acquisition processes, as for instance in remote sensing applications, streaks represent a serious source of impairment. The streaks have the structure of parallel lines with irregularly distributed gaps. The streak direction can be brought to coincide with the horizontal or with the vertical axis. Many streak suppression methods, including the one described here, can be considered as variants of the box-filtering technique (McDonnell, 1981). For better matching the streak structure, let us choose a rectangular m x n observation window U,centered on the current pixel ( x , y ) . Assuming horizontal streaks, n should be chosen to be sensibly greater than the streak width, while m should be equal to the estimated shortest streak segment between gaps. The method is based upon the assumption that the deviation between the average gray value ,uuover U and the average ,uo over the current line through (x, y ) (with m x 1 pixels) is an error due to a streak. The gray value f(x, y ) is therefore corrected by subtracting the error, thus obtaining the enhanced gray value u(x, y): U(XY

Y ) = f(XY Y ) +

(,uu

- Po).

Analogous considerations apply to vertical streaks; in this case p0 must be computed over a 1 x n column containing (x, y). Figure 8 shows an example of streak elimination in the watermark of an ancient manuscript. Here the streaks are caused by the structure of the sieve used for producing the paper.

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IV. SHADINGCOMPENSATION

a. Definition of the Problem. An image can be impaired by a background intensity function (shading), superimposed on the useful image contents information, and containing only very low spatial frequency components. The origin of the shading can be manifold: The sensitivity of the imaging device is not uniform over the whole image field. In particular, the response of the optical system is often stronger at the center than at the border. This induces a dome-shaped background shading. Even if the scene is uniform (e.g., a neatly written text on a white paper sheet), uniform scene illumination can be attained only under photographic studio conditions. The background of drawings, manuscripts, and other similar images is in practice not sufficiently uniform. The elimination of this spurious shading can represent a most decisive enhancement of the image quality, especially in view of binarizing the image signal, as shown schematically by Fig. 9. If the image contents is intrinsically a two-level one (e.g., text, drawings), a binarization with fixed threshold can lead to serious errors in the presence of a pronounced background shading. As the background signal b(x,y) contains spatial frequencies of the order of magnitude of the image size, i.e., of the lowest possible spatial frequency, it can be extracted from the image signal f ( x , y ) by means of a spatial low-pass filter with a properly chosen cutoff frequency. Fahnestock Ideal binary signal White

Constant

.threshold Black

Background

m U 1 -~

Result of a binarization with fixed threshold

FIGURE 9. Schematic representation of the problem arising in the binarization with a fixed threshold of an almost ideal binary image signal (top) in presence of shading and noise (middle). The binarized signal (bottom) does not reproduce the original information. (From Zamperoni, 1991.)

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and Schowengerdt (1983) give the following general formula for the output signal u(x, y) after a high-frequency enhancement:

where p(x, y) is the local mean gray value in the observation window. The high-pass filtering function is realized as a weighted point-to-point difference between the original image and the output of a linear low-pass filter. A simplified variant of this formula, containing only one parameter k instead of A ,B, and C, can be derived by imposing the constraint of preserving the original gray value dynamic range. Setting C = 0 and f ( x ,y) = p(x, y) for constant gray-value regions, one obtains A = 1/B. The unique parameter k is in the following relation with A and B: k = 1 - B and A = 1/(1 - k). Further, the background signal b(x,y) is not necessarily obtained by means of a linear low-pass, since there are other smoothers with advantageous properties, as for instance the median filter. These considerations suggest the use of the modified background subtraction formula u(x, Y )

1

=

[ f ( x ,y ) - k * b(x,y ) ] l-k

with 0

I

k < 1.

For the special case of b(x,y) = p(x, y), Fahnestock and Schowengerdt (1983) give the frequency response of the background subtraction filter for different values of k and for different window sizes. With decreasing window size, the filter’s lowest no-attenuation spatial frequency grows, and the peak-to-peak ripple of the filter transfer function decreases. On the other hand, an increasing value of k enhances the attentuation of the low spatial frequencies. The difficulty of implementing a linear low-pass filter with very low cutoff frequencies lies in the time-consuming convolution with averaging kernels of big size. The computational complexity can be partially reduced by splitting the averaging operation into two consecutive one-dimensional passes with kernels of n x 1 and 1 x n pixel [OQn)], instead of performing one pass with an n x n kernel [O(n2)]. Another compromise approach for determining the background shading by linewise image averaging has been proposed by Dapoigny et al. (1991). The length n of the one-dimensional smoothing n x 1 filter can be adaptively optimized dependent on the line contents. Much faster smoothing filters for computing the background image b(x,y) can be realized by using the recursive technique proposed by Deriche (1990). The implementation of these recursive filters is rather complex, and it requires a number of auxiliary image memories. Their smoothing effect depends only upon the choice of one parameter, and it can be made

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equivalent to that of a nonrecursive filter with a very large window, while the computation time is very low and independent of the intensity of smoothing. Beside the linear approaches, there are nonlinear techniques aiming at the extraction of a background image b(x,y ) which should be more suitable for yielding the shading function to be subtracted. Some of them will be described in the following.

b. Background Extraction by Linear Regression. A relatively simple and fast method for extracting the background information is to approximate it with a piecewise linear two-dimensional gray-value function g(x,y ) of the spatial coordinates x and y . The whole image is subdivided into Q x Q square fields, each of L x L pixels, where L is a power of 2. As shown in the lower left of Fig. 10, g(x,y) is then the juxtaposition of the linear partial backgrounds of the single fields. The value of Q should be

FI~URB 10. Example of shading compensation by background subtraction. Upper left: Original watermark image. Upper right: result of the background subtraction with the background extraction method of gray-value tracking with lag. Lower left: Background extracted by piecewise linear regression in 32 x 32 fields. Lower right: Result of the shading compensation by subtraction of the background shown at the lower left, with a weighting factor of k = 0.6.

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IMAGE ENHANCEMENT

adapted to the rate of variation of the background over the image field: The less uniform the background is, the higher Q should be. For a background with very slow variations, Q can be set equal to 1. Inside of each field F,,, 1 Ic, r IQ,the gray-value function f(x, y ) is approximated by a separate linear function

- X c r ) + BcAY

+ Ccr, where A,,, B,,, and C,, are constants, and (X,,, I-&) is the center of Fcr. Choosing the coordinate axes in this way simplifies the computation of A,,, B,,, and Ccr.Minimizing the mean square error [g,,(x,y) - f(x,y)]’ over each field, one obtains gcr(X,Y) =

1

Ccr

=-

-

X-r)

C f ( ~Y ), ,

LZFEr

with

U

=

C (X

- 0.5

- X,,)’

4,

=

C ( y - 0.5 - I&)’. Fcr

In other words, one obtains the plane coefficients by convolving the image with three kernels, two consisting of the local coordinates, and the third being an averaging kernel. The enhanced image u(x,y) results from a weighted difference between the original image and the background: 1

U ( X , Y ) = -[for,u) -

1-k

k * g(x, u)l.

Figure 10 shows at the upper left the watermark of an ancient manuscript, and at the lower left the background obtained with L = 32. The discontinuities at the field borders of the background image can be smoothed with a fast small-window (21 x 21 in this case) separable averaging filter before performing the image subtraction, whose result is displayed at the lower right of the same figure. Subtracting a background function, as described earlier, which substantially spans the same gray-value range as the original gray-value function may cause too strong an enhancement of the high spatial frequencies, contained in the differencef(x, y ) - k * g(x, y ) , if k is chosen too near to 1. For obviating this effect, Lamure et al. (1989) propose the use of a modified background function g‘(x, y), obtained from g(x, y ) by shifting it downwards, in the gray-value scale, until it darkest point attains a gray value

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of 0. The value of Q is taken equal to 1. The enhanced, background-free image signal u(x, y) is given by u(x, Y ) = f ( x , Y ) - g(x,Y ) +

min

(X,Y) E 4 1

M X , Y)l.

c. Background Extraction by Gray- Value Tracking with Lag. Also, the method developed by Voss and SuDe (1991) aims at determining the image component with very low spatial frequencies, without making use of a timeconsuming large-window averaging filter. The main idea is that of shaping a background function b,(x,y), which slowly tracks the gray-value function f ( x , y ) , scanned for instance linewise from left to right, in such a way that sudden variations of f ( x , y) are not reproduced. This is obtained by means of a technique similar to delta modulation. If x and y are discrete coordinate values, b,(x,y) is given by bl(x - 1 , ~ +) 1 if bdx, Y ) c f ( x , y ) , bdx - 1, u) - 1 if bdx, Y ) > f(x,Y ) , bdx - 1, Y ) if bdx, Y ) = f(x u). The streaks in the scanning direction, affecting b , (x , y ), which are due to the lag effect, are compensated by performing two horizontal scans (left to right and right to left) and two vertical scans (bottom to top and top to bottom), thus obtaining four background signals b i ( x , y ) ,i = 1, ..., 4, with obvious modifications of the preceding formula. The final background image b(x,y), to be used for the weighted subtraction fromf(x, y) as in the previous approaches, is obtained as b ( x , y) = min (bi(x,.Y)). i = 1...4

The upper right of Fig. 10 shows an example of background subtraction for the watermark image at the upper left, with a weighting factor of k = 0.6.

d. Weighted Unsharp Masking (Jackson and Kaye, 1982). The conventional unsharp masking is a well-known method for enhancing high spatial frequencies by subtracting from the original image f ( x , y) a blurred version b(x,y) of the same. If b(x,y) is obtained by means of a linear low pass, as for instance by determining the unweighted average gray value inside of the observation window U,the unsharp masking is equivalent to a linear high pass. Now, let us consider an ideal gray-value step function with step height A , and an edge pixel p = (xo,yo). Unweighted averaging in the neighborhood of p results in a function b(x,y) having approximately the same value on both sides of the edge. Therefore, the subsequent weighted subtraction [l/(l - k)] * [ f ( x , y ) - kb(x, y ) ] will not eliminate the step function in the neighborhood of p .

IMAGE ENHANCEMENT

25

The zonal filtering technique improves the unsharp masking by introducing into the averaging operation the “hard” weights Wu depending upon a threshold T:

With this modification, if A > T, the step is preserved in the background to be subtracted, and it is thus eliminated from the filtered image. The method proposed by Jackson and Kaye (1982) is a modification of the zonal filter, which utilizes “soft” weights, given by a continuous monotone function w(d), with d = If(x,y) - f(i,j)l, decreasing from w(0) = 1 to w(G) = 0, if G is the full gray-value range. For obtaining better results, the authors propose to make use of the following variant:

P

for 0

Id IT ,

f o r d > T, where the parameter q controls the rate of fall of w(d) for d > T. e. Robust Background Estimation with Rank-Order Statistics. Although smoothing, which has been the subject of Section 111, can be considered as the opposite task of background subtraction, these tasks both need to perform first the estimation of a locally dominant gray value u(x, y) inside a more or less large observation window. For smoothing purposes, the window size tends to be smaller than for background compensation, where the window size may coincide with the whole image. For both applications, a robust estimation of the dominant gray value, i.e., an estimation which is not influenced by outliers, is of great importance. For instance, the method described by Chochia (1988) can be used for background extraction in the presence of impulsive and of random noise, if observation windows of proper size are used. The method of Chochia considers the current pixel p = ( x , y ) as the center of two windows, V and U C V , of different size, containing A4 and m pixels, respectively. U,e.g., with m = 5 , is expected to be homogeneous, while V is considerably larger and can also contain several homogeneous regions. The background image pn(x,y) results, at the nth iteration, from a converging median filtering process. This process is initialized by estimating the gray value pL,(x,y) by means of a locally adaptive k-nearest-neighbor filter acting upon the elements of U:

1

P ~ ( X , Y )= -

1

Q ( i , j ) tz u

f(i,j)

with If(x,y) - f(i,j)l < 2oU,

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where Q is the number of pixels satisfying the preceding condition, and 0,” is the variance in U.This filter performs an edge-preserving smoothing, as it averages f ( x , y) only with neighbors that are estimated to belong to the same region as (x, y), since their gray-value differences, with respect to f ( x , y), do not exceed a given threshold. Before the second filtering step, the integer ranks rl and r,, with 1 Irl < r, 5 M , are chosen empirically, and u,(x, y) is determined on the basis of the following law:

ifcr9

Y)

u1

=

f(rl)

iff(r,) c P ~ ( xY, ) < f ( r 9 if I C ~ ( XU) , 5 f(r,) if ~u (x, U) 2 f ( r 9

9

9

where f ( r ) , r = 1, ...,A4 are the rank-ordered gray values of V . The subsequent filtering steps are iterations, in successive passes of a median filter performed upon those gray valuesf(i,j) of V that meet the following condition: U , , + ~ ( X , Y )=

medianIf(i,j)l: [W)E VI A [ I f ( i , j )- U,(X,Y)~c A = 20,

where 0, is the variance in V . Depending upon the application and on the window size used, the filter result can be considered as a smoothed image with sharp edges, or as a background extracted from the image. In both cases, the preservation of the edge sharpness and the robustness with respect to long-tailed and impulsive noise are a consequence of the fact that this filter is an adaptive rank-order filter, although the filter function is not expressed explicitly in terms of a rank-order filter.

V. LOCAL CONTRAST ENHANCEMENT A most frequent source of image impairment is the poor contrast with which high spatial-frequency detail is reproduced. This is often due to the decrease of the imaging system’s modulation transfer function at higher spatial frequencies. Equalizing the modulation transfer function by means of a linear filter mostly does not represent a satisfying solution, because this would enhance the random and discretization noise more than the useful image contents at high spatial frequencies. An enhancement may also be useful when there is no appreciable noise caused by the imaging system, and an enhancement of the local contrast is desired for a better visual image assessment-for instance, in medical diagnostics. A similar situation occurs

27

IMAGE ENHANCEMENT

in listening to music, where the listener can choose a less faithful but more pleasant reproduction, for instance by raising the treble. This section describes several nonlinear local contrast enhancement methods, which generally perform better than the linear ones, and yet do not require a higher computational complexity. A representative set of contrast enhancement methods has been tentatively chosen out of the vast literature on this subject, and these methods have been subdivided into classes characterized by typical approaches. However, a neat distinction between classes is not always possible. Similarly, it is sometimes not easy to distinguish, based only on the effects produced upon an image, between a contrast enhancement and a shading compensation method. The latter is mostly associated with the subtraction of an image background, i.e., it implies generally a global gray-value shift, which eventually undergoes a partial compensation, while in the former the local average is not altered. In practice, the borders between these two cases are rather fuzzy. A . Local Range Stretching and Rank Transformation A contrast enhancement can be obtained by stretching the local grayvalue range in the observation window U to the full gray-value scale 0, ,G - 1. Several variants of this method are described in the literature, as for instance by Alparslan and Ince (1981), Fahnestock and Schowengerdt (1983), Yaroslavsky (1985), and Kim and Yaroslavskii (1986). The main difference between these variants regards the shape assumed by the local histogram after the stretching operation. In the first two methods just referenced, this shape is substantially preserved after the stretching; in the latter ones, a histogram equalization also takes place.

...

a. Hktogram Stretching with Shape Preservation. By ordering the N gray values f ( x ,y), with (x, y) E 17, one obtains the rank-ordered gray values f C i ) , i = 1, ...,N, with hi)5 f(i+l). The local extrema are therefore fcl) and f C m .If f ( x , y) is the current gray value and u(x, y) the filter’s result, the transformation

otherwise, linearly stretches the local histogram without altering its shape, except for the gaps between the histogram bins caused by the stretching. The size of U plays a decisive role as far as the intensity of contrast enhancement is concerned. For each window position for which f ( x , y) is equal either tofC,,

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or to f&, u(x, y) is equal to 0 or to G - 1 , respectively. Since the ranks 1 , ...,N are equally distributed, independently of the local gray-value distribution, the probability (equal to 2/N) of u(x, y) being identical to a local extremum grows with decreasing window size. Asymptotically, for a 2 x 1 window, one would obtain a binarized image, characterized by a maximum of local contrast. Variants of this basic technique are described by Alparslan and Ince (1981) and by Fahnestock and Schowengerdt (1983).

b. Histogram Stretching with Equalization. Let us consider the rank r, 1 I r IN, off(x, y) in the set of the ordered gray values of U ;r is defined by means of the scores, u and w, of the pixels of U with gray value inferior to and, respectively, equal to f(x, y):

r=u

+ integer(w/2).

If we perform the transformation

r- 1 u(x, y ) = - (G - I), N- 1

-

the gray values of f ( o are mapped into the gray values G-1 N - 1 (i - 1)

for i = 1,

...,N,

obtained by subdividing the full gray scale into N - 1 regularly spaced intervals. Thus, u(x,y) is the gray value of this set that has the same rank r as that of f(x,y) in the set of the &)'s. Since the ranks are equally distributed (see earlier discussion), the local histogram of the transformed gray values is also equally distributed. This is true in the same sense as discussed in Section II,B in relation to the histogram equalization, i.e., taking account of the histogram gaps between bins. As for the influence of the window size, it is the same as described earlier in the case of histogram stretching with shape preservation. Figure 11 shows at the upper left an original remote sensing image, and in the other quadrants the result of a contrast enhancement with equalization, with window sizes of 7 x 7 (upper right), 13 x 13 (lower left), and 21 x 21 (lower right).

c. Blurred Weighted Adaptive Histogram Equalization (Gauch, 1992). In the methods described previously, for adjusting the intensity of the enhancement effect deriving from the local histograms stretching, the user has no other freedom than to vary the window size. Moreover, all the gray values contained in the window contribute with the same weight to determine the gray-value transformation function; if the window size is

IMAGE ENHANCEMENT

29

F i a m 1 1 . Example of the contrast enhancement obtained by histogram stretching with equalization on the remote sensing image of the upper left, with various window sizes: 7 x 7 (upper right), 13 x 13 (lower left), and 21 x 21 (lower right). (From Klette and Zamperoni, 1992.)

considerable, it would better conform with the image structure to assign a decreasing influence to pixels which are located farther from the current pixel. The method of Gauch (1992) aims at obviating these disadvantages. The approach used for performing the local equalization is the same as in Section I1,B. It utilizes the cumulative gray-value distribution function IfL@), where the integer g (0 Ig IG - 1) is the gray value, as the input/output gray-value transformation function for determining the enhanced gray value u(x, y). Here, however, HL(g) is computed in an L x L window U,centered upon the current pixel (x, y). In order to take account of the pixel locations inside U, the gray values . f ( i , j ) , ( i , j ) E U, are weighted with weights w(i, j) shaped according to a Gaussian function G(i,j ) , with G(i = x , j = y) = 1, and G ( i , j ) = 0 at the borders of U. The amount of contrast enhancement can be varied by computing HL(g) using, instead of the local histogram hL(g), a smoothed histogram hk(g). The basic idea is that, if h t ( g ) would be perfectly smooth, HL(g)would be the identity transformation, and no gray-value modification would take

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place. By smoothing hL(g) more or less strongly, all the intermediate enhancement effects between full and no enhancement can be attained. The smoothing is performed by convolving hL(g) with a one-dimensional Gaussian kernel K&) depending upon the scale parameter q. For each image point (x,y ) the following procedure, depending upon the user parameters L and q, must be carried out: 1. Compute the local histogram: f o r p = 0, ..., 1 - G laL@) 2. Compute the smoothed histogram: hL(g)=h,(g)*K,(g) f o r g = O , ..., 1 - G . 3. Compute the cumulative distribution: %(g) = c:=oh;(s). 4. Determine the transformed gray value: U(XY

u) = (G- 1)

*

HL(.f(X,Y)).

The histogram equalization approach, as described in Section II,B and in this section, takes account of objective gray-value differences, but not of the subjectively perceived contrast. The contrast sensitivity of the visual system with respect to the relative gray-value difference Ag/g between two regions depends in reality upon the luminosity go of a uniform background, in which the two regions are embedded. This senisitivty is a maximum for g = go, and decreases monotonically for other values of g. Mokrane (1992) has developed a modified histogram equalization method that aims at a uniform subjective contrast discrimination over the whole gray value range, based on experimentally determined sensitivity curves. In complex scenes, the background luminosity is set equal to the global average gray value. B. Inverse Contrast Ratio Mapping

The consideration of the sensitivity of the visual system is at the basis also of the inverse contrast ratio mapping approach (Jain, 1989): The visibility of a pattern depends on the ratio a / p between local standard deviation and local average. For enhancing the visibility of low-contrast patterns, the original gray value f ( x , y ) is scaled by a factor proportional to the local value of p(x, y ) / o ( x ,y ) . After this local scaling, performed with each pixel, it is necessary to carry out a global scaling in order to match the gray-value range of the filtered image to that of the original image. For this aim, the resulting gray values are multiplied by the ratio M J M , between the global average gray values of the original and of the filtered image. Since Mz can be determined only after the image filtering scan, a second scan is necessary for the final scaling.

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IMAGE ENHANCEMENT

The window size L represents a user parameter that determines the intensity of the enhancement effect. The operations to be performed for every pixel (x, y ) in order to compute the filter output u(x, y) can be summarized as follows: 1. Initialize:

2. First pass:

Update: 3. Second pass:

Mi = O , u‘(x, y ) =

M2=0. P(X,Y )

1

+ a, Y)

MI := MI + f ( x , y ) ,

M1 M2

M2 := M ,

+ u’(x,y).

-

u(x, y ) = - u’(x, y).

An example of the enhancement effects obtainable by means of the inverse contrast ratio mapping is shown in the upper half of Fig. 12. The original microscopic cell image at left has been processed with a 7 x 7 window, and

the result is displayed at right.

FIGURE12. Upper row: Enhancement of the microscopic cell image at left by means of the inverse contrast ratio mapping with a window size of 7 x 7. Lower row: Edge-based adaptive contrast enhancement of the original upper left image with a window size of 9 x 9 and r = 0.5 (at left). The resulting image has been further de-enhanced with the same method, by choosing r = 3 and a window size of 11 x 11 (at right). (From Klette and Zamperoni, 1992.)

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C. Extremum Sharpening

The extremum sharpening technique was investigated very early in image processing by Kramer and Bruckner (1979, who applied it to line-drawing reconstruction, and who analyzed its convergence properties after a number of iterations. Later, Lester et al. (1980) used it in the enhancement of biomedical images for diagnostic purposes. The extremum sharpening can be considered as a local binarization: The operator output u(x, y ) assumes either the value of the local maximum f ( N ) or that of the local minimumf(,, in the N-pixel observation window U, depending upon which one of these values lies closer to the gray value of the current pixel f(x, y). The floating binarization threshold is implicitly set at V ( N ) + f(l))/2:

The effects of the extremum sharpening upon a one-dimensional gray-value function are illustrated qualitatively in Fig. 13 for various window sizes. Local extrema act as attractors for neighboring gray values, and step edges are preserved. Their visibility is also enhanced, since the intermediate gray values of a transition are eliminated. Unlike with median filters, spikeshaped patterns can also be preserved, if the window size L is small enough. For larger window sizes, a spike remains unchanged only if there is no other

b

- f

--

4

-____-

F : *A :

__3

L=4

U

LJJ I

L

FIGURE13. Effects of the extremum sharpening operator with various window sizes L upon some standard one-dimensional gray-value functions. Preservation and enhancement of small-size patterns (a), and of ramps (b). (From Klette and Zamperoni, 1992.)

IMAGE ENHANCEMENT

33

FIGURE14. Example of a remote sensing view (upper left) processed with the extremum sharpening operator with various window sizes L . Upper right: L = 3; lower left: L = 7; lower right: three-level variant, with L = 11. (From Klette and Zamperoni, 1992.)

spike with higher contrast in U (see Fig. 13a). As shown in Fig. 13b, monotone portions off(x, y ) of size greater than U,as for instance ramps or power or exponential functions, are just shifted by L / 2 pixel by the operator. The coarseness of the resulting image, i.e., the degree of elimination of small details, grows with the window size L. This is demonstrated by Fig. 14, showing at the upper left an original remote sensing view, and the resulting of the extremum sharpening with L = 3 (upper right) and with L = 7 (lower left). The lower right image was obtained with L = 11, and with an operator variant characterized by a more gradual enhancement of the local contrast. Considering the average gray value W in U,this variant performs a three-level quantization:

otherwise.

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Fairfield (1990) developed an enhancement filter (the toboggan algorithm) which has the same aims as the extremum sharpening filter, but which follows another principle. Considering a 3 x 3 observation window, centered upon ( x , y ) , the filter output u(x,y) assumes the value f ( i ,j ) of that pixel ( i , j ) E U for which a measure of discontinuity d ( i , j ) is a minimum in U. For instance, the gradient d ( i , j ) can be used as a measure of discontinuity:

D. Adaptive Contrast Enhancement in the Neighborhood of Edges

In the approach of Beghdadi and Le Negrate (1989), a particular importance is attributed to the contrast enhancement in the neighborhood of edges. Thus, it is a locally computed edge value E that controls the amount of contrast enhancement. The local contrast C, which has to be eventually enhanced, is a measure of the deviation between E and the current gray value f ( x , y ) . First, a transformed (eventually enhanced) contrast value C' is computed. Then C' is introduced into the inverse formula, which gives the current gray value as a function of the contrast, for determining an enhanced output gray value u(x,y). This results in a higher edge contrast than in the original image. Let us consider an L x L observation window U,centered upon the current pixel (x,y ) ;the elements of U are defined by ( X , Y) = (x - k + i, y - k + j ) , 1 Ii , j IL, with k = (L + 1)/2. The edge value is the normalized and weighted gray value in U,where the weights dij are the values of an edge gradient operator, computed in (x - k + i, y - k + j ) : r

E=

1-1

1 1 Edij ij

~

-

ldijf(x-k+i,y-k+j)

for21i,jsL-

1.

ij

The variation range of i a n d j is limited as shown because the computation of an edge gradient, as for instance of the Prewitt operator considered next, requires that a 3 x 3 window U,, c U be centered upon ( X , Y). The Prewitt-operator-based expression for dij = d,, is d,,

=

max(If(X - 1 , Y + 1)

+ f ( X , Y + 1) + f ( X + 1 , Y + 1)

- f ( X - 1 , Y - 1)-f(X,Y-

l)-f(X+

If(X-1 , Y + l ) + f ( X - l , Y ) + f ( X -f(X+ 1,Y+ l)-f(X+

l,Y)-f(X+

1 , Y - 1)1, 1 , Y - 1) 1 , Y - 1)ll.

IMAGE ENHANCEMENT

35

The local contrast C is defined as the Canberra distance between the actual gray value and the edge value:

The inversion of this formula gives f(x, y ) as a function of E and of C :

The contrast enhancement is attained by transforming the observed value of C into a new value C ’ , C’ 2 C , by means of the concave characteristic C’ = C‘, with r < 1 a real number. r is a user parameter by which the enhancement intensity can be adjusted. The enhanced output gray value u(x,y) is thus obtained by means of the formula reported earlier, which gives f(x, y ) , by substituting C with C ‘ . A contrast attenuation can also be obtained, by choosing r > 1 . An example of the effects of this operator is given by the lower half of Fig. 12. The microscope image at the upper left has been processed with a window size of L = 9, and r = 0.5. The result, displayed at the lower left, shows a strong enhancement of the epithelium cells with respect to the background. For attenuating this effect, this image can be furtherly processed for a contrast de-enhancement, with L = 1 1 and r = 3, thus obtaining the image at the lower right of Fig. 12. From the description of this method, it emerges that the enhanced contrast value C’ is very sensitive to the current gray valuef(x, y). If f(x, y ) is strongly affected by noise, or if it is an outlier, it can simulate higher edge values than it is correct to assume. Further, if we choose a large window size, the value off(x, y ) can be very little representative of the gray-value population inside U.Considering these limitations, some improvements have been introduced by Le Negrate et al. (1992). In the preceding formulas, f(x, y ) has been substituted by a weighted averagef,(x, y ) , obtained by convolution with a low-pass kernel, which reflects the spatial dependence between neighboring gray values. A bimodality analysis, conducted on the processed pictures, shows the benefits of this operator for image segmentation pre-processing, especially if the latter is performed by aid of multilevel thresholding met hods. Alternatives to the mapping function C‘ = C‘ for the contrast enhancement are discussed by Yu and Mitra (1993). In particular, they suggest an iterative law of dependence between Cand C ’ . If C; denotes the value of C’

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after the tth processing iteration, the following polynomial expression and initialization values are used for deriving C; : C;+, = (1 - a)C;

+ a(C$

with C A

=

C,

where a is a parameter between - 1 and 0. This causes the contrast enhancement to be weak when C is low, and strong when C is high. Moreover, Yu and Mitra (1993) also proposed other laws of dependence between C and C ’ , in which the relation C’ 2 C does not hold over the whole range of values of C, as is the case for the law C’ = C‘, but where there is an enhancement range (where C ’ 2 C ) , and a de-enhancement range (where C’ IC ) . The basic method of Beghdadi and Le Negrate (1989) has been taken as a basis also by Dash and Chatterji (1991), who obtained a diminution of the computation time, without a substantial performance degradation, by aid of an interpolation technique. The image is subdivided into Q x Q nonoverlapping square regular blocks CR,,, 1 Im , n IQ, called contextual regions. Then the mean edge gray value and the enhanced contrast value are computed for the whole contextual region, and they are used for determining the enhanced gray values in a mapping region MR,, centered upon CR,,, but of double side length. This processing step can also be sped up by making use of look-up tables that take account only of the gray values actually occurring in MR,,. The final enhanced gray value u(x, y ) is then determined as a combination, by means of a bilinear weight function, of the gray values of the four mapping regions in which the point (x, y ) is included.

E. Contrast Enhancement Based on a Pyramidal Image Model A most sensitive point in image enhancement, as in other fundamental image-processing operations, is the problem of specifying the scale level at which are situated the patterns to be enhanced. This determines an empirical choice of the observation window size, a choice which is necessarily not an optimum for all the image’s patterns that need to be enhanced. Pyramidal image data structures and pyramidal algorithms, described for instance by Rosenfel and Kak (1982), have the advantage of making it possible to take account of informations extracted from intermediate scale levels, and to use them, e.g., for adaptively optimizing the parameters of low-level operators. A pyramidal data structure, illustrated summarily in Fig. 15, consists of arrays of gray values g(pk) of pixel Pk at different resolution levels Lk , k = 0, ..., K . Level Lo consists of the n x n pixel of the original image,

31

IMAGE ENHANCEMENT

FIGURE15. Schematic representation of a pyramidal image data structure with four resolution levels L,, k = 0,..., 3 = K . *, p k : generic pixel at level L,; 0,p y l , , i = 1, ..., 4: “sons” of p k ; 0,p $ i l , i = 1, .. ., 4 : “fathers” of p k .

while level Lk has 2K-k x 2K-k elements. The gray values g(pk) of Lk are generally obtained as linear combinations, with weights wl,. ., w4, of subsets (which may also be not disjoint) of four pixel (“son”) gray values of the next lower level Lk-l, as shown schematically in Fig. 15 for K = 3 and k = 1:

.

4

dpk)

=

c wi

*

g(pfL1)-

i= 1

Similarly, the background B(pk) function for pixel pk at Lk can be thought

of as a linear interpolation, with weights ct/;, of the grey values of the spatially nearest pixel (“father”) of the next higher level Lk+l: 4

B(pk) =

c

*

g(pf!l).

i= I

In the pyramid-based contrast enhancement method of Jolion (1993), the local contrast C(p,) at level Lk is defined as the ratio g(pk)/B(pk) between gray value and background. Similarly, as in other methods already examined, the contrast enhancement is obtained through the following steps: 1. Choose the appropriate pyramid level Lk . 2 . Determine c ( p k ) = g(pk)/B(pk).

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3. Determine the enhanced contrast value c ’ ( p k ) as a function C’ = d(C) of C ( p k ) ,such that

c

i f C = 1,

C’
if C < 1,

C’>C

i f C > 1.

C’ =

This can be realized, for instance, by means of the power function C’ = C‘, with r > 1 and real. 4. Obtain the enhanced gray value g’(pk)by using the inverse formula g’(pk) =

C’(Pk) ’B(pk)*

5 . Repeat the same procedure for all the pixels pk of level Lk.

One advantage of the pyramidal approach is that the intensity of the enhancement effect, which depends upon the value of the exponent r, can be made dependent on the pyramid level-for instance, it can be made stronger at the lower pyramid levels, where the finer detail is present, by taking r = 2(1 - k / K ) .

VI. DETAIL ENHANCEMENT Image analysis is a generic expression, mostly designating a variety of tasks to be carried out all at the same time. One of these tasks is the preservation or the enhancement of high spatial-frequency patterns-for instance, those representing human artifacts such as roads, channels, buildings, or urban settlements, embedded into a natural background in remote sensing views. However, a high spatial-frequency content is no peculiar feature of detail patterns, being shared with many natural textures. While image analysis requires mostly that the texures should be smoothed, in view of the uniform labeling of homogeneous-textured regions, spatially ordered detail patterns should mostly be made more visible, and more easily segmented, by means of some adequate enhancement technique. Thus, it is evident that the feature which discriminates between high spatial-frequency patterns to be smoothed and those to be enhanced is rather some measure of the spatial order of the pattern under consideration. Within the scope of this task, some detail enhancement methods do not attempt an explicit definition of the local degree of order, but try to preserve this order by considering the gray-value function in different onedimensional subsets of pixels. Other methods rely upon the extraction of an “anisotropy” or “local spatial order” feature, and use it for controlling

IMAGE ENHANCEMENT

39

the parameters of an adaptive filter. Finally, other methods are based on the mathematically morphology operators, and achieve the detail preservation by making use of shape-adapted structuring elements. The presentation of some typical examples of these approaches will be the subject of the next subsections. A . Detail-Preserving Multistage One-Dimensional Filters

The main idea of multistage one-dimensional filters, introduced by Nieminen et al. (1987), and further developed especially by Arce and Foster (1989), by Wang and Wang (1990), and by Wang (1992b), is that the advantages of the nonlinear smoothing performed by the median filter can be combined with the preservation of thin line-shaped patterns by making use of a set of one-dimensional observation windows. The final filter outputs results mostly from a data-dependent selection or combination of the partial filtering outputs, obtained on each one-dimensional window. Let us consider the L x L observation window U,centered upon p(x,y ) and represented in Fig. 16a, and the four one-dimensional subwindows U,, , a = 1, ..., 4 . It is evident that, especially if the window size is small, the intersection of U and a thin line through p will mostly coincide with one of the subwindows U,,.Now let us consider the output p,, of a smoothing operation performed on U,,.More generally, p,, can be considered as the , supposed to be uniform and affected by estimation of the gray value of U,, noise having a certain distribution. Although the optimum estimates differ , the average and the depending on the gray-value distribution inside U,, median Mu represent good estimates in several concrete cases.

b

a

m-'2 2 2 0 2 2 2

Lpixel

U

FIGURE 16. (a) Observation window U and its one-dimensional subwindows U , , . .., 4 , considered in multistage filters. (b) Determination of the T = 2(L - 1) rectilinear one-dimensional subwindows U, ( t = 1, ..., T ) containing the current pixel p , which can be inscribed in an L x L window. Two subwindows, for t = 2 (0)and for t = 9 (0). are shown. (From Klette and Zamperoni, 1992.) u = 1,

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In hybrid multistage filters, a normal distribution is assumed, and the filter output u(x, y ) is obtained as follows: u(x, y ) = med ian lz, a = 1, ..., 4 )

Median multistage filters, whose output is u ( x , y ) = median(M,, a = 1, ..., 41

are superior from several points of view: The median is a robust estimator in presence of outliers. In fact, if p lies on a thin line, all the subwindows U, not lying in the line direction feature a uniform distribution with a strong outlier: the line pixel p . It is easier than in the two-dimensional case to discriminate between line segments (to be preserved) and short strokes (to be suppressed) on the basis of their length, and to choose the proper window size for this. In fact, a one-dimensional median filter with a window size of L leaves monotone sample sequences of length rn 1 (L + 3)/2 unchanged. Thus, the median multistage filter eliminates strokes up to a length of rn pixels if its window size is made equal to L > 2rn - 3, with L odd. A step edge (if preserved) is not blurred by the median filter, because the median, as a rank-selection filter, does not introduce new intermediate gray values not present in the original signal. In an adaptive variant of the median multistage filter of Presetnik and FilipoviC:(1988), the first stage is substituted by the search of the subwindow U, with a minimum of absolute average deviation from the subwindow mean gl;: u(x, y ) = M , ,

for which D,

=

min (0,) (I=

with 0,= (idE

u,

Ig(i,j) -

1 ... 4

El.

Standard-deviation controlled subwindow selection approaches of this kind are shared by several authors and have been tested successfully in a number of practical applications to the line-preserving smoothing of remote sensing views, interferograms, and microscopic images. Detail enhancement can be considered in many cases as the enhancement of patterns whose gray values represent a minority in the gray-value distribution inside of U. This is the basic concept of the max/min-median multistage filter of Wang and Wang (1990). In the first filtering stage the

41

IMAGE ENHANCEMENT

subwindow medians M a , for a = 1 , ..., 4, and the median Mo over the whole U are computed. The filter output is then made equal to that extreme value of the set (U,) which has a stronger deviation from M,: Max W , Y )=

Min

if [Max - M,I 2 (Min - M,I, otherwise,

with Max = max ( M a ) a = 1 ...4

and

Min

=

min { M a ) . a=1...4

Figure 17 shows, on the right, two examples of detail enhancement performed by means of the max/min-median filter on the original images on the left. Among the numerous multistage median filter structures described by Arce and Foster (1989) together with their statistical and deterministic properties, those based on sets of one-dimensional subwindows seem to be especially useful within the scope of a detail enhancement task. Other filters, which utilize two-dimensional cross-shaped subwindows, are more

FIGURE 17. Example of two images processed with the max/min-median filter. Upper row: Watermark, 5 x 5 window. Lower row: Synthetic aperture radar image, 3 x 3 window, two iterations. (From Klette and Zamperoni, 1992.)

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advantageous for edge-preserving smoothing purposes. The two filters (I) u(x, y ) = median(Min, Max,f(x, y ) ) , (11) u(x, y ) = median(median(M, ,M , ,f(x, y ) ) ,median[M,, M4 ,f(x, y ) ] ] have been demonstrated to be equivalent by Arce and Foster (1989). They attain a good compromise between an effective noise smoothing and the preservation of both bright-on-dark and dark-on-bright linear patterns; on the other hand, they perform less well than the max/min-median filter as far as detail preservation is concerned. The reason is the inclusion of the original gray valuef(x, y ) among the arguments of the median operation at both filtering stages. The values Max and Min, defined as before for the max/min-median filter, can be assigned to the output u(x, y ) only if f(x, y ) is an outlier which deviates significantly in the same direction both from Min and from Max, also inside the subwindows U,. On the basis of the multistage median filter just described, several adaptive extensions have been developed. Generally, in the proposed adaptive variants the filter output is selected between the output of the basic multistage median and the median over the whole window U,depending on a user parameter which determines the maximum length of linear strokes that should be deleted. The aim of the multistage median filter generalization of Wang (1992b) is to provide, by means of user parameter, more precise control of the sieving of short lines oriented in one of the four main directions. In the first stage of the generalized multistage median filter, the gray value f(.). with rank r (1 Ir IL ) is extracted from the L - 1 rank-ordered gray values I... ~ f ( ~of - the ~ subwindow ) ~ U; = (U,- p ) (a = 1, ..., 4), i.e., from the subwindow Ua without the current pixel p , which, being common to all the Uu’s,can be neglected. Thus, the ordinary multistage median filter is just a particular case of the general one, for r = (L - 1)/2. Except for substituting the median with the general rank r, the first and the second stage are analogous to the ordinary multistage median filter. The filter output is u(x, y ) = median[Max, Min,f(x, y ) ) with Max = max [f(.)@] u=1

...4

and

Min

=

min a=l

...4

[f(r)u).

The filter properties with respect to the preservation or deletion of linear strikes of given length can be controlled through the choice of the user parameter r in the following way. Let us consider for simplicity a binary image with gray values 0 and 1. If any subwindow contains a bright line

IMAGE ENHANCEMENT

43

(or a stroke) of at least Q pixel length (Q < L ) , this line will be preserved by assigning to the parameter r the value L - Q + 1. In this case, both Max = f& , for one of the values of a, and f ( x ,y ) are equal to 1, and the majority count of the second stage yields u(x,y) = 1. Similar considerations hold for the dark line case. In this way the parameter r can be used for controlling the filter’s sieving behavior with respect to short segments. B. A n Extended Set of One-Dimensional Subwindows for Multistage Filtering

Most of the multistage filters described by several authors consider only the four one-dimensional subsets U,, ..., U, in the main directions, shown in Fig. 16a. However, and especially if the window size is not very small, the four main directions alone may not take account of important linear patterns to be enhanced whose orientations lie between the four main directions. As illustrated in Fig. 16b, in an L x L window can be inscribed T = 2(L - 1 ) digital straight line segments of length L and containing p . The performance of all the multistage filters considered up to now, and also of the adaptive rank-order filters to be described in the next subsections, can be sensibly improved by taking account (for instance, in the computation of Max and Min) of all the orientations associated with digital straightline segments going through p . In the following, a simple control structure for addressing the elements of each one of the T one-dimensional subwindows U 1 , ..., U, will be described. This control structure is independent of the multistage filter type with which it is used. With reference to Fig. 16b, each subwindow U,is associated with a value of the counter t = 1, ..., T. Let z = 1, ..., L be the running index of the pixels of a subwindow. These pixels can be addressed by building two twodimensional L x T look-up tables Dx(z, t ) and Dy(z, t ) , containing, for each pixel, the relative coordinate increments with respect to the current pixel p = ( x , y ) . The look-up tables can be computed by means of the following procedure:

k = (L - 1)/2 For t = 1 to t = L , compute: Ay=l+k-t For z = - k to z = k, compute: DY(z, t ) = ( - ( Z A y ) / k ) Dx(z, t ) = Z , (where ( ) means “nearest integers of”)

44

PIER0 ZAMPERONI

For t = L + 1 to t = T , compute: Ax=t-L-k For z = - k to z = k , compute: Dx(z, t ) = < - ( Z A x ) / k ) , DY(z, t ) = Z . Figure 16b shows two examples of digital straight-line segments, for t and for t = L + 2, with L = 7.

=

2

C. Detail Enhancement by Means of Adaptive Extremum Sharpening Filters The contrast enhancement properties of the extremum sharpening filter have been examined in Section V,C. The extremum sharpening filter performs a local binarization, which assigns to the current pixel (x,y ) the gray valuef(,, of the local minimum (in the N-pixel observation window U ) or the valuefcN, of the local maximum, depending on whether the current gray value f ( x ,y ) lies closer to f(l)or to f c N ) . This filter enhances any local contrast, without regard to the shape features of the underlying grayvalue pattern; this can give rise to artifacts in the form of spurious edges and inconsistent flat regions. However, the properties of the extremum sharpening filter can be usefully exploited by making its behavior adaptively dependent upon locally extracted features, which detect the presence of a pattern to be enhanced. The two following subsections describe two examples of adaptive extremum sharpening filters, which have proved to perform a very effective detail enhancement. The filters differ only in the local feature that controls the filter’s adaptation, i.e.: 1. The local anisotropy degree, for the enhancement of thin linear patterns. 2. A local measure of the degree of spatial order, for enhancing not only lines, but, more generally, any pattern with spatially ordered structure. 1 . Anisotropy-Controlled Adaptive Extremum Sharpening Filter

A local degree of anisotropy a can be defined axiomatically as a quantity between 0, for a perfectly isotropic window (a gray-value terrace), and 1 , for a maximally anisotropic pattern, as for instance for a one-pixel thin straight line. With reference to the one-dimensional subwindows U,, ..., UT represented in Fig. 16, where T = 4 in Fig. 16a, and T = 2(L - 1) = 12

45

IMAGE ENHANCEMENT

in Fig. 16b, let p t , t anisotropy measure a =

Max - Min Max + Min +

=

E

1, ..., T , be the average gray value in CJ,. The

with Max = max ( p t ) ,Min t = I ... r

=

min ( p t ) ,

t= 1

... T

where E is a very small quantity preventing a division by zero, has been shown to be particularly effective compared to several other investigated measures (Zamperoni, 1992). Some of its advantages are (i) a assumes the value 0.5 for an ideal step edge; (ii) a increases with decreasing average local brightness, in accordance with the human perception of the details of an image (see also Fig. 21). In order to preserve the details, the filter must be of the rank-selection type, because, depending on the value assumed by the control parameter a, it should be able to select and to assign to the output a “minority” gray value, as that of a thin line containing p . This cannot be accomplished by a median (“majority”) filter or, generally, by an alpha-trimmed-mean filter. Therefore, the gray values of CJ must first be rank-ordered; one obtains the ordered set f(l), ...,f“). The filter output u(x,y ) is then set equal to f(T), where r is a rank between the median m = (N+ 1)/2 and 1 if f ( x , y ) is closer tof(,) than to fcN);otherwise, r is between m and N. The difference r - 1 (or N - r) is inversely proportional to the anisotropy measure a:

r=

[

(m + - 1)) ( m - a(m - 1))

i f f @ ,Y ) > ( f ( N ) otherwise,

+ f(1))/29

where ( ) means “nearest integer of.” Thus, the filter performs as a median filter in isotropic neighborhoods, and as an extreme value filter in presence of a strong anisotropy, with all the intermediate stages. The upper row of Fig. 18 shows, on the right, an example of the application of this filter, with a window size of 7 x 7, to the remote sensing view on the left. One can notice that, in spite of the considerable window size needed for smoothing purposes, the anisotropic thin-line details are preserved. 2. Adaptive Extremum Sharpening Filter Controlled by a Measure of the Local Spatial Order Based upon the same idea as the previous filter, this filter aims, more generally, at enhancing all the patterns with a spatially ordered structure, as for instance pattern b in Fig. 19d, and not only the line-shaped ones, as for instance pattern a of the same figure. For this aim, it is necessary to define opportunately a “degree of local spatial order” Q, with 0 5 Q 5 1, to be formally substituted for a in the preceding formula giving the rank r of the selected filter output f(r).

46

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FIOURE18. Examples of the application of adaptive extremum sharpening filters to the remote sensing view of an urban area (upper left) and to an infrared image of a street scene (lower left). Upper right: Anistropy-controlled filter with a 7 x 7 window. Lower right: Filter controlled by the local degree of spatial order with a 9 x 9 window.

Two approaches to a definition of Q have been introduced and investigated by Zamperoni (1993). In one of these approaches Q is determined on the basis of the second-order statistics of the local gray-value distribution inside U,since the first-order statistics alone does not take account of the spatial relationships between gray values. With reference to Fig. 19a, let us consider the distribution of the absolute differences d(c, n) between the gray values of pixels A and B of U at a distance c and at an orientation of n n / 4 . The basic idea of this approach, supported by theoretical and empirical considerations, is to consider the degree of spatial order related to some kind of cluster separation measure in an opportune space, spanned by the features d(c, n), for various values of the parameters c and n. Figure 19b shows an example of a two-dimensional space of this kind, spanned by the absolute differences d(1,o) between the gray values of horizontal neighbors and d(1, e) between vertical neighbors. Several cluster separation measures have been proposed in pattern recognition, but most of them are based upon the ratio of “between cluster spread” to “within cluster spread.”

47

IMAGE ENHANCEMENT

C

&,

d

>

0

0

- Pattern a

----

E(d)

D

*

0

Pattern b

FIGURE19. Illustration of the definition of the local degree of spatial order. (a) Observation window U with two pixels, A and B, at distance c in the direction at an angle of n/4. (b) Feature space spanned by the absolute gray-value differences d ( l , 0 ) between horizontally neighboring pixels, and d ( l , e) between vertically neighboring pixels. (c) Example of the local gray-value density function q(d) of the absolute gray-value differences d between neighboring pixels. (d) Anisotropic (a) and isotropic (b) binary patterns with a maximum degree of spatial order. The histogram of the quantity d and the average p of the distribution are also shown. (e), (f), (g) Examples of binary patterns with a decreasing degree of spatial order. For each pattern, the histogram is shown and the location of the average p is indicated.

In our case, for reasons of computational simplicity, this model has been utterly simplified: Without distinguishing between the orientations of neighbor pairs, the quantities d( 1,o) and d( 1, e) have been merged into the unique feature d, and its distribution q(d), for 0 I d ID = hN)- f(,), has been considered. As the cluster separation measure, and thus as the

48

PIER0 ZAMPERONI

degree of spatial order, has been taken the normalized standard deviation a of d : a 2a Q=--

amax

-

D

*

In Fig. 19d-g are shown several typical binary patterns, together with their distributions and their averages p. The patterns a and b of Fig. 19d both feature the highest degree of order, although the former is maximally anisotropic, and the latter is perfectly isotropic. The values of Q for the patterns of Fig. 19e, 19f, and 19g are 0.98, 0.86, and 0.66, respectively. An example of the performance of this filter, with a window size of 9 x 9, applied to an infrared view, is shown in the lower half of Fig. 18.

D . Detail Enhancement Methods Derived from the Mathematical Morphology The mathematical morphology (see Haralick and Shapiro, 1992, for a tutorial introduction) provides a most efficient low-level image processing toolbox, based on the fundamental operations of erosion and dilation and on their combinations. It is well known that the basic morphological operations for gray-value images are the maximum and the minimum inside the observation window U,which plays the role of a flat structuring element. Thus, the morphological operators for binary images can be regarded as a particular case of the same operators for gray-value images. This technique gives the user the possibility of solving complex image analysis of pattern selection and enhancement problems, by developing an opportune sequence of erosions, dilations, and point-to-point operations between images, and by choosing at each time the proper structuring element. The leading criterion in developing the solution for a concrete problem is to exploit the selective reversibility of the mathematical morphology operators, i.e., the fact that, in the general case, the dilation does represent the erosion’s inverse operation (and vice versa) only for certain patterns contained in the image, depending upon the structuring element’s shape and upon the image contents. Thus, the task to be carried out can be generally formulated as follows: Establish an operation sequence which, seen as a whole, should be reversible with respect to the patterns to be extracted or enhanced (at the end these patterns are preserved) and irreversible with respect to all that is not interesting to the user (at the end it is eliminated). This is an implicit formulation of the problem, and therefore one which is not easy to put into practice without skill and experience.

49

IMAGE ENHANCEMENT

The soft morphological filtering, described by Kuosmanen et a/. (1992) and by Kuosmanen (1993), aims at giving the user the possibility of gradually varying the performance of the morphological basic operators by introducing the following extensions, which confer on the soft morphological filters some useful detail-preserving properties not possessed by the conventional morphological filters: The Max and Min operators are substituted by a general rank selection operation: The filter output is the rth smallest gray value of the set of the operator’s arguments. Combining the morphological operators with the weighted rank-order filtering technique, the set of the operator’s arguments is extended beyond the set of the gray levels of U. Each gray level of U is taken with a given multiplicity q, which can be assigned to a gray value on the basis of its spatial position inside of U , 4ij 0 f ( x

+ i, y + j ) = f ( x + i , y + j ) , ...,f ( x + i, y + j ) ,

1

L

qij times

or, on the basis of its rank r,

-

4rOfcr) = f ( r ) ,

*-*,&)*

4r times

The basic operations of soft morphological filtering, i.e., the soft erosion and the soft dilation, will be now defined assuming, for simplicity, a pointsymmetrical structuring element B with its reference point located in the center of symmetry. Figure 20 illustrates the effects of soft morphological filtering upon a binary image f ( p ) = f ( x , y ) , shown in Fig. 20b. Figure 20a shows the simplest structuring element, i.e., the “unity circle” in the 4-metrics. In the soft morphology, the structuring element B is subdivided into centerA C Ba nd border B\A. The definition of the border follows directly from the specification of the center, which does not comprise necessarily all the points of B having no 4-neighbor in the complement of B. For instance, A can consist only of the central point of B . Further, the integer parameter r , 1 5 r 5 IBI, where lB( represents the number of elements of B, must be specified. Let Bp represent the structuring element, placed with its reference point on the pixel p . The soft erosion of f ( p ) by the structuring system [B, A , r] is denoted and defined by f ( p ) O [B,A, r]

=

rth smallest value of the set

( r 0f ( a ) : a E A P )U [ f ( b ):b

E

(B\A),).

50

PIER0 ZAMPERONI

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

..

0

00

0000 0 . 0 0 0 0000

0..

00.

a

0-0 0

b

C

.... .... . . . . .... ........ .......... .... ...... .... .... . ..... ....... ...... .... . . . .... ...... .... . .... .... .... .... 00

0 0 0

00

0

0.. 00.0

0..

0

0.00

-

0.

0

0

0

0..

0 0

e

f

06.

0.

00.0.

0..

0

a

0.0 ..0.0. 0 0 0.0

0

0

0

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0 0.0

00 O..O.O.Q 00000

0

0

0.00

. . 0

0

0

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0..

00.

0.

00.

0000

00

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0

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06.

0000. 00.

0

0 .

0

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0

n

0 00 0 . 0

00.

0 0 (3. 0.0. 0 0.. 00.

00..

0

m

I

k

0..

0.. 0 O..O 0 00.0

0

. . 0 0

0000

0 0 0 0

0.

0

0

. 0

0000

0.

00 0

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0

0

0

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0 00 0 - 0

0..

i

h

9

0.

.0.

0

0 . . 0 0 0

0

0

0

0..

0

r

P

FIGURE 20. Basic operations of the soft mathematical morphology, performed on a binary pattern. (a) Structuring element B ; its center A is here defined as the central pixel 0. (b) Pattern under examination. (c), (d), (e), (f) Resulting patterns (0) of the soft erosion f ( p ) Q [B,A , r] with r equal to 1, 2, 3, 4, respectively. (g), (h), (i), ( j ) Resulting patterns (0 and 0 ) of the soft dilation f@) @ [B, A , r] with r equal to 1, 2, 3, 4, respectively. (k), (I), (m), (n) Resulting patterns (0) of the soft opening f ( p ) o [B, A , r] with r equal to 1 , 2, 3, 4, respectively. (o), (p), (q), (r) Resulting patterns (0 and ol of the soft closing f@) [B, A , r] with r equal to 1, 2, 3, 4, respectively. (From Kuosmanen, 1993.)

Similarly, the soft dilation of f ( p ) by the structuring system [B,A , r ] is denoted and defined as follows f(p) 0 [B,A , r] = rth largest value of the set

u (f(b) :b E (B\A),).

[ r Of@) :a E A P )

It can be easily shown that standard morphological operations are particular cases of the soft operations with B as a structuring element, if r = 1 o r i f A = B.

IMAGE ENHANCEMENT

51

Figures 20c-f demonstrate the gradual erosion effect produced upon the binary pattern of Fig. 20b with the structuring system of Fig. 20a and obtainable by varying the parameter r : -Fig. 20c: f(p) 0 [B, A , 11 Pixels with fewer than four 4-neighbors are deleted. -Fig. 20d: f(p) 0 [B, A , 21 Pixels with fewer than three 4-neighbors are deleted. -Fig. 20e: f ( p ) 0 [B, A , 31 Pixels with fewer than two 4-neighbors are deleted. -Fig. 20f: f(p) 0 [B, A , 41 Pixels with fewer than one 4-neighbor are deleted. It is evident that the soft erosion can preserve thin line structures and suppress spike noise at the same time; this is not possible by means of the conventional erosion. The corresponding gradual dilation effect, with the same binary image and the same structuring system, is illustrated by Figs 20g-j: -Fig. 20g: f(p) 0 [ & A , 11 Background 4-neighbor become object pixels. -Fig. 20h: f ( p ) 0 [ & A , 21 Background Cneighbors become object pixels. -Fig. 2Oi: f(p) 0 [B, A , 31 Background Cneighbors become object pixels. -Fig. 20j: f ( p )0 [B, A, 41 Background 4-neighbors become object pixels.

pixels with at least one pixels with at least two pixels with at least three pixels with at least four

In analogy to the conventional mathematic morphology, the soft opening defined as a soft erosion followed by a soft dilation (with the same structuring system), and the soft dosing (0) is defined as a soft dilation followed by a soft erosion: (0) is

f@)o[B,A,rI

=

cf(P)

0 [&A,rl) 0 IB,A,rI,

fm .[ B , A , r l = CTm 0 [B,A,rl) 0 lB,A,rI. Figures 20k-n and 200-r show the results of opening and closing, performed upon the binary image of Fig. 20b with the Structuring system of Fig. 20a, and with growing values of the parameter r = 1, ...,4. In general, the modifications produced by the soft morphological operators upon the original image become stronger (approaching those of the conventional morphological operators) with decreasing values of r. This behavior can also be explained considering the soft morphological operators as special cases of weighted rank-order filters.

52

PIER0 ZAMPERONI

A generalization of the soft morphological filters, requiring the specification of an additional integer parameter s, with 1 Is < r (and s = 1 if r = l), has been developed by Kuosmanen et al. (1992), with the aim of giving a further degree of freedom for achieving a better discrimination between detail and noise. The generalized soft erosion is defined as f(p) 0 [B,A , r, s] = rth smallest value of the set lr 0f ( a ) :a E AP1 U Is 0f(b) : b

E

(B\A)pl,

and the generalized soft dilation is defined as f(p) 0 [B,A , r, s] = rth largest value of the set

( r 0 f(a) : a E A P )U (s o f(b) :b E (B\A)P1. The detail preservation properties of the soft morphological filters have been investigated by Kuosmanen (1993), and those of the generalized soft morphological filters by Kuosmanen et al. (1992). A number of detailpreserving properties of these filters, for a proper choice of the parameters r and s, have been put into evidence, as for instance: The suppression of impulse consisting of arbitrarily shaped blobs C of ICI pixel, by means of soft closing (for positive impulses) or opening (for negative impulses) with the structuring system [B,A , r ] , if r is made to meet the following requirement:

The fact that generalized soft morphological opening and closing preserve infinite edges. the preservation of infinite corners under some conditions to be met by generalized soft opening and closing. Generalized soft morphological opening preserves infinite valleys. The same holds for the closing, provided that the valley is at least as wide as the diameter of IB I.

VII. LINEPATTERNS ENHANCEMENT This section presents some methods, of various provenance, whose explicit aim is the enhancement of line-shaped patterns. It is, of course, very problematic to trace a sharp border between contrast enhancement (see Section V), detail enhancement (see Section VI), and line enhancement.

53

IMAGE ENHANCEMENT

In fact, the domains of application for the methods described in Sections V and VI and in this section are partially overlapping. In the presence of complex natural scenes, even a skilled user cannot always classify clearly a problem as a pure contrast-, detail-, of line-enhancement problem. The criterion used for including in this section some interesting methods known from the literature is that the authors explicitly describe their main motivation to be the enhancement of line-shaped patterns. Three restrictions will be introduced within the scope of this section; (i) It is concerned with line patterns occurring in the primary image, but not with those of edge maps, obtained by means of edge detection operators; (ii) it does not deal with the Hough transform for straight lines, which is a special global non-iconic technique, treated exhaustively in textbooks, as for instance by Leavers (1992); (iii) it does not describe line-following methods for gray-level images, as for instance in the work of Groch (1982) or of Berthod and Serendero (1988), or agglomeration methods for gradient directions with line fitting, as in Khan et al. (1990). In the following subsections some typical approaches to line enhancement, presented approximately in order of growing complexity, are described summarily, with the aim of pointing out their basic ideas of general interest, rather than the application-specific refinements. A . Iconic Maps of a Local Anistropy Measure A very simple but effective approach to line enhancement in gray-value images, is to build an iconic map of an opportunely defined scalar a!, a “measure of the local anisotropy,” and to represent this scalar with a symbolic gray value at the corresponding image’s location. In Section VI,C,l, the following definition of a! has been proposed: a1

=

Max - Min Max + Min + E

with Max = max ( p t ) ,Min f = 1 . ..T

=

min ( p t ) , ... T

t= 1

where pl,. . . , p T are the average gray values in the one-dimensional subwindows U1, ..., U T , shown in Fig. 16b. An alternative definition of a! is a!z =

Max - Min Max+& *

The dependence of a!] and a2 upon the local midrange gray value p = (Max + Min)/2 is evidenced by Fig. 21, where A is equal to (Max - Min). Both a1 and a2 increase with decreasing p, in conformity with the visual perception of details. However, this dependence is less pronounced for a2, which in practice has proven to be more effective than a].

54

PIER0 ZAMPERONI

0

A 2

G-F A

G

FIGURE21. Law of dependence of the local anistropy measures a1and a2upon the midrange gray value p in the observation window, where A is the local contrast (maximum minus minimum gray value).

There are two possibilities of building an iconic map of a: 1. If we wish to enhance dark and bright lines as well, then the quantity to be mapped into the output image is just u(x, y ) = (G - 1) a(x, y ) , where G - 1 represents the white level. 2. If we wish to enhance only bright lines on dark background, a(x,y) has to be used as a coefficient of attenuation of the original gray value f ( x , y ) : u(x, y ) = af(x,y ) . For enhancing dark lines, first the negative image f - ( x , y ) = G - 1 - f ( x , y ) must be generated; then one can proceed with f - ( x , y ) as in the previous case.

-

Figure 22 shows an example of extraction of bright and dark lines from a remote sensing view. With images of this type, containing important linear

FIGURE22. Line enhancement by computation of an ionic map of a local anisotropy measure of the original remote sensing image at left, with a 9 x 9 window. (From Klette and Zamperoni, 1992.)

55

IMAGE ENHANCEMENT

patterns at intermediate orientations between the four main ones, it is particularly advantageaous to make use of the one-dimensional subwindow subsets of Fig. 16b instead of the reduced set of Fig. 16a. B. Line Detection by Aid of Linear Filters

The most straightforward linear approach to line detection is to convolve the image with a template of the line pattern-for instance, for the ideal horizontal bright line (gray value 1) on dark background (gray value 0), reproduced on the left-hand side of Template 1. Since the convolution product with the template should be zero in correspondence with a constant gray value, it is desirable to use a zero-mean template, as for instance the one reproduced on the right in Template 1, obtained by convolving the left one with the Laplacian operator. Let us denote by Lh the result of convolving the image with the zero-mean kernel of Template 1, and by L , , L,, and L , the convolution products with similar kernels in the vertical, diagonal positive, and diagonal negative directions. Then a line-enhanced image L can be obtained by combining the partial results in the following way: L

= max[lLhl,

ILA, IL,,I, IL,II.

In many cases, the linear convolution may not be satisfying, because templates such as Lh react not only to lines, but also to edges and, even more strongly, to isolated points. Therefore, Rosenfeld and Kak (1982) propose the following nonlinear modification. Let us consider the detection of horizontal lines. With reference to the pixel gray values in the 3 x 3 window (Template 2), let Lh be equal to the convolution product with the 3 x 3 mask, constituted by the central part of the 5 x 5 template displayed

~

~ _ _ _ _

... 0

0

0

0

0

...

...

0

0

0

0

0

...

...

0

0

0

0

...

...

1

1

1

1

1

...

... 1 1

1 1 1

...

. . . -2 -2 -2 -2 -2 . . .

...

0

0

0

0

0

...

...

1

1

1

1

1

...

...

0

0

0

0

0

...

...

0

0

0

0

0

...

0

Ideal line model

Zero-mean line template

TEMPLATE 1.

56

PIER0 ZAMPERONI

TEMPLATE 2.

on the right in Template 1 , only if the following condition is met: (p4

> pl)

A (p4

>

A

> p2)

> p7)

A

A (pS

>

A (pS

>

Otherwise, we set Lh = 0. The same condition is imposed upon L , , which is then combined with Lh as before, in order to obtain the final result L . Simple line detectors like the one just described, having a purely highpass character, often turn out to be inadequate in the presence of noise, because they tend to react to noise in flat regions by producing spurious line fragments. In order to improve this situation, it is advisable to combine the high-pass function with a smoothing of the highest spatial frequencies, so that the overall filter function is rather of a band-pass type. The approach proposed by Awajan et al. (1987) is based upon two sets of linear templates, one for smoothing, and the other for line detection, but the template choice is performed adaptively, in dependence on the local image data. The smoothing, which constitutes the first step, is performed by means of a set of four 5 x 5 templates, M I , ..., M,, while for the line detection eight templates, S1, . .,S8,are used. The first elements, M , , M 2 , and S , , ..., S,, are shown in Template 3. The remaining templates are obtained by rotations of 90" and 45", respectively. Convolving the image with the smoothing template Ma (a = 1, ...,4), one obtains the weighted average ma,whereby at the same time the standard deviation su of the weighted gray values with respect to mais computed. Then the image is smoothed using, for each point, the oriented template Mh (1 Ih I 4), for which s, is a minimum for a = 1, ..., 4 . The next step is the line detection by means of the template set S , , ...,&. The convolution of the smoothed image with each one of these templates yields the set of gray values u l , ..., 248 , and as a final result is taken that value u k for which u k = max ( 4 1 .

.

a=I

...a

The Canny/Deriche approach to edge detection represents a decisive step towards a methodical framework for designing optimal linear filters for detecting patterns which can be described analytically. Canny determined

51

IMAGE ENHANCEMENT

0

0

0

0

0

0

0

0

0.5

0.5

0

0

1

1

1

0

0

0

1

1

0.5

0

0

0

0

0

1

1

1

0

~~

0.5

1

1

1

0.5

0

1

1

1

0

2

2

2

2

2

0

1

1

1

0

0.5

1

1

0

0

0

1

1

1

0

0

0

0

0

0

0.5

0.5

0

0

0

0

0

0

0

0

s1

M2

Mi 0

0

0

0

0.5

0

0

0

0.5

2

0

0

0.5

2

0.5

0

0

1

2

2

0

0

1

2

0.5

0

0

1

2

0

0.5

1

2

1

0.5

0

1

2

1

0

0

1

2

1

0

2

1

0

0 1

10.5

2

1

0

0 1

1 0

2

1

0

0 1

0

0

0

0

2

0.5

0

0

0

2

0.5

0

0

1 2 0.5

sz

s3

0.5

s4 .

TEMPLATE 3.

the optimum shape of a convolution kernel for detecting noisy step edges. The joint optimization criterion used by Canny takes account of the following aspects of edge detection: (i) minimum signal-to-noise ratio; (ii) minimum variance of the position of the convolution product’s peak; (iii) the edge value must have only one peak. Once the optimum onedimensional kernel is determined, a two-dimensional separable kernel has been derived. Deriche’s contribution was the development of a recursive separable filter having the same impusive response as the Canny filter. The advantage is that one can obtain the same smoothing effect as with a nonrecursive filter of arbitrary window size by adjusting only one parameter. The filter structure and the execution speed are always the same. The Canny/Deriche technique has been extended to the detection of other patterns, as for instance to ramps by Petrou, and to lines by Ng and Petrou (1991) and by Ziou (1991). While the first authors followed Canny’s linear nonrecursive filtering approach, the last one adopted Deriche’s recursive scheme, which also ensures a high execution speed for large smoothing windows. The theoretical foundations of Ziou’s method are rather complex, and they can be reproduced here only summarily, but the practical steps necessary for the algorithm’s implementation are rather simple to describe and to carry out.

58

PIER0 ZAMPERONI

0.0106

I

0.0066Q

0.00331

0 t

-700

-0.00265 FIGURE23. Profile of the one-dimensional line detection convolution kernel, with different values of the parameters a and w. (a) (Y = w = 0.01; (b) a = 0.02, w = 0.01. (Reprinted from Pattern Recognition, 24(6), Ziou, D., Line detection using an optimal IIR filter, pp. 465-478, Copyright 1991, pp. 465-478 permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington OX5 IGB, U.K.).

The one-dimensional line detection function f(x) used by Ziou (1991) is zero-mean and symmetrical about the line’s center, and it depends upon the parameters c, w, and CY :

j(x) = [-ca sin(wlx1) + cwcos(wIxI)l e-OllXl. The profile of this function, shown in Fig. 23, is close to that of the second derivative of a Gauss function. The two-dimensional separable filter for detecting vertical lines results from the product of the detection function f(x) and of a smoothing function g(y) in the orthogonal direction:

and the complete masks X ( x , y ) in the x-direction and Y(x,y) in the y-direction are the products

-

X(x, Y ) = f(4 g(A

and

Y(x, u) = f(y) * dx).

These expressions of f(x) and g(x) have been chosen by Ziou not only with the point of view of matching the profile of a line, but also so that they can each be easily implemented by two purely recursive (and therefore

IMAGE ENHANCEMENT

59

strictly causal) filters in two opposite scanning directions, namely left-toright and right-to-left for realizing the mask in the x-direction, and top-tobottom and bottom-to-top for realizing the mask in the y-direction. The transfer function t(z)in the complex variable z that realizesf(z) can be split into the sum of the two causal transfer functions t-(z) and t+(z):

In correspondence of these transfer functions, the outputs y+(n)and y-(n) of the two opposite horizontal scans at the nth sample are given by the filter equations

r+(n)= aox(n) + a , W - 1) - b , y + ( n - 1) - b,y+(n - 21, y-(n) = a2x(n + 1) + a3x(n + 2 ) - b l y - ( n - 1) - b 2 y - ( n + 2 ) . Similarly, the smoothing function g(x) can be realized by the sum of two causal transfer functions with opposite scan directions, where the coefficients a,, a , , a 2 , a3 are substituted by a,, a s , (26, a,. In Ziou (1991) are given the numerical values of all the coefficients needed for the filter realization: a0

cw

01

- c(w cos w

02

a, - sob,

a3

- a0 b2

a4

c2

as

( -c2 cos w

a6

as - C 2 b l

a7 bl b2

+ a sin w) eFa

+ c1sin w) e-O1

-c2b2 - 2 e-O1 cos w e-2a

CCY C1

c2 c

a2

+ w2

cw

+ w2 (1 + bl + b2)(a2+ w2) 2a e-OLsin w + w - w e-2a a2

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PIER0 ZAMPERONI

Running the filter requires altogether eight one-dimensional image scans, namely four for detecting horizontal and four for vertical lines. For each direction two opposite scans are performed for each of the two separable functionsf (detection) and g (smoothing). Thus, buffer image memories are required for storing intermediate results. The number of memories can be minimized by multiple use of the same buffer for different passes; as this minimization depends on the system used, it will not be treated here, and we shall consider for clarity as many distinct buffers D , D+,D- ,P , P+,P- , H , and Vas necessary. The filtering process of an M x N image B(i,j ) consists of the following steps:

...,M and j = 3, ...,N , compute D + ( i , j )= u,,B(i,j) + a l B ( i , j - 1) - b l D + ( i , j- 1) - b 2 D + ( i , j- 2).

For i

=

1,

For i = 1, ...,M a n d j = N - 2, ..., 1, compute D-(i,j ) = a2B(i,j

+ 1) + a3B(i,j + 2) - b l D - ( i , j + 1) - bzD-(i,j + 2).

For i = 1, ...,M and j = 1, ...,N , compute

D ( i , j ) = D + ( i , j )+ D - ( i , j ) For i = 3,

..., M and j

=

1, ...,N , compute

P+(i,j)= u4D(i,j)+ a,D(i - 1 , j ) - b,P+(i - 1 , j ) - b,P+(i - 2 , j ) . For i

=

M - 2, ..., 1 a n d j = 1, ...,N , compute

+ 1 , j ) + u,D(i + 2 , j ) - b l P - ( i + 1 , j ) - b2P-(i + 2 , j ) . For i = 1, , ..,M and j = 1, .,.,N , compute

P-(i,j)

=

u,D(i

H(i,j ) = P-(i, j ) + P+(i,j). The image H ( i , j ) represents the first partial result of the filtering in the horizontal direction. The second partial result, i.e., the image V ( i , j ) , is obtained by repeating the steps described earlier, by swapping the coordinates i a n d j . The final result, i.e., the filtered image A ( i , j ) , is one of several possible combinations of the partial results, as for instance, A ( i , j ) = max(H(i,j), ~ ( i , j ) ~or A ( i , j ) = J H 2 ( i , j )+ V 2 ( ij, ) .

The physical meaning of the user parameters a and w is evident from the expression of the detection functionf(x). a determines the rate of fall of f(x), i.e., the smoothing effect perpendicularly to the line profile. Thus, a can be used for adjusting the minimum width of the lines to be detected. The parameter w determines the oscillating frequency of f ( x ) , and therefore the distance of spurious maxima which could be erroneously interpreted

61

IMAGE ENHANCEMENT

as detected lines. In Ziou (1991), the influence of the parameter values is discussed and some hints to their choice are given. A good compromise is obtained by choosing a = 1 and w Ia,but generally the parameter values can be well matched to various situations. The approach of Ng and Petrou (1991) is very similar to that of Ziou, as it utilizes the Canny detection function optimization technique, applied to a parameterically defined line profile, with the difference that its realization is performed with an FIR instead of an IIR filter. The parameters of the line profile model a(x) are the half of the line width d, the line boundary sharpness parameter s, and I = 10/d. The following line profile modeling function has been chosen: A 3cosh(sx) - A , cosh(Ix) + A ,

for x 2 d, for d > x 2 -d,

A , e" - A , esx

for - d > x,

A , e-IX

,

with the following values of the constants A , . . .,A S: =

[s2(1 - e-") - I'( - e-")]-',

Al

=

Aos2sinh(sd),

A,

=

Ao12sinh(sd),

A,

=

A ~ I ~ ~ - ~ ~ ,

A , = AOsZe-ld, A S = A,(? -

12).

The parameters d and s, although they are not critical, should be chosen by the user depending on the features of the lines to be detected. The optimization following the Canny method leads to the following optimal convolution filter functionf(x) in the direction of the line profile:

K , euxcos(m) + K2 euxsin(crx) + K3 eTuxcos(cux) +K4.e-uXsin(cux)+AlK,e'x-A,K6eSx for - d > x > - w , f(x) = N , euxcos(crx) + N, ear*sin(cux) + N3 e-ax cos(cux) + N4 e-ux sin(m) + A3K6 cosh(sx) - A4K5 cosh(Ix) + N5 for 0 1 X > - d, f(-x) for x 2 0.

I

The values of the parameters K , , ..., K 6 , N , , ..., N,, a, and w are tabulated in Ng and Petrou (1991) for various values of s = 2, ..., 7 . Typical values of a and w are 0.1 and 4, respectively. The line scale parameter d depends upon the user's choice, and is essential for the piecewise definition of f(x).

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PIER0 ZAMPERONI

The image is convolved with the one-dimensional detection function f ( x ) in two orthogonal directions, for instance horizontal and vertical, and the resulting line images can be combined, as in the method of. Ziou, by the maximum function or by the root of the sum of the squares. Suppression of non-maxima results in a one-pixel wide line pattern. This method gives very good results in the detection of regular (d fairly constant) line textures, as for instance fingerprints or tissue weave patterns.

C. Topographical and Morphological Approaches to Line Enhancement Within the scope of the task of detecting linear patterns in remote sensing views, Destival and Le Men (1986) investigated the performances of two very different approaches, namely the ridge-and-valleys approach of Haralick (1983), and the use of a well-known operator of mathematical morphology, called the “top hat” operator (Haralick and Shapiro, 1992).

a. Topographical Approach. The first approach has been named “topographical” because it considers the gray-value function as a terrain relief, where the bright and dark lines play the role of ridges and valleys, respectively. For detecting ridges and valleys, the gray-level function f ( x ,y ) is regularized and by fitting it with a cubic surface

+ k2x + k3y + k4x2 + k,xy + k s y 2 + k,x3 + ksx2y + k9xy2 + k , , y 3

f ( x , y ) = kl

with a coordinate system (x,y ) centered on the current pixel (0,O).The values of the coefficients k , , ...,klo have been determined by convolving the image with 5 x 5 kernels, specified by Haralick (1983), and determined by means of a regression technique with discrete orthogonal polynomials, described in detail by Haralick (1982). To put better in evidence the direction-dependent features of the surface f ( x ,y ) in the neighborhood of (0, 0), let us consider the polar coordinates p, a,defined by x

=

psina,

y

=

pcosa.

In correspondence with ridge or valley points, the first directional derivative f,l(a,p = 0) is zero in a direction a, for which the second directional derivative fi(a,p) is a positive or a negative maximum. Although there are some problem cases arising from this geometrical description of a gray-value function, which have been treated by Haralick (1983), the topographical approach can be quite effective, especially for sufficiently regular gray-value surfaces. Substituting the polar coordinates into the

63

IMAGE ENHANCEMENT

expression of f ( x , y), one obtains:

+ Bp2 + C p + k,, with A = k7 sin3 a + k8 sin2 a cos a + k9 cos2 a! sin a + kloc0s3 a!, B = k4 sin2 a! + k5sin a! cos a + k6 cos2a,

f ( a , p ) = Ap3

C

=

k2sin a! + k3 cos a.

The first two directional derivatives in direction 3Ap2 + 2Bp

fp’(a,p)

=

f’(a!,p)

= 3Ap

a!

are

+ C,

+ B.

For detecting ridges and valleys, one must determine the values a0for which the second derivative, which represents the surface’s curvature, is zero at the current point, i.e., a. is a solution of

a/aa
0. A ridge or valley is detected if the other condition is satisfied: f,’(a0,p) = 0

=

for p = 0, e.g., p

c 0.5 pixel.

6. Top-Hat Transformation. The top-hat transformation is a simple but effective operator of mathematical morphology (for a tutorial introduction see Haralick and Shapiro, 1992), which can be utilized as a sieve for extracting from a gray-tone image only the regions with a size inferior to a given threshold d . Within this scope, the size of a rectangular region can be considered roughly as the length of its shorter side. Thus, the top-hat transformation extracts not only the lines thinner than d pixels, but also circular blobs of diameter less than d , and correspondingly thin “arms” or “necks” of bulky regions. It performs as a line detector only if regions of the type just mentioned are not present. Let us consider bright lines on a dark background. Eroding the image r times with the 8-neighbor unit circle as a structuring element means to perform a minimum operator on a 3 x 3 square window. This operation deletes thin regions (in the sense specified earlier) less than d = 2r + 1 pixel wide. A subsequent performance of r dilations (maximum operator) with the same structuring element (opening) returns the original image, with the exception of the deleted regions, which have disappeared irreversibly. Subtracting the opening’s result from the original image yields all the deleted regions. Denoting by A‘” and A‘-‘) the dilation and, respectively, the erosion iterated r times upon the image A , the result B of the top-hat transformation for detecting (i) bright lines, and (ii) dark lines is

(i) B

=

A - [,4-‘)]@),

(ii) [A(‘)](-‘)- A .

64

PIER0 ZAMPERONl

FIGURE24. Example of application of the top-hat transformation to a remote sensing view (upper left) for extracting regions of various sizes d . If TH, is the result of the top-hat transformations with r iterations of erosion and dilation, the following results are displayed: Upper right: T H , , d = 3 ; lower left: TH, - TH, (only regions with d = 7); lower right: TH, - TH, (only regions with d = 9). (From Klette and Zamperoni, 1992.)

Fundamental properties of the morphological operators ensure that the result of these point-to-point differences between images contain only nonnegative gray values. Figure 24 shows an example of application of the top-hat transformation to a remote sensing view (upper left) with different values of r, for extracting regions with a given size d. With r = 1 (upper right), d is equal to 3. In the lower row the difference TH, - TH,,, between the result TH, of the top-hat transforms with different values of r has been performed for extracting only the regions with the size d = 2(r + 1) + 1. Lower left: r = 2; lower right: r = 3.

VIII. ENHANCEMENT OF BINARYIMAGES

From the point of view of several important basic image processing methods, as for instance of mathematical morphology or of rank-order and

IMAGE ENHANCEMENT

65

nonlinear filtering, binary images can be regarded as particular cases of gray-tone images. Thus, there is in principle no need to deal separately with the same procedures under the constraint that only two gray values, namely 0 for the background, and G - 1 (or, normalized, 1) for the objects are present in the image. However, in some cases it is more simple to describe some algorithms in terms of logical operations between binary gray values; in other cases, the enhancement procedures operate on typical shape descriptors, as for instance on the contour chain code (Freeman, 1961), which have no counterpart in gray-tone images. An important constraint, to be satisfied in numerous applications, is that the result of a binary image enhancement method should be a binary image. This condition is not met, in the general case, by linear filters and by many nonlinear ones, as for instance by L-filters, i.e., by linear combinations of rank-ordered gray values. The contrary is true for rank-selection filters and for quadratic filters. Thus, a reason for preferring binary-image-specific variants of general methods could be that they comply with this requirement. Also, the kind of enhancement that can be performed upon a binary image is only partially the same as the gray-tone images. There can be no local contrast or detail enhancement, because the step edges already have an ideal shape in the impaired image. On the other hand, t o perform a noise reduction means to eliminate binary noise specks within the scope of a given criterion for discriminating between information blobs and noise specks, as for instance the blob size or the shape factor. This section aims at giving an overview of some typical approaches to binary image enhancement with methods of nonlinear filtering, of mathematical morphology, and of contour coding. A. Rank-Selection and Majority Filters

Non-adaptive rank-selection filters, whose output is the rth smallest rankordered gray value of the observation window, represent in several cases a simple and effective binary image enhancement method. This is the case when statistics both of the useful information and of the noise are sufficiently stationary over the whole image under consideration. If we have, for instance, typewritten documents with nearly constant stroke width and noise specks of limited size, the rank value r alone can be sufficient for discriminating between useful pattern and noise, in an adequate observation window of L x L pixels. This principle will be now illustrated by means of some examples. Figure 25a schematically illustrates an image model in which the useful signal is typically represented by line strokes of width not inferior to W

66

PIER0 ZAMPERONI

L=5 stroke width w=3

L=3 r=3

no

L = 5 r=10

R -

r L=5 r=5

Central pixel deleted by D(r) ?

L = 3 r=3

Y es

Yes

Yes

Central gap filled by D(r) ?

no

FIGURE2 5 . Noise cleaning in binary images which comply with a stationary model of the stroke width and of the noise. (a) Models of the stroke width and of the noise in a 5 x 5 window. (b) Examples of patterns of various sizes L in which an object pixel is deleted or preserved, according to the estimation based on the noise model. (c) Examples of patterns of various sizes L , in which a background pixel remains unchanged or is converted to an object pixel, according to the estimation based on the noise model.

pixels, and the noise is sparse, with a probability of occurrence p < pr = W/L, where L is the window size, Then, a non-adaptive rank-selection filter E(r), which outputs the rth smallest gray value, with r = 1 + (L - W)L = (1 - p,)L2, statistically eliminates the noise and leaves the useful signal unchanged. Let us consider the patterns of Fig. 25a as statistically representative for the useful information and for the noise of a line image, with L = 5 , W = 3, a n d p < 0.6. Then, taking r = 11 will preserve the lines and suppress the noise. Without loss of generality, one can restrict the values of r to the range 1 Ir I(L2 + 1)/2. The filter E(r) can be regarded as a generalization of the erosion operator of the mathematical morphology, which is a particular case with r = 1. Analogously, the other basic morphological operators of dilation, opening, and closing can be generalized as follows:

67

IMAGE ENHANCEMENT

0

Erosion

0

Dilation Opening Closing

-+

-+

+

+

E(r) D(r) O(r) C(r)

rank r rank s

=

L2 - r

+1

sequence E(r), D(r) sequence D(r), E(r)

Generally speaking, E(r) preserves (erases) patterns consisting of at least (less than) L2 - r + 1 object pixels; D(r) preserves (erases) patterns with at least (less than) r object pixels. The examples shown by Figs. 25b and 25c illustrate some typical effects of selective pattern enhancement, obtainable with rank-selection filters. In Fig. 26, some effects of the composite operators O(r) and C(r) are compared with those of conventional opening and closing, with regard to noise cleaning and shape preservation. In Fig. 26a, single noise pixels are also merged into connected blobs by conventional closing. In Fig. 26b, with conventional opening, single-pixel holes are merged together into connected holes by the first step (erosion). The operators C(2) in Fig. 6b and O(2) in Fig. 6c, with L = 3, perform a better discrimination between noise and patterns to be preserved. A similar approach to the application of extended morphological operations to binary image enhancement has been developed by van den Boomgard (1989) in terms of threshold logic operations. Result of conventional closing: A and B Result of C(2) with L = 3 and r = 8: only B

A

noise

B

Result of conventional opening: right object with hole Result of O(2) with L = 3 and r = 8: right object, no hole

FIGURE26. Comparison of the noise cleaning effects obtained with the conventional morphological opening and closing, and with the extended variants of these operations.

68

PIER0 ZAMPERONI

Like the preceding method, the spot and blob noise reduction procedure for binary images of Ray (1988) is based on a pixel count. In fact, with binary images the median filter is equivalent to a majority count, and the other rank-selection filters can be considered as modified majority counts. In the approach of Ray, the deletion of the current pixel p , with gray level 1, as a potential noise pixel is subject to the result S of a count of the 1-pixels on the border of a window containing p and also other pixels, whose gray values are not examined. Generally, the considered windows are not symmetrical with respect t o p ; their size is m x n , with m and n between 3 and 5 . For the 3 x 3 window, the sums over the north/south border and over the east/west border are also evaluated. The current pixel p is deleted (its value is set to 0) if at least one of the following conditions is met: S=0

for n

=

m

=

3 (only north/south or east/west borders)

S

I1

for (m < 4) v ( n < 4) (complete border)

S

I2

for ( m > 3) A (n > 3) (complete border)

In this method, the discrimination between information and noise relies entirely upon the fact that the noise blobs are expected to be entirely contained by a window of size less than or equal to 5 x 5 . B. Enhancement of Intrinsically Binary Images by Polynomial Filtering

An intrinsically binary image is an image whose contents are bilevel because the reproduced real scene contains only two light intensity levels, namely that of the background and that of the objects-for instance, in documents, drawings, or in some workpieces. However, after image acquisition, all the gray values can be present in the digitized image; this is mainly because of the unequal scene illumination and the band-limited transfer function of the imaging device. The enhancement to be performed on this type of image aims at restoring as well as possible its bilevel structure, i.e., at eliminating noise and other low-contrast gray-value variations, and at enhancing the high-contrast variations, including impulses. The use of polynomial filters, such as those of the quadratic type, for image enhancement has been the object of numerous studies. In particular, Ramponi and Fontanot (1992), starting from the results of their previous work, concentrated upon the application of quadratic filters to document image enhancement, and upon their design. Given a 3 x 3 observation window F(p,), centered onp, = (x, y ) , the output g(p,) of a mixed (type one and type zero) quadratic filter is a weighted sum of gray values f ( p i ) ,

69

IMAGE ENHANCEMENT

of squared gray values f(pi)’, with ( p i )E F(po), and of products of 8-neighbors’ gray values f ( p i ). f ( p j ) ,with p i , p j E F ( p ) and i f j :

where d is a scaling factor between the linear and the quadratic component. The linear term in this equation represents a weighted averaging component, which exerts a noise-smoothing action independently of the local signal dynamic range. The quadratic component gives more pronounced enhancement of the large signal variations associated with the graphic information of the document (edges, lines, etc.), independently of their spatial frequency contents. Sufficient criteria for determing the filter coefficients hi, wii, and wij have been formulated by Ramponi and Fontanot (1993). These criteria are based on the requirement of preserving the image signal dc-level, of preserving the shape of a positive spike, and of having an isotropic filter response. The first two conditions yield

C hi = 1,

C wij = 0,

and

woo = 1 - ho,

wii = - h .1 .

ij

i

The remaining degrees of freedom of the filter coefficients are eliminated by imposing some isotropy conditions to the filter response on the 3 x 3 window support. For this purpose, the coefficients wij are subdivided into two disjoint sets, So and S1. The former set contains weights of the type woi, i.e., relative to a gray-value product, having the current pixel gray valuef(po) as one of its factors. The latter set comprises the other weights wij. So consists of eight elements, four related to horizontally or vertically neighboring pixels, and four related to diagonally neighboring pixels. For S1, consisting of 12 elements, the same classes contain eight and four elements, respectively. The filter’s isotropic behavior is enforced by giving all the coefficients of the same set and subset the same value w;”, w:“, w,d’, and w p , where the indices hv and di stand for “horizontal/vertical” and “diagonal,” respectively, with obvious meaning of the symbols. Then the second constraint specified earlier can be rewritten as w;.

+ w,d’ + 2 w p + w;Ii = 0.

A further set of conditions, allowing univocal determination of the weights w i j , is derived from the constraint of equal response to diagonal and to horizontal/vertical input patterns, together with the last two conditions:

w;u

=

w;,

Wd 1i

= -wf,

w:u

=

- w,hu/2.

70

PIER0 ZAMPERONI

Altogether we have four conditions imposed on four quantities to determine. Having determined all the weights wij with i # j , the other coefficients h,, hizo and wij can be derived from the sufficient conditions just formulated WOO

=

1 - ho,

wii

=

- h .I ,

woo +

C

wii =

1 - ho -

i#O

C hi = 0 . i#O

The scaling factor d controls the blend ratio between the smoothing linear low-pass component and the quadratic detail-enhancing component. It can be determined by imposing the condition that a one-pixel impulse of height b, superposed on a background of gray level a, should be considered as a detail to be preserved, while smaller impulses should be considered as noise to be smoothed: g(po) = hob

+ (1 - ho)a + [(l - ho)(b2- a 2 ) ] / d= b

+

d

=

6.

Experimental results obtained by Ramponi and Fontanot (1992) with this method have put into evidence the noise-reduction and detail-enhancement behavior of this mixed linear/quadratic filter. To obtain the best performance, it is necessary that the filter input be a zero-mean signal. Considering the first-order statistics of (the negative of) a typical document image, the results make it evident that the average gray value, i.e., the background level to be subtracted in order to obtain a zero-mean signal, is very near to zero. In this case, a is also equal to zero. If this assumption does not apply, either the value of a must be determined globally, or, if the background is not sufficiently homogeneous, a background image can be extracted by means of a piecewise smoothing technique.

C. Enhancement Methods Based on Contour Chain Processing The contour chain code (Freeman, 1961) is an exact description of a binary object, consisting of its contour path step sequence. On the basis of the 8-metrics, each step is represented by a direction code 0, ..., 7 , as shown in Fig. 27a. Figure 27b gives an example of an object and of its contour code chain, beginning at point C . The advantage of the contour code is that it makes it possible to perform even complex object shape transformations and to extract shape features by the aid of simple numerical manipulations of the contour code symbol chain with list-processing algorithms. This speeds up the execution time, as no image data need to be accessed. In principle, all the operators defined on the pixel matrix can also be performed on the contour code, because no information loss occurs as a consequence of contour coding.

71

IMAGE ENHANCEMENT

a

b

5

6

7

6666000013443210445

C

‘s

d 177507543 17755

q$

16666

0 666 7 66 5 5 4 6 5 5 5 5 4 4 3 5 4 4 4 U U 5 5 5 5 5 4 4 4 4 4 4 4

5 5

0

FIGURE27. Contour smoothing in binary images by means of contour chain transformations. (a) Step directions of the Freeman chain code. (b) Binary object and its contour code. (c) Determination of the residuum (dotted line) of the path A B by means of transformations of the contour chain. (d) Smoothing of an object’s contour with the residuum determination rule. (From Zamperoni, 1991.)

As far as image enhancement is concerned, contour chain processing methods can be used for enhancing the contour smoothness by eliminating irregularities such as those shown on Fig. 27d, which are due to discretization noise. This kind of enhancement is important within the scope of contour parsing into digital straight-line segments, because the number of digital straight-line segments is unnecessarily increased by the irregularities. The algorithm for contour smoothing is only a special case of a more general algorithm for determining the residue, i.e., the shortest path between the extremes, A and B, of a path described by a chain code (Freeman, 1961), as illustrated in Fig. 27c. The residue determination method described here, a modified version developed by Zamperoni (1991), consists of a look-up table of substitutions of contour step pairs by other contour steps. The elements of each pair do not need to be contiguous in the contour chain. Substitutions are undertaken as long as there are contour step pairs S 1 , S2 for which the values of M-rn and of rn, with m

=

min(S,, S2J,

M = m a & , S2J,

72

PIER0 ZAMPERONI TABLE I

THESLJLWITLJTION TABLE M - m

m

Modified contour steps

0 1 2 2 3 3

odd even odd even

no modification no modification m+l,m+l m + l m + l m+2 delete m and M m-1 m-2 m-1,m-1 m-1 no modification

4

-

5 5 6

odd even odd even -

6 7

match with the entries of the substitution table reproduced in Table I. Figure 27c shows an example of application of the residue determination rule. In the example of Fig. 27c, the maximum distance of two steps forming a pair, inside of the chain code, is subject to no constraint; thus, the resulting dotted path A B has the absolute minimum length. However, for contour smoothing purposes it is opportune to consider pairs of steps contained in the same window of +-ncontour steps. The value of n must be chosen as a compromise between the intensity of tolerated shape modifications and the desired smoothing effect. In the example of Fig. 27d, n has been chosen equal to 1. After manipulating the contour chain code with the rules of Table I, and under the constraint n = 1, one obtains the contour chain code at the bottom of Fig. 27d. Not only has the number of digital straight-line segments been reduced from four to two, but also what looks like discretization noise has been eliminated from the contour.

D. Smoothing of Discretization Noise by Aid of the Distance Transform Discretization noise, which has also been mentioned in connection with some of the methods presented in this section, can be very annoying and difficult to remove without suppressing at the same time important shape features of the image to be binarized. The reason is that the binarization enhances the visibility even of very low-contrast gray-level fluctuations in proximity to step or ramp edges if these fluctuations cross the binarization

IMAGE ENHANCEMENT

73

threshold. Introducing a hysteresis into the thresholding operation mostly helps only to obviate the effects of small oscillations. Taking a closer look at the shape of the contour between object A and background B after binarization, one can often observe irregularities of the type illustrated in Fig. 28a, with dents and inclusions due to the presence of noise or of shading effects, which affect gray levels in proximity to the binarization threshold. These impairments can seldom be eliminated by means of the basic morphological opening and closing operations, as shown by Figs. 28b and 28c, where the 8-metric unit circle has been used as the structuring element. An efficient procedure for enhancing the shape quality of the image after binarization, developed by Arcelli and Sanniti di Baja (1988) and improved by Ragnemalm (1991), utilizes information derived from the distance transformation for coping with the problem just outlined. This procedure will be now illustrated, for the sake of clarity, using the distance transform based on the 8-metrics, and the improvements introduced by Ragnemalm (1991) by using the Euclidean distance will be mentioned afterwards. The distance transform (see Haralick and Shapiro, 1991) assigns to each object’s pixel the value of its minimum distance from the background, on the basis of a given metric. Analogously, the background’s pixels can be labeled with the distances from the object. Algorithms for performing the distance transformation have been described by Rosenfeld and Kak (1982). Figure 28d shows the labels assigned in the region near the edge, with the convention of assigning negative values to the distances of the background pixels. Within the scope of this method, a distance threshold d, which influences the contour smoothing degree, is chosen by the user; then the set E of pixels of A and B with distance values less than d in the module is considered. Taking d = 2, one obtains the set E represented in Fig. 28e. By means of a new distance transformation, performed upon the set E , its elements are subdivided into those nearer to A and those nearer to B (on the basis of the chosen metrics), marked with o and, respectively, with x in Fig. 28e. The final border is then obtained by asigning the pixels marked with o to A , and those marked with x to B . Pixels having the dsame distance from A and B (marked with = in Fig. 28e) can be assigned to either set making use of an empirical criterion. A comparison among Figs. 28b, 28c, and 28e shows the contour shape improvement obtained by this method. The improvements introduced by Ragnemalm (1991) are twofold. In the first place, the Euclidean distance has been preferred to the 8-metrics, because it corresponds much better to the intuitive concept of distance and to the requirements posed by practical applications. In fact, the disadvantages of the 8-metrics, where unit balls correspond to squares of

74

PIER0 ZAMPERONI Background B

blank: x:

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x x x * x x x x x x x x x x x * x x x x x

x x x x

Object A

a

X X

x x x x

b

x x x x

x x x x

x x x x

x x x x

x x x x x x x x x x x x x x - x x x x

x x x x

x x x x

Result of opening on f i g u r e -an : p o i n t s of A d e l e t e d by opening

-----

-- -

-

-

x x x x

C

x x x x

x x x x

x x x x

x x * *

Result of c l o s i n g on f i g u r e "a" * : p o i n t s added t o A by c l o s i n g

and from A (neg.1 9 = -1 8 = -2

2 1 9 8

2 1 9 8

d

2 1 9 8

2 1 9 8

x x x x * * x x x * * x

x x x x

x x x x

X * * X * *

x * * * * x x * x

Distances from B (pos

x x x x

x x x x

2 2 2 1 9 8

2 1 1 1 9 8 8

2 1 9 9 9 9 8

2 2 1 1 1 1 9 8

2 1 1 1 9 9 9 8 8 8 8

2 1 9 1 9 9 9 9 9 9 8

2 1 9 1 1 1 1 9 1 9 8

2 1 1 1 1 9 9 9 9 9 8

2 1 9 1 9 9 1 1 9 8 8

2 1 1 1 1 9 9 9 9 8

2 2 2 2 1 9 9 8 8 8

2 1 9 8 8

2 1 9 8

2 1 9 8

2 1 9 8

x : nearer t o A

x x x x x x x x x x x x x x x A x x x x x x = x x x x x x x x = = = = = = = x ~ x x x x

o : nearer t o B = : equidistant

o o o o o o o o o o o o = o o o o o o 0 0 0 0 0 0 0 0 0

Set E

e

0 0 0 0 0 0 0 0 0 0 0 0 0

B

FIGURE28. Smoothing of the discretization noise in binary images with the distance transform. (a) Example of an edge between a binary object A and the background B, affected by discretization noise. (b) Morphological opening of A with the 8-metrics unit circle as structuring element. (c) Morphological closing of A with the 8-metrics unit circle as structuring element. (d) Map of the 8-distances from A and from B in the object/background border region. (e) Improved smoothed border, obtained by attributing to A and to B the pixels of the transition region on the basis of their distances.

IMAGE ENHANCEMENT

75

conventional geometry, become evident even after a superficial look at Fig. 28. Fast algorithms yielding good approximations of the Euclidean distance have been developed by Borgefors (1986) are are widely used. The second improvement consists of a series of techniques for speeding up the computation of the Euclidean distance by making use of a propagation method and of contour lists. More sophisticated enhancement methods for binary images, which cannot be considered as low-level iconic operators, and which therefore go beyond the scope of this work, are known from the literature. At least two important examples should be mentioned at this point: 1 . The investigations of Schonfeld and Goutsias (1991) aim at an optimal restoration of noisy binary images by means of morphological filters of the most general type, defined by the attributes “increasing” and “idempotent.” The optimization criterion is the least mean of a setdifference distance function, which can be regarded as a distortion measure. For describing the binary noise, a statistical model known as germ-grain process is assumed. The optimum morphological filter preserves the key geometrical and morphological image features, but eliminates the noise. 2. The approach of Wolberg and Pavlidis (1985) is the binary equivalent of the stochastic relaxation with annealing technique of Geman and Geman (1984), which has raised a great quantity of investigations in the field of gray-value image restoration. This technique performs a maximum a-posteriori estimation of each pixel of a noise-affected image, by evaluating a local potential function determined by the local pixel pattern. The performance of this method has been investigated for eliminating binary white noise from binary images.

REFERENCES Alparone, L., Carla, R., and Puglisi, C. (1992). Proc. IGARSS ’92, 899. Alparslan, E., and Ince, F. (1981). ZEEE Trans. SMC-11,376. Arce, G. R., and Foster, R. E. (1989). IEEE Trans. ASSP-37,83. Arcelli, C., and Sanniti di Baja, G. (1988). In “Proc. of 9th International Conference on Pattern Recognition,” Rome, p. 948-950. Awajan, A., Mignot, J., Rondot, D., and Stamon, G. (1987). Proc. MARI ’87, Paris 320. Beghdadi, A,, and Le Negrate, A. (1989). Computer Vision, Graphics, and Image Processing 46, 163. Berthod, M., and Serendero, M. A. (1988). In “Proc. of 9th International Conference on Pattern Recognition,” Rome, p. 456. van den Boomgaard, R. (1989). In “Progress in Image Analysis and Processing” (V. Cantoni et al., eds.). World Scientific, Singapore, 111-118.

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Borgefors, G. (1986). Computer Vision, Graphics, and Image Processing 34, 344. Chehdi, K., Sabri, M., and Corazza, M. (1991). Traitement du Signal 1991, 63. Cheong, L. C., and Morgera, S. D. (1989). IEEE Trans. ASSP-37, 580. Chochia, P. A. (1988). Computer Vision, Graphics. and Image Processing 44, 21 1. Dapoigny, R., Diou, A., Dumont, C., and Voisin, Y. (1991). Traitement du Signal 8, 115. Dash, L., and Chatterji, B. N. (1991). Pattern Recognition 24, 289. Deriche, R. (1990). IEEE Trans. PAMI-12, 78. Destival, I., and Le Men, H. (1986). In “Proc. of 8th International Conference on Pattern Recognition,” Pans, 856. Fahnestock, J. D., and Schowengerdt, R. A. (1983). Optical Engineering 22, 378. Fairfield, J. (1990). In “Proc. of International Joint Conference on Pattern Recognition,” Atlanta, 712. Freeman, H. (1961). In “Proc. of National Electronics Conference,” Chicago, 421. Gauch, J. M. (1992). Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 54, 269. Geman, S . , and Geman, D. (1984). IEEE Trans. PAMM, 721. Gonzalez, R. C., and Wintz, P. (1987). “Digital Image Processing.” Addison-Wesley, Reading, MA. Groch, W.-D. (1982). Computer Graphics and Image Processing 18, 347. Hall, E. L. (1979). “Computer Image Processing and Recognition.” Academic Press, New York. Haralick, R. M. (1982). In “SPIE Conference Proceedings on Robot Vision,” Arlington. Haralick, R. M. (1983). Computer Vision, Graphics, and Image Processing 22, 28. Haralick, R. M., and Shapiro, L. G. (1991). Pattern Recognition 24, 69. Haralick, R. M., and Shapiro, L. G. (1992). “Computer and Robot Vision.” Addison-Wesley, Reading, MA. Imme, M. (1991). Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 53, 204. Jackson, P. H., and Kaye, G . (1982). In “Proc. ISMIEEE-82,” Berlin, 10. Jain, A. K. (1989). “Fundamentals of Digitial Image Processing.” Prentice-Hall International, London. Jensen, J. R. (1986). “Introductory Digital Image Processing.” Prentice-Hall, Englewood Cliffs, N.J. Jolion, J. M. (1993). In “Image Processing: Theory and Applications” (G. Vernazza, A. N. Venetsanopoulos, and C. Braccini, eds.), 205. Kautsky, J., Nichols, N. K., and Jupp, D. L. (1984). Computer Vision, Graphics, and Image Processing 26, 271. Kahn, P., Kitchen, L., and Riseman, E. M. (1990). IEEE Trans. PAMI-12, 1098. Kim, V., and Yaroslavskii, L. (1986). Computer Vision, Graphics, and Image Processing 35, 234.

Klette, R., and Zamperoni, P. (1992). “Handbuch der Operatoren fur die Bildbearbeitung.” Vieweg, Wiesbaden. KO, S. J., and Lee, Y. H. (1991). IEEE Trans. CS-38, 984. Kramer, H. P., and Bruckner, J. B. (1975). Pattern Recognition 7, 53. Kuan, D. T., Sawchuk, A. S., Strand, T. C., and Chavel, P. (1985). IEEE Trans. PAM1-7,165. Kundu, A., Mitra, S. K., and Vaidyanathan, P. P. (1984). IEEE Trans. ASSP-32, 600. Kuosmanen, P., Koskinen, L., and Astola, J. (1992). In “Proc. of 11th International Conference on Pattern Recognition,” Den Haag, 236. Kuosmanen, P. (1993). Dept. of Mathematical Science, Univeristy of Tampere, Report A 270. Lamure, M., Milan, J., and Nicoloyannis, N. (1989). In “Acta Stereol., Proc. ECS 5,” Freiburg. 609.

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Le Negrate, A., Beghdadi, A., and Dupuisot, H. (1992). Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 54, 13. Leavers, V. F. (1992). “Shape Detection in Computer Vision Using the Hough Transform.” Springer-Verlag, Berlin. Lee, J. S. (1980). IEEE Trans. PAMI-2, 165. Lee, J. S. (1981). Computer Vision, Graphics, and Image Processing 17, 24. Lester, J. M., Brenner, J. F., and Selles, W. D. (1980). Computer Graphics and Image Processing 13, 17. Leu, J. G. (1992). Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 54, 497. Mallikarjuna, H. S . , and Chaparro, L. F. (1992). IEEE Trans. PAMI-14, 671. McDonnell, M. J. (1981). Computer Graphics and Image Processing 17, 65. Melsa, J. L., and Cohn, D. L. (1978). “Decision and Estimation Theory.” McGraw-Hill, New York. Mokrane, A. (1992). Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 54, 171. Moloney, C. R., and Jernigan, M. E. (1989). In “Proc. ICASSP-89,” Glasgow, 1433. Ng, I., and Petrou, M. (1991). In “Progress in Image Analysis and Processing 11” (V. Cantoni et al., eds,). World Scientific, Singapore, 183. Nieminen, A., Heinonen, P., and Neuvo, Y. (1987). IEEE Trans. PAMI-9, 74. Pitas, I., and Venetsanopoulos, A. N. (1990). “Nonlinear Digital Filters: Principles and Applications.” Kluwer, Boston. Pizer, S. M., Amburn, E. P., Austin, J. D., Cromartie, R., Geselowitz, A., Greer, T., ter Haar Romeny, B., Zimrnerrnan, J. B., and Zuiderveld, K. (1987). Computer Vision, Graphics, and Image Processing 39, 355. Presetnik, F. F., and FilipoviC, M. (1988). In ‘‘Signal Processing IV: Theories and Applications” (J. L. Lacoume, et al., eds.), 651. North Holland, Amsterdam. Ragnemalm, I. (1991). In “Progress in Image Analysis and Processing 11” (V. Cantoni el al., eds.). World Scientific, Singapore, 83-90. Ramponi, G., and Fontanot, P. (1992). Signal Processing 33, 23. Ray, S. (1988). Pattern Recognition Letters 7, 9. Rosenfeld, A., and Kak, A. C. (1982). “Digital Picture Processing.” Academic Press, New York. Schreiber, W. (1978). Proc. IEEE 66, 1640. Schonfeld, D., and Goutsias, J. (1991). IEEE Trans. PAMI-13, 14. Voss, K., and SiiRe, H. (1991). “Praktische Bildverarbeitung.” Hanser, Miinchen. Wang, D. C. C., Vagnucci, A. H., and Li, C. C. (1981). Computer Graphics and Image Processing 15, 167. Wang, X., and Wang, D. (1990). IEEE Trans. SP-38, 1473. Wang, X. (1992a). IEEE Trans. SP-40. 482. Wang, X. (1992b). IEEE Trans. Image Processing 4, 543. Wolberg, G., and Pavlidis, T. (1985). Pattern Recognition Letters 3, 375. Wu, Y., and Maitre, H. (1992). Optical Engineering 31, 1785. Yaroslavsky, L. P. (1985). “Digital Picture Processing.” Springer, Berlin. Yu, T. H., and Mitra, S. K. (1993). I n “Image Processing: Theory and Applications” (G. Vernazza, ed.), 75. Elsevier, Amsterdam. Zamperoni, P. (1989). In “Proc. ICASSP ’89,” Glasgow, 1401. Zamperoni, P. (1991). “Methoden der digitalen Bildsignalverarbeitung.” Vieweg, Wiesbaden. Zamperoni, P. (1992). Digital Signal Processing 2, 174. Zamperoni, P. (1993). In “Proc. of 14th GRETSI Colloquium,” Juan-les-Pins, 543. Ziou, D. (1991). Pattern Recognition 24, 465-478.

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ADVANCES IN IMAGING AND ELECTRON PHYSICS. VOL . 92

Electromagnetic Propagation and Field Behavior in Highly Anisotropic Media

.

CLIFFORD M KROWNE Microwave Technology Branch. Electronics Science & Technology Division Naval Research Laboratory. Washington. D .C.

I . Introduction . . . . . . . . . . . . . . . . . . . . . . I1 . Materials . . . . . . . . . . . . . . . . . . . . . . . 111. Conductivity Tensor for Carriers and Fields . . . . . . . . . . . . IV . Anisotropic Energy Bands . . . . . . . . . . . . . . . . . . V . Quantum Phenomena from Magnetic Fields . . . . . . . . . . . . VI . Band Structure Anisotropy Effect on Conductivity Tensors . . . . . . . VII . Variational Formula for the Propagation Constant . . . . . . . . . VIII . Finite-Element Formulation for Propagating Structures . . . . . . . . IX . Planar Guiding Structures . . . . . . . . . . . . . . . . . . A . Normal Mode Fields . . . . . . . . . . . . . . . . . . B. Transformation Operator Matrix P . . . . . . . . . . . . . C . Dyadic Impedance Green’s Function Construction . . . . . . . . D . Strip Surface Currents . . . . . . . . . . . . . . . . . . E . Dyadic Admittance Green’s Function . . . . . . . . . . . . . F . Slot Surface Fields . . . . . . . . . . . . . . . . . . . G Anisotropic Determinantal Equation . . . . . . . . . . . . . X . Magnetoplasma Permittivity Tensor . . . . . . . . . . . . . . XI . Chiral and Chiral-Ferrite Media . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . B. Theoretical Foundations . . . . . . . . . . . . . . . . . C Vector Helmholtz Equations . . . . . . . . . . . . . . . . D . Dyadic Green’s Function . . . . . . . . . . . . . . . . . E . Dispersion Relations . . . . . . . . . . . . . . . . . . F . Electric Field Polarization . . . . . . . . . . . . . . . . G . Conclusions . . . . . . . . . . . . . . . . . . . . . XI1 . Vector Variational and Weighted Residual Finite Element Procedures for Highly Anisotropic Media . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . B. Vector Helmholtz Sourceless Equation . . . . . . . . . . . . C . Gyroelectric-Gyromagnetic Variational Analysis . . . . . . . . . D . Isotropic Gyroelectric and Gyromagnetic and Optically Active Variational Analysis . . . . . . . . . . . . . . . . . . . . . . E . Complex Anisotropic Variational Analysis . . . . . . . . . . . F . Weighted Residual Analysis of Non-Hermitian 6 x 6 Tensor Medium . . G . Conclusion . . . . . . . . . . . . . . . . . . . . . XI11 . Ferrite Media . . . . . . . . . . . . . . . . . . . . . . A . Introduction . . . . . . . . . . . . . . . . . . . . . B. Governing Helmholtz Wave Equation . . . . . . . . . . . . .

.

.

I9

80 81 83 85 88 90 93 94 95 95 102 103 106 108 108 112 114 117 117 118 122 123 125 130 132 132 132 135 136 139 142 145 158 158 158 159

Copyright 0 1995 by Academic Press. Inc . All rights of reproduction in any form reserved ISBN 0-12-014734-3

.

80

CLIFFORD M. KROWNE

Finite-Element 2-D Equations . . . . . . . . . . . . . . . Interfacial Boundary Conditions . . . . . . . . . . . . . . Assembly Process . . . . . . . . . . . . . . . . . . . Determination of Circuit Parameters . . . . . . . . . . . . . G. Conclusions . . . . . . . . . . . . . . . . . . . . . XIV. Numerically Calculated Results for Guided-Wave Structures . . . . . . . A. Semi-infinite Planar, Rectangular, Circular, and Planar Microstrip and Slotline Structures . . . . . . . . . . . . . . . . . . . B. Conclusions . . . . . . . . . . . . . . . . . . . . . XV. Overall Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . C. D. E. F.

162 167 179 181 183 183 183 196 200 201

I. INTRODUCTION It is the purpose and thrust of this survey chapter to encourage and foster an atmosphere of research and development based upon the properties of anisotropic semiconductors under applied static magnetic fields, anisotropic chiral materials, full bianisotropic media, anisotropic ferrite under applied static magnetic fields, and as yet undiscovered materials which result in additional or induced anisotropy. The consequent nonreciprocal microwave and millimeter-wave device properties resulting from the induced anisotropy are the focus here. It is of immense technological interest to implement control components in planar configurations compatible with hybrid and monolithic circuits used in commercial and military systems. Heterostructure semiconductor devices may provide an alternative class of structures displaying nonreciprocal behavior relying upon periodic variation of the semiconductor band structure over spacings characterized by hundreds of angstroms. Much attention will be given to the fundamental underlying physics of carrier transport and electromagnetic properties, since this is where the necessary changes and innovations to the fields of the magnetoplasma effect, chiral phenomena, full bianisotropic media, and gyromagnetic effect in microwave and millimeter-wave devices may occur. Some of the most recent and advanced numerical simulation results will be presented for the magnetoplasma effect. It is apparent that much of the basic magnetoplasma physics work occurred in the 1940s through the 1960s (see these references as indicated by the titles of the papers; about 135 references) in unbounded media, unbounded and waveguiding structure work in equal amounts in the 1960s through the 1970s (about 75 references), unbounded planar and bounded planar waveguidesin the 1980s through the early 1990s (about 50 references). From 1960 onward, about 70% and 30% took place, respectively, in the 1960-1970 and 1980-1990 periods, with nearly half above 30 GHz.

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

81

11. MATERIALS Some of the common semiconductor materials associated with magnetoplasma studies and devices are InSb, Te, GaAs, Si, and Hg,-,Mn,Te. Table I lists some of the properties of these semiconductors. The data provided in this table are for T = 300K room temperature, x = 0.4 for for low T. Cd,Hg,-,Te, and hole and electron mobilities p h and P, &, N,, N,, m*,x , Eg, EF, A D , a, 5, p, Eo p ,A , , are, respectively, the density, relative dielectric constant, conduction and valence band edge density of states, relative effective mass, electron affinity, direct bandgap, Fermi energy, Debye length, lattice constant, momentum relaxation time, mobility, optical phonon energy, and mean free distance. One of the characteristic measures in the examination of magnetoplasma behavior is the cyclotron frequency o,= eB/rn*, (1) which, in comparison to a ferrite material with a Larmor precession frequency of (Brazis et al., 1979a) oL= eB/m,, (2) is much larger since the free space electron mass me is much larger than the electron effective mass m*. Here e is the charge of an electron and B the magnetic field magnitude. For hole carriers instead of electron carriers, the same reasoning applies. Operation of solid-state plasma microwave/ millimeter-wave devices up to 1,000 GHz has been postulated theoretically. However, a recurrent problem with the solid-state devices has been the dissipative loss associated with the carrier scattering in the lattice. A momentum relaxation time t characterizes this scattering. This can be ameliorated somewhat by the reduction of lattice temperature, but the desire for room-temperature operation of circuits makes this an unattractive option except in specialized cases such as those found in radio astronomy. Cyclotron effective mass can be a tensor, in which case a generalization to (Ashcroft and Mermin, 1976)

rn*(&,k,)

hZ aA(&,k,) 2n a&

=-

(3)

occurs, where k, represents the arbitrary B field direction and A the k-space area enclosed by the orbit in its plane. In this formula E is the electron energy. Another important measure in the study of magnetoplasma effects is the plasma frequency = ne2/m*&,

(4)

82

CLIFFORD M. KROWNE

TABLE I" PROPERTIES OF SEMICONDUCTORS USEDIN ~~AONETOPLASMA STUDIES

P (dcm') &

Nc N"

Si

Ge

2.33 11.9

5.33 16.0

2.8 x 1019 1.04 x 10'9

1.4 x 1019 6.0 x 10l8

Te 6.25 &I = E, =

55

32.5

rn*

x

e h (v)

Eg (ev) EF(ev) n, (1/cm3) A Df m )

rn: = 0.19 rn& = 0.49 4.05 1.12

= 0.98, = 0.16,

1.64, 0.082 0.044, 0.28 4.0 0.66

1.45 x 10" 24 5.431 2.5 x 10-3

2.4 x 1013 0.68 5.646 1.0 x 10-3

1,500 450 0.063

3,900 1,900 0.037

e

76

105

h

55

a( ) T

rn: rnk

6)

0.0145 0.40 0.334 6.0 x 1015

p (cm2/V.s)

e h Eop(ev)

1,(A)

InSb

GaAs

5.78 17.7

5.32 13.1

2.8 x 1019 1.04 x 1019

4.7 x 1017 7.0 x 10l8 0.067 = 0.082,

0.0145 0.40

rn:

4.59 0.17

rn& = 0.45 4.01 1.42

5.76 x 1014T1.5exp(-0.129/kBT) 6.479

1.79 x lo6 2,250 5.653 10-8

1,250 76 55

9.86 5.7pm) = 7.6 = 1.0pm) = 4.5 2.7s x 1017 2.75 x 1017 rn$(EF)= 0.00948

n(A n(A

=

rn, = rn, = 0.064, rn3 = 0.69 0.013 0.0298 4.746 1.0 x lo-'' (10 K) 9.0 x lo-'' (10 K)

8x 1.8 x 80,000

Bi

8,500 400 58

9 x 107(4.2/T)' 6 x 106(4.2/T)'

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

83

TABLE I-continued Cd,Hg,-,Te 7.13 15.3

0.006 (x = 0.2) 0.008 (x = 0.196, T = 24 K) 0.426 3.6 x loi4 7x

6.47 (x = 0.2, T = 77 K, n = 10,ooo (x = 0.2)

'The data provided in this table are for T = 300 K room temperature, x = 0.4 for for low T. P , E , N , , N,, m*, x , Eg, EF,A,, 0 , r, P, Eop. A,, are, Cd,Hg,-,Te, k h and respectively, the density, relative dielectric constant, conduction and valence band edge density of states, relative effective mass, electron affinity, direct bandgap, Fermi energy, Debye length, lattice constant, relaxation time, mobility, optical phonon energy, and mean free distance.

which is a function of the carrier concentration n and the dielectric constant E . Conductivity effective mass can be a tensor, in which case a generalization from (3) to

(5) oIB

1

=--

a2&

h2 aka ak,

occurs. 111. CONDUCTIVITY TENSOR FOR CARRIERS AND FIELDS The interaction of a system of carriers with a microwave field varying as exp[-i(wt - 41-11can be conveniently described in terms of the complex conductivity tensor of the electron gas. We begin by considering an electron gas consisting of a single type of carrier characterized by an isotropic effective mass. In the framework of this model, which corresponds quite well to many narrow-gap semiconductors in the extrinsic range, the solution of the Boltzmann equation which is applicable under well-known conditions

84

CLIFFORD M.KROWNE

yields a conductivity tensor in the following form (Brazis et al., 1979b):

where B is the magnitude of the dc magnetic field B, Bi,j,kare its Cartesian components, and

ne2 c1=

2

(

f

)

1 + (w,2)2 '

ne2

c 2

= m*

(

),

w,2f3

1

ne2

+ (w,f)2

c 3=

(

w,f2

)

112* 1 + ( w , f ) 2 (7)

In (7), ( F ) indicates an average over the energy,

j; de F ~ ~ / ~ ( d f ~ / d e ) (F)= j; dE E3/2(df0/de) 9

fo is the Fermi-Dirac distribution function, and ?(&) =

7(E)

1 - iw7(~)'

(9)

Furthermore, m* is the effective mass; o is the signal radian frequency; 7 is the momentum relaxation time, which can depend on energy E ; n is the electron concentration; 6, is the Kronecker delta function; &,k is the antisymmetric unit tensor; and we define the cyclotron frequency parameter w, = eB/m*, where e is the charge of the free carrier in question (thus, e > 0 and w, > 0 for holes; e < 0 and w, < 0 for electrons). It is convenient to choose a coordinate-system such that B points in the positive z-direction. We then obtain a conductivity tensor in the gyrotropic form axx

axu

am

0

:),

6 ,

where axx= ayy, axu= - aYx, (1 -

i07)7

Note that reversing B reverses the sign of ow.

(10)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

85

This tensor can be diagonalized in a "rotating" coordinate frame,

[

1 $ ( i + i j ) , E ( i- i j ) ,

(12)

so that the current in this frame can be written as

where the tensor components a, are

The quantity J+ represents here the amplitude of the vector 1

J, = J+ -(i + ij)e-"'",

\Iz

(15)

i.e. , the amplitude of the circularly polarized current; similarly, E+ denotes the amplitude of E,, etc. The subscript + indicates that the current is polarized clockwise when viewed from the origin toward the positive z-direction in a right-handed coordinate frame, i.e., in the same sense as that of the cyclotron motion of a negative charge when B is taken along the positive z-direction. For a system consisting of several types of carriers, the preceding results can easily be generalized because of the additive nature of currents arising from each carrier type. Thus, for s types of carriers, the components of the conductivity tensor become s

aij=

c 08)

(16)

I= 1

where 08) is the contribution of the lth type of carrier, calculated as described earlier. IV. ANISOTROPIC ENERGY BANDS Microwave propagation is affected by the nonspherical band characteristics of many semiconductors including Si, Ge, and narrow-bandgap semiconductors such as PbTe and Bi,_,Sb,. Such semiconductors are described by an anisotropic carrier effective mass, and some of them also by an anisotropic lattice dielectric constant x1 (Brazis et al. , 1979b). These effects

86

CLIFFORD M.KROWNE

can be included in the complex dielectric tensor &

= &.(X1

+ i&),

where a is the conductivity tensor of the semiconductor. Various semiconductors are characterized by different types of electron energy spectra such as parabolic and nonparabolic, and by different forms of constant-energy surfaces such as ellipsoids, warped ellipsoids, and warped spheres; therefore, a general discussion of conductivity anisotropy would be difficult. Particular forms of the microwave conductivity tensor for a number crystals (Ge, Si, Bi, graphite, and other elemental and compound materials) have been discussed in connection with earlier investigations of cyclotron resonance and helicon resonances. Here we choose the many-valley model with ellipsoidal constant-energy surfaces which is applicable to Si, Ge, PbTe, and PbTeBi,-,Sb,. The contribution of a single valley to the ac conductivity can be found by solving the equation of motion

d(m --

- v) - e(E + v x B ) - v m - v

dt

and using the relation for the current density j = a - E = nev,

(19)

where v is the collision frequency tensor. The nth valley conductivity for a parabolic band model with the electric field varying as exp(-iwt) and the dc magnetic field B is a=

(v

-

ne2 iw)m + e n ’

(20)

where 0 -B,

B2 (21)

Bi are the components of the dc magnetic field in the chosen coordinate system. When the collision frequency is small (vij < a),there occurs the cyclotron resonance at w = w,, where

ELECTROMAGNETICPROPAGATION AND FIELD BEHAVIOR

87

is the cyclotron frequency, I is the unit vector in the direction of the dc magnetic field B, and Det rn is the determinant of the effective mass tensor rn defined in a chosen coordinate system which in general does not coincide with the prinncipal axes of the mass ellipsoid. Note that in the many-valley model with N ellipsoids, there are N species of carriers characterized by different cyclotron effect masses. If some of the ellipsoids make the same angle with B and have identical principal components rnl, rn2, rn3, they are called equivalent. One can take the four electron ellipsoids in PbTe or Ge with B applied along the [ l l 11 direction as an example in which one ellipsoidal valley gives electrons with low cyclotron effective mass rn, = mT, and the other three valleys are equivalently tilted to the first to give m, = rnT.\/2rn,/(rn, cos2 e + t?&Tsin2e) > mT, (23) where B is the angle between the axis of revolution of the ellipsoid and the magnetic field (0 Q 0.4n for B 11 [lll]). For collision-dominated media (vij> a), it is convenient to introduce the inverse mobility tensor y = p-' = rn * v/e (24) and the cyclotron mobility

in analogy with the cyclotron effective mass. With these notations and for B 11 z , the conductivity tensor for a single ellipsoidal valley is

The second term in o has a maximum at pcBz = 1 . The limit vij % w is applicable to many narrow-bandgap semiconductors at temperatures above that of liquid nitrogen, and for the frequency region below 10 GHz. When the collision frequency is scalar, the p-approach can be extended to include effects of inertia by simple transformations pij -,pij/(l yij -, yij(l

-

iwz),

iwz),

(274 (27b)

where z = v-I. A more microscopic treatment of conductivity can be obtained by using directly the Boltzmann equation, and moments of the Boltzmann equation.

88

CLIFFORD M. KROWNE

-

For hole carriers using the k p perturbation theoretical framework, the energy vs. k band structure is given by

h2 2m

E(k) = - - [ A k 2

+ Cz(kzk; + k,k: + k;k:)].

f .\jB2k4

(28)

Using this band structure model, the light and heavy holes in Ge and Si can be described applying the Boltzmann theory or Shockley’s integral for warped surfaces. The parameters A, Byand C are given approximately as Ge Si

A 13.1 k 0.4 4.0 f 0.1

B

C

8.3 f 0.6 1 . 1 f 0.4

12.5 f 0.5 4.1 f 0.4

(29)

The nonparabolic nature of the conduction bands has been examined in InSb at very high B fields by applying fields between 60 and 300 kG, and studied between wavelengths of 10 and 40pm. The results must be interpreted in terms of resonance in the quantum limit between Landau states of low quantum number. Energy band structure expressions are expressible using an effective mass equation such as the Klein-Gordon equation for relativistic electrons:

Energy level separation is determined by the difference between the nth and (n - 1)th levels. Approximate expressions for effective masses and g factors result from this energy band model:

V. QUANTUMPHENOMENA FROM MAGNETIC FIELDS

The Hamiltonian of an electron in a magnetic field gives rise to harmonic oscillator-like levels. In a semiconductor, the Hamiltonian in the effective mass approximation employing the Landau gauge A = H(0, x , 0 ) looks like (Lax and Mavroides, 1960) 1 2m*

XY,,= -[p,2 + (p,

-

m*o,x)’

+ p,2]~,,.

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

89

The solution to the quantum-mechanical problem is

:>

(

c n = n + - h a c + -PZ 2m*'

(33)

In a semiconductor, the full wave function is a product of the envelope function, which is the solution of the effective mass Hamiltonian and extends through the crystal, and a band edge function which is characteristic of the unit cell, namely, Win

(34)

= Wn(r)Ui,

where i refers to a particular band. There are two classes of selection rules for transitions between Landau levels, intraband and interband. The matrix element can be written as

1

Win(P * e)Wjm =

ice. 1 +1 Leu uiuj dr

Wn(P * e ) w m dr

crystal

u i ( p * e ) u j dr

y ncy,dz.

(35)

crystal

The selection rule for the first is intraband or cyclotron resonance transition with the selection rule An = f 1 , and the second is the interband transition with An = 0. These two phenomena are complementary and together have proved to be very powerful tools for studying the band properties of semiconductors. A dramatic example of cyclotron resonance at submillimeter wavelengths is the polaron resonance in CdTe. An electron in a magnetic field in a polar semiconductor introduces additional terms which alter the transition energies between quantum levels. Perturbation theory (a variational approach could also be done) leads to the quantized energy level expression

en = (n + 1/2)hwc +

sin-' 1/2)~w,

a(Aw)3'2

4(n +

d y . (n

+ 1/2)hwc

(36)

Expansion to higher-order terms of the cyclotron frequency shows that the electron-phonon interaction, as represented by the electron-phonon coupling coefficient a!, introduces nonparabolic terms. The variation of the effective mass with magnetic field is given approximately as

)'.

m* Fitting this equation to data yields theory of the polaron.

a!

(37)

= 0.4, which confirms the Frohlich

90

CLIFFORD M. KROWNE

VI. BANDSTRUCTURE ANISOTROPY EFFECTON CONDUCTIVITY TENSORS The effect of anisotropy in band structure on the conductivity tensor can be demonstrated by treating electrons in the conduction band in Ge, for which the surfaces of constant energy in momentum space consist of four ellipsoids of revolution oriented along the [l 1 11 directions (Donovan and Webster, 1963). Thus, choosing the principle axes of an ellipsoid as the coordinate system, the energy E may be expressed in terms of the momentum p as

where the principal effective masses are m, = m2 (transverse) and m3 (longitudinal). We denote the time-dependent electric field by Gi and a static magnetic field by Hi (i = 1,2,3). The contribution to the current density from one ellipsoid is given by

where fo is the Fermi distribution function and f is (Gaussian units)

4=-

er mlD(l

e'5

+ ior)

+

c(1

+ ior)

[P;2.

P3H2] 1713

plus two further terms obtained by cyclic permutation of the suffixes, with D = l + (

er

7["'+-+-].

(1 + i o r )

11111113

Hi m3ml

H: m1m2

The current density can be written in tensor form as

The sii form may be examined by looking at sll: Sll =

--

s

-

4n3h3 dEm, D(1

er

+ ior)

[I+*( er y]dp. mlm3 c(1 + ior)

(41)

91

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

Considering only lattice scattering, the relaxation time is

where a is a function of temperature and k is Boltzmann's constant. Intervalley scattering is neglected. View the electron gas as nondegenerate, with the distribution function

where n is the carrier density for a single ellipsoid. Using the notation

E x=kT'

u=-

4ne2a 37P2m3'

k =m L , m1

the first integral in sll leads to the result

[x2(x+a2&

+ a2Q2)- iawx3/2(x + a202- a2Q2)] (x - a2w2 + a2Q2)

--x

e dx, (47)

where

Parameter Q is the cyclotron resonance frequency associated with the ellipsoid for a general orientation of the magnetic field, and has the value eH/m, c for H = (0, 0, H3). The integral part of sllis defined as a(o,H). Other sik(w,H) require that two more integral expressions be defined as

j

OD

Y ( W H )=

0

x2(x - 3a202 + a2Q2)- iuax3/2(3x - u 2 0 2 + a2m2)] --* x(x - 3a2w2+ u2m2)2+ a202(3x- a2w2+ a2m2 )2 e dx* (50)

a,/3, y , after partial fraction expansions, can be written as combinations of

92

CLIFFORD M . KROWNE

where

Total current density is found by the following procedure. Transformation to the cubic axes of the crystal and summation over the four ellipsoids is carried out, using indices 1 , 2, 3 to signify cubic axes. For each ellipsoid an appropriate back transformation must be performed on H to compensate for the effect of rotating the field along with the axes when transforming to the ( 1 , 2 , 3 ) system. This causes a different contribution from each ellipsoid to any given tensor component, which is obtained as a sum over the four possible values of aj(o,H), q,&(o,H), and y j ( o , H), the qj coefficients dependent upon H. Different values of aj(o,H), etc., occur as a consequence of the four cyclotron frequencies Qj into which the earlier Q expression transforms. We find QI(W, H)

=

1 2

- [Z~(O + Q)

+ Z ~ ( O - Q)]

ia 2

- -[(a+ Q)Z,(o 1

B(O, H) = -[(w + Q)Z2(0 2Q

i + -[Z,(W 2aQ

+ Q) + (0 - Q)Z,(o + a) - (0 - Q)Z2(0

+ a) - Z,(w

- Q)],

- Q)],

(54)

- Q)]

(55)

93

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

The four cyclotron frequencies are

0,= L[(K + 2)IH12 + 2(K

l)(HIHz + H2H3

+ H3H1)]1/2, (57a) O2 = A[(K + 2)IHI2 + 2(K - 1)(-H1H2 - H2H3 + H3H1)]1/2,(57b) n3= L[(K + 2)IHI2 + 2(K - 1)(H1H2- H2H3 - H3H1)]1/2, ( 5 7 ~ ) Q4 = L[(K + 2)IH12 + 2(K - 1)(-H1H2+ H2H3 - H3H1)]1’2, (57d) -

with 1/2

and the coefficients qj are 41 =

(K - 1 W I + Hz)+ (K + 2 w 3

q 2 = (K

- 1)WI

Y

(59a)

H2) + (K + 2 w 3 ,

(59b)

+ (K + 2)H, , + Hz) + (K + 2)H,.

(59d

-

q3 = (K - 1)(-H1 - H,) 44

=

(K - l)(-H1

(59d)

Other components of S, are found by cyclic permutation of the subscripts on H. One can introduce a system of axes ( x , y , z ) such that the incident radiation is propagated along the positive z direction, and the electric vector is initially parallel to the x axis. Relative to these axes, the magnetic field has the form (0,0, Ho),and the current density components can be put down as

It remains to transform from the previous (1,2,3) system to the (x, y, z) system.

VII. VARIATIONALFORMULA FOR THE PROPAGATION CONSTANT In order to derive an approximate expression for the propagation constant in a solid-state plasma waveguide, consider Maxwell’s equations and eliminate H (Kuno and Hershberger, 1967a). We find

VxVxE

=

02poeE - jmp0i3 * E.

(61)

Scalarly multiplying this by E* gives

-

-

E* V x V x E = 02poeE*- E - jopoE* (6E).

(62)

94

CLIFFORD M. KROWNE

The wave propagating in the positive z direction can be written as Substituting this field into the previous equation multiplied by the conjugate field and integrating over the cross-section S of the waveguide, making use of vector identities and the divergence theorem, produces k2

1is

(a, x e:) r

- (a, x el) dS +

s s.

(V, x e:)

- (V, x el)dS

r r

where C is the contour along the waveguide wall enclosing the cross-section S , and n is the outgoing normal unit vector. Looking at the case where a longitudinal B field is applied, quasi-TE modes exist. Choose a trial field el such that the z component of el is negligibly small everywhere in S and that the tangential component vanishes at the waveguide walls, i.e., el a, = 0 in S and n x el = 0 on C. Then solving for k finally yields

This variational formula can be shown to be stationary provided that = 0 on C. This formula is in particular applicable to cylindrical symmetry problems.

n x el

VIII. FINITE-ELEMENT FORMULATION FOR PROPAGATING STRUCTURES

The finite element method is an extremely general numerical technique suited to solving electromagnetic field problems having irregular geometric configurations and anisotropic substances. For a layered field problem recently considered in the literature (Mohsenian et al., 1987), two-dimensional finite-element equations were derived (although the problem is strictly

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

95

one-dimensional). The electric field E, used was chosen as the dependent variable for the finite-element analysis. This component is an uncoupled component, and the other field components are expressible in terms of E, using Maxwell’s equations. A Galerkin formulation was employed, yielding

where the Ni(x,y ) are the finite-element interpolating (shape) functions for the problem, and the index i ranges over those nodal points at which no geometric boundary conditions are imposed for E,. The interpolating functions are used to approximate E,(x, y ) in the following manner: E,(x, y ) about equal to [N(x,y)][E,],where [N(x,y ) ] is the row vector of interpolating functions and (E,) is the column vector of nodal point values. An application of the divergence theorem yields the finite-element equations in the form

The i, j-th element in the coefficient matrix [TI is given by

The line integral in (67) is to be evaluated around the boundary C of the area A under consideration, with (n,, n,,)being the components of the unit normal vector to C. The finite-element equations are assembled from the element contributions to (67), the boundary integral being evaluated around the boundary C, of each element.

IX. PLANARGUIDINGSTRUCTURES A . Normal Mode Fields Each layer has four eigenfunction field solution sets (Krowne, 1984a). Superposition of these four normal mode sets of field components constitutes the actual total field solution obeying all boundary conditions (BCs). Here the normal modes are found for the mth layer without regard to the BCs. Imposition of various BCs as part of the overall propagation constant y determination is done in the discussion on Green’s function construction.

96

CLIFFORD M. KROWNE

Time harmonic, plane guided-wave solutions proportional to expuot - yz) are assumed. The propagation constant y = a + j/3 makes the wave + z-direction propagating if > 0. Insertion of the time harmonic nature of the plane wave into Maxwell’s two curl equations creates the single sourceless matrix equation LTVL = j ~ V i .

(69)

Magnetic current sources are not considered in the treatment in this paper, and electric current sources are included by discontinuity BCs at interfaces between layers. VL and V i are column vectors containing electromagneticfield components in rectangular coordinates both tangential to the parallel interfaces (xz-plane) and parallel to the y-axis. VL consists of the electricfield E and magnetic-field H components. Vi consists of the electric displacement-field D and the magnetic displacement-field B components:

VL =

’

Vi =

Primes indicate that these vectors contain x, y, and z spatial variation with the z spatial variation to be dropped shortly. Operator LT is a 6 x 6 matrix composed of single partial-derivative operators. LT can be expressed as

where submatrix L 1 is a 3 x 3 matrix

0 L1 =

a

-

az

--a az

a

G a

0 --

ax

(73)

ELECTROMAGNETICPROPAGATION AND FIELD BEHAVIOR

97

To remove explicit z spatial dependence from the sourceless matrix equation, V, and V, are defined as V, = V; eYz,

(74a)

VR = V i eYZ.

(74W

The uniform (in the z-direction) guided-wave problem with this last prescription can now be solved in transformed space by going from x direct space to k, reciprocal space. The one-dimensional Fourier-transform pair ( f , f ) is defined as

j

00

f ( k x,Y ) =

A x , u) e-jks dx,

(7W

-00

wheref(x, y) is any real space variable. Using the last two equations allows conversion of the guided-wave problem as given by the original souceless matrix equation into the FTD (Fourier-transform domain)

L,VL

(76)

=j0VR,

where tildes denote FTD variables and

The Fourier-transformed electromagnetic fields

v;.= [Ex Ey FR = [ f i x

fiy

vL and vRare

Ez fix

fiy

fiz]Ty

(79d

Bx

By

a]'.

(79b)

fiz

The medium of each layer can be characterized by a single 6 x 6 constitutive tensor k in direct space. k relates VR to VL: V; = kv;,

(80)

98

CLIFFORD M. KROWNE

where

;I.

.=[;,

2 and C; are, respectively, the permittivity and permeability tensors. E^ can lead to electric birefringence in uniaxial and biaxial dielectric crystals. Gyroelectric nonreciprocal behavior can be induced through 2. This may occur by applying a magnetic field to a doped semiconductor layer in a Faraday, Voigt, or mixed configuration. Similarly, using p, magnetic birefringence can be obtained in a dual fashion in uniaxial and biaxial magnetic crystals. Gyromagnetic nonreciprocal properties can be had by applying a magnetic field to a ferrite material, for example. p^ and p^' tensors are responsible for optical activity. k could create nonlinear effects by being dependent on the field vector VL. k could also be dependent on the coordinates x and y. It is possible that k might be dependent on both the field vector and coordinates. In order to maintain a linear problem, k is treated as a constant tensor. x-coordinate variation of k cannot be built back into the problem solution by such an assumption, but y-coordinate variation can be by slicing the layer with ky-dependence into thin sections. Fourier-transforming the last equation and using the VL, definition yields VR =

kVz.

(82)

Placing this result into the FTD sourceless matrix equation produces the equation which must be solved for the normal-mode field vectors,

The V2 and V5 vector components of VL (i.e., Ey,R,,) are algebraically expressible in terms of the other field components using rows 2 and 5 of the last equation, giving 6

-yV4

- jkxV6 = jo

C m2ic, i= 1

The solution of this pair of equations is 6

6 = C aij(l - S,,j)(l j= 1

- S,,,)C,

(84a)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

where

are Kronecker deltas. The aij coefficients are

Here the mu are the & tensor elements. Rows 1 , 3,4, and 6 of the sourceless matrix FTD equation are first-order linear differential equations, dv6

yV5 + - = j w dY

dv4

--

dY

6

C mlic, i= 1 6

+ jk,v5 = j u C

-

dv3

6

dY

i= 1

-yv2 - -= j w C dvl

- - jk, dY

ntjic,

i= 1

6

m4ic,

v2= jo C m6ic. i= 1

(9W

100

CLIFFORD M. KROWNE

Using (85) to eliminate equations produces

v2 and v5 from these last four simultaneous

0 0 0 1 d 0-1 0 0 G&lo 0 - 1 0 [l

Lo

0

0 1

Here, the matrix elements rb of R’ are

i = 1,2,3,4.

(93)

Equation (91) reduces to a more convenient form when both sides of it are multiplied by the prefactor matrix S, found on the left-hand side. Since S,S, = Sp” = I, the identity 4 x 4 matrix, (91) becomes - -“ - RCp. .iadu

(94)

Cp is the four-element column vector in the FTD having only interface

tangential field components:

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

101

R is composed of matrix elements r, related to the R' matrix elements by

r.. = r!. ,,

i

= 1,4,

r.. U = -r&,

i

= 2, 3,

j = 1,2,3,4.

Translate y into the mth local layer shifted coordinate system: m-1

Y ~ = Y C - hj,

(97)

j= 1

where hj is the j t h layer thickness and y h = 0 corresponds to an interface. The solution to (94) in the yh coordinate system can be written as 4(yh) = ejkyyh4(0).

(98)

Substituting (98) into (94) produces

where the attached i subscript's meaning will be clarified soon. Equation (99) is an homogeneous equation in four unknown 4 vector components. It has four normal-mode vector solutions &(O), constrained by the requirement that

t ]

Det-I-R

=O

to assure a solution. Equation (100) generates four k,,eigenvalues kYi.These kYi values are placed in (99) to find the individual c#q(O) normal-mode vectors at yh = 0. The normal-mode vector 4Y(yh) at y A > 0 is found from dT(0) by multiplication with a 4 x 4 matrix characterizing the mth layer medium:

mrh) = JwtPMv9.

(101)

Here, pm(y6) = ly"(o)Km(r6)w"(o)-', K$ = 6, exp(jk$yh), ly"(0) =

r4w> 4 m 4x-9 4w-91.

(102)

(103) (104)

The superscript m in (101)-( 104) emphasizes the specialization of the eigenvector solution to the mth layer. Y"'(0) is a 4 x 4 matrix constructed out of the four normal-mode vectors. +Y
102

CLIFFORD M.KROWNE

B. Transformation Operator Matrix P In the ith layer, the solution of the differential equation in the y-direction can be put down as (Mostafa et al., 1987a)

%57

= P(Yf)W)

(105)

where P ( y i ) is the state transition matrix of the ith layer given by ~ ( y f=)eAyi,

(106)

and O(0) is the state vector at the interface yf = 0, i.e., at the bottom of the ith layer. The state transition matrix P is 4 x 4 and is recognized as a transformation operator which transforms the field from yi = 0 to the field value at yf within the ith layer. The transformation operator P ( y ) can be calculated by first finding the transverse eigenvalues kyi of A and then applying the Cayley-Hamilton theorem to the matrix A. As will be proven shortly, P ( y ) can be expressed as 3

C aiAi.

~ ( y=)

i=O

In general, the eigenvalues may be degenerate. The unknown coefficients

ai, which are functions of y , can be found by solving the system of linear equations 3

ekyiY =

C aikij, i=O

j = 1,2,3,4.

(108)

Fur the case of repeated (degenerate) eigenvalues, the derivatives of this equation are applied. The characteristic polynomial of an n x n matrix is defined as A(A)

=

IA - All

=

(-A)"

+ cn-lA"-' + + CIA + co = 0. * * a

(109)

This equation is satisfied by the eigenvalue Ai. Now if P(A) is a scalar polynomial of degree m and Pl(A)is another polynomial of degree n where n c m, then P(A) can be written as P(A) = Q(A)Pi

+ R(A).

(1 10)

The quotient Q(A) is a polynomial of degree m - n, and the remainder polynomial R(A) is of degree n - 1. If we choose P(A) to be any analytic function and P i @ )to be the characteristic polynomial, then

P(Ai) = I?(&).

(1 11)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

103

This is the Cayley-Hamilton theorem, which states that every matrix satisfies its own characteristic equation, that is, A(A) = [O].

(1 12)

Making use of the theorem, and using a matrix function argument in (1 1l), one can express any analytic function of a matrix P(A) as P(A) = U,I

+ a l +~a 2 ~+* - - - + a,_,A"-'.

(1 13)

The coefficients ai are the same for the corresponding eigenfunction equation P(&) = a,I + a11i + a21: + .*.+ (1 14) If the n eigenvalues are distinct, we get n independent algebraic equations that determine the coefficients ai . However, if we have repeated eigenvalues l i of a multiplicity order of m i , then we have dj/dAjA(A)lAi= 0, where j < m ; therefore

djR(1) d1j

IAi

-

IAi

--dJP(A) , - dAJ

j = 1,

..., mi - 1.

(1 15)

We obtain an mi set of linearly independent equations for Ai for degeneracy. Thus, a full set of n equations is always available to find ai. If we choose P(A) = eAy,where A is 4 x 4, with repeated eigenvalues A , of multiplicity m, = 2, then

C. Dyadic Impedance Green's Function Construction Boundary conditions on perfectly conducting metal strips are used to impose the necessary constraints for Green's function determination in the FTD (Mostafa et al., 1987a). At the interface y = hi in the FTD, fiX(hT)- fiX(h7)= - j z ,

(1 17a)

fiz(h;) - fiz(h7)= j;.

(117b)

Jzand J, are the unknown Fourier-transformed x and z current components at y = hi. The tangential electric field components EZ,Ex must be continuous along the interface. The

+ and - superscripts denote, respectively,

104

CLIFFORD M. KROWNE

just above and below the interace y = hi.Boundary conditions could be expressed in the vector form

The Green function formulation for a two-layerec. medium with current strips located at the interface will be derived here for demonstration purposes. Generalization to multilayers with several current strip interfaces is a straightforward procedure that follows the derivation here. The vector field &(h;) can be related to &(O) through the transformation operator P, i.e., & I ; ) =

P(l)(h1)&O).

(119)

The boundary conditions at the interface y = hl can be written as

Again, P(2)(h,)transforms the field vector &(h{) just above the interface y = hl to give the field vector at the top plane y = hl + h, such that

where Pc2l)is Using the abbreviation P(29(hl+ h,) = P(2)(h2)6P(1)(hJ, a 4 x 4 matrix, this last equation can be rewritten as

r

o

i

105

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

fix(0),fiz(0), fix(hl + h2), and fiz(hl + h2) are the tangential magnetic fields at the ground and top plate, respectively. Note that the boundary conditions at y = hl h2 have already been satisfied in this last form. From (122) we can express fix(0),fiz(0)in terms of J,, j z .Also, from the first and second rows of (120) we can express the slot field Ex&), &(Itl) in terms of fix(0),fiz(0),which are

+

fix(())= [S;(pg')pg- p(2l)p(2) 14 24) -

j ~ ( (21)P(2) ~ 2 4 13 - P{;')Pi;))]Det,

(123a)

fiz(0)= [.fx(Pi;l)Pii) - P(21)P,(;4)) 23 - Jz(P{;')Pi;) - P$gl)Pii))] Det, (123b) 23 - Pi;1)&1), Det = P{i1)@21)

(124)

giving

+ P;:)fiz(o), Ez(hl) = P$i)fi*(O)+ P$)BZ(O). Ex(hl)= P#fix(0)

(125a) (125b)

The resulting relationships define the impedance-type Green's function G in the FTD which relates the field components at the interface to the interface currents at y = hl : Ex,

=

EJan) =

6 1 1 ( ~ ,a n ) & ( a n )

+ 6 1 2 ( ~ a, n ) j z ( a n ) ,

( 126a)

6 2 1 ~ a~n 9) J x ( a n )

+ 6 . 2 ~a~ n)9 Jz(an),

(126b)

where the elements G, are given as (a,= kxn)

d,, = [Pfi)(P$il)P$) - Pfil)P$))+ Pi:)(Piil)P$) - P(21)Pi:))] 23 Det-', (127a)

6,, = - [p{!)(p$l)p(2) 14 23 ) + P14 (1) (p(21)pi;) 13 - p$:1)pt;))]Det-l, 13 - p(21)p(2) (127b) (1) P(2l)p(2)- P@')P{:))] Det-1, 621= [Pi:)(pi:')P(2) 14 - P(21)P(2) 14 24) + P24 ( 13 24 23 (127c) (1) p(21)p(2)- p(21)p(2) (1) p(21)p(2) - @1)@;))] Det-l, 6 2 2 = -IP23 ( 24 13 14 2 3 ) + P24 ( 13 23 (127d) where Pcm)ijdenotes the ijth element of the rn matrix, rn For a three-layer structure, P(')(h1)is replaced by

=

1,2,21.

PC21)= P(2)(h2)P(1)(h1), and P(2)(h2) is replaced by P")(h3).Also, PC2')(hl+ h2)is replaced by

P'3

=

P'3'(h,)~"(h2)~'''(h 1).

106

CLIFFORD M. KROWNE

Making use of these replacements, the three-layer Green's function is

6,, = [Pt;')(P$:')P::)- Pijl)P$:))+ Pj;')(Pt;')P$)- P$ll)Pt:))])/Det, (128a)

6,, = - [P$:')(Pg1)Pj;) - Pl:')P$l)) + Pl:')(Pf;')P$;) - P$;')P{;))]/Det, (128b)

GZl = [p$;l)(p$:l)p::)- p (143 1 ) p (243 ) ) + P g l ) ( P # l ) P $ ) &22

= -IP23

(21) p ( 3 1 ) p O ) ( 24 13

-

p{:l)pO)23

)

+

P(31)Pi:))]/Det, 23 ( 128c) Pgl)(P$;l)P$:) - P$;')P$)]/Det, -

(128d) Det = p t j l ) p ( 3 1 ) - p ( 3 1 ) ~ ( 3 1 ) . 23

13

24

(129)

Expanding the strip currents in (126) in terms of suitable basis functions, applying Parseval's theorem, and using a Galerkin approach, a determinantal equation for the propagation constant can be written based on the fact that the current expansion coefficients are not a trivial null set. D. Strip Surface Currents

For even modes with n, strips (Mostafa et al., 1987a), the FTD strip surface current expansions are (130a) (130b)

(131a) (131b) where n,, = n , / 2 , (n,- 1)/2 for even and odd numbers of strips, respectively, and n, , n, are the number of basis functions for j , and J , . Here the spacing factor sj is the distance from the origin to the center of thejth strip and an = (_2n l)n/2b, n n / b for even and odd modes, respectively. The quantities r e , to, ti, , 4, are single microstrip even and odd basis functions of j , and j,, respectively.

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

107

We will prove that the FTD form of the even mode currents is correct as provided. Currents J,, J, over any number of strips can be written as

The first sum term in these two equations accounts for those strips with x > 0, while the second sum term accounts for the strips with x < 0. The quantities t, tl are particularized for even and odd modes and indexed over thejth strip. We have J,(x) = J,(-x), (133a)

J,(x)

=

-JA-

x),

(133b)

which results in aJ. = c:J ’

bJ. = d:J ’

e.J = g:J ’

f. J = - h .J .

( 134)

The basis function td(x), for example, can be expressed as tej(x) = te(x -

sj).

(135)

By expressing the other basis functions in the same form, J,, J, can be rewritten as nss

J,(x) =

C

b j [ t e ( x - sj)

+ te(x + sj)l + bj[to(x - sj) - t o @ + sj)ll,

j =1

(136a)

j= 1

(136b)

If the Fourier transforms are taken for both sides of these two equations, the currents J, and J, can be set down in the Fourier-transformed domain. It is easy to show that a Fourier shifting relationship fe(x

* sj) = e*jansjfe(x)

(137)

holds. Identical transformation forms occur for the other basis functions. Applying the FTD to J, and J, then produces

108

CLIFFORD M. KROWNE

These equations are considered for only one basis function over each strip. Another n, or n, summation must be added to allow for many basis functions.

E. Dyadic Admittance Green's Function An admittance type anisotropic dyadic Green's function, in the spectral domain, is used to relate surface current J' to the tangential electric field at the conductor slot interface (Krowne et al., 1988a). The one-dimensional finite Fourier transform of an arbitrary two-dimensional spatial function f(XY Y ) is b/2

S(an Y ) = 9

-b/2

f ( x , y ) e-j"fi dx,

(139)

where y = interface and an = (2n - l)n/b, 2 n d b for even or odd modes, respectively, with n = any integer. Thus,

G ; ~ ( Yan>gx = G;~(Y [email protected](an) , + 6 1 2 ( ~ ,a n @ z ( a n ) , (140b) dyadic 6' = 6-', 6 being the impedance anisotropic dyadic. Slot electric & =

field components are expanded in terms of complete trigonometric basis function sets satisfying the edge condition. As for the impedance Green's function, a Galerkin approach is used to obtain a determinantal equation for the propagation constant.

F. Slot Surface Fields For single slot even and odd modes with respect to E, , we use, respectively, the basis functions (Krowne et al., 1988a)

E,,(x) = q,,(x)

=

R-'sin

(141a)

L

3

E,,(x) = tem(x)= R-' cos (2m - 1)and

,

(141b)

EXm(x)= qorn(x)= R-' cos

(142a)

E,,(x) = to,(x) = R-' sin

(142b)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

for 1x1

5

109

w/2 with m = 1,2, ..., and

[

(31 2 1/2

R = 1-

(143)

and the right-hand sides of E,, and Ezm are zero for 1x1 The finite Fourier transforms of Exmand Em are

+ Jot$

> w/2.

- (2m - l)n)] 2

(144b)

for the even mode, and

-

Jot$

- mn)]

(145b)

for the odd mode. Jo is the Bessel function of the first kind and zeroth order. Total slot spectral fields are then (146a)

(146b)

110

CLIFFORD M.KROWNE

For coupled slots, the Fourier transformed electric fields can be found by the ensuing procedure. Within the slots (field zero otherwise), write nX

ExW

=

C [ a m l t l e m ( x + s) + b m l t t o m ( x + s)I m= 1 nX

+ C

[amzftem(x - s)

+ bm2qom(x

- S)I,

(147a)

m=l

nz

Ez(x) =

C

+ s) + d m l t o m ( x + s)I

[~mItem(x

m= 1

m=l

Here the first summed term accounts for the slot located at x = -s, (w + w1)/2, whereas the second term accounts for the slot located at x = +s. Basis functions qem, qom,re,, tom are even and odd single slot functions. For the coupled-slot even-mode case, s=

(1 48a)

(148b)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

111

Inserting these coefficients into the original E , ( x ) formula gives nz

EzW =

C [ ~ m l [ t e m +( ~s) - CemW - S)I + d m l [ t o m ( ~+ s) + t o m ( x - ~ ) I I . m=l (153)

Following a similar procedure for E J x ) produces

am]= - a m 2 9

( 154)

bml = bm2,

and "X

W X )=

C m=

1

( a m i [ t l e m ( x + s) - tlem(X

- s)I

+ bml[tlom(X + s) + tlom(x - @]I. (155)

By employing the Fourier shifting relationships

f e , o ; m ( X + s)

=

e~anste,o;m(an),

(156a)

+ s)

=

e~V'*nst?e,o;m(an),

(156b)

t?e,o;mW

where the arguments of the left-hand side variables are present to identify the spatial variables transformed, the final forms of spectral slot electric fields are stated, using the last E , ( x ) and E x ( x ) forms, as nx

Ex(4

=

C [ah, sin(ans)t?em(an)+ bh1 co~(ans)t?om(an)I, m=l

(157a)

nz

Ez(an> =

C [CAI C O S ( a n s ) f e m ( a n ) + m=l

sin(ans)fom(an)l. (157b)

Here the primed coefficients are related to the old coefficients as follows:

ahl

bhl

= 2jaml;

=

2bm1;

chl = 2cml;

dh, = 2jdml. (158)

For the coupled-slot odd-mode case,

E,(x) = -E,(-X),

(159a)

E,(X)

(159b)

=

Ex(-x).

Using the same reasoning process as for the even-mode case, nx

Ex(an) =

C [GICOS(ans)t?em(ay,)+ bL1 sin(ans)t?om(an)l, (160a) m=l nz

&(an>

=

C [GIsin(ans)fem(an) + dA1 cos(ans)fom(an)l, m=

(160b)

1

where

a&

=

2aml;

bkl

=

2jbm1;

cdl

=

2jcml;

dA1 = 2dm1. (161)

112

CLIFFORD M. KROWNE

G . Anisotropic Determinantal Equation

Coupled-slot anisotropic determinantal equation is derived in the following manner (Krowne et al. (1988a). First find a Parseval theorem. At the interface of interest, write (162a)

A(n) = &an) = Letting A

=

[

b/2

A(x) e-jas dx.

(162b)

g or f ,

If one considers g = J, or J,, and f equal to the tern (or qem)or torn (or qom) parts (indicate parts by primes) of E, or Ex expansions, noting that f ( - x ) = +f(x) depending on symmetry, the assumption of perfect conductors leading to complementary current and field parts makes the right-hand side of the preceding equation zero. Now examine the two admittance dyadic Green’s function equations relating coupled slot and surface currents by the anisotropic G dyadic. Multiply the first equation by ijLj and the second by <$, tij;then the spectral sum (we can employ n = 1,2, ..., due to symmetry if desired) is m

m

1

n = -m

1 -m

iiij

G ; ~ ( Yn)E,(n) , = 0,

j = 1,2, ...,n,;

(164a)

m

m

n=

c

n)g,(n) + n = -m

ii~jG;i(~,

iiLj

c

6;,(v,n)E,(n) + n =

-m

t?;j

GiZ(y, n)E,(n) = 0,

j

=

1,2, ...,n,; (164b)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

113

Treat the even-mode coupled-slot situation by placing EJn) and EJn) into the preceding four expressions, grouping spectrally summed terms using q& = sin(ans)dej, #A, = c o ~ ( a ~ s f~~ ) ~=~ cos(ans)cej, ~, < L ~= sin(ans)coj, and defining (165a) (165b) (16%) (165d) (165e) (165f) (165g) (165h) (165i) (165j) (165k) (1651) (165m) (16511) (1650) (165P)

114

CLIFFORD M. KROWNE

one obtains

“‘1 2 @[“‘1 5 @[“‘1

9 s{y[ $’] +

m=l

bm1

S{r[

m=l

dA1

+

m= 1

bA1

=

0,

= 0,

...,n,,

(166a)

j = 1,2,..., n,,

(166b)

j = l,2,

dA1

m=l

where S { y , Sir, Siy,and Sir are n, x n,, n, x n,, n, x n,, and n, x n, sized matrices, and

Consequently, the anisotropic determinantal equation can be expressed in compact form as

where (169ab) or, in the most abbreviated form, as

s[

;]= o .

It is the solution to Det S(y) = 0 which we seek, y = y’, y-. For the oddmode coupled-slot situation, the same procedure applies if all sin(ans) and cos(a,s) factors are interchanged in the Ximqpand double-primed notation adopted for v, and w,.

X. MAGNETOPLASMA PERMITTIVITY TENSOR

E^ tensor will be derived here which is inserted into the 6 x 6 constitutive tensor fi (Krowne, 1984a). A Drude model is employed to describe the individual electron motion in the semiconductor by (Krowne et al., 1988a)

dv m*v m*- = q(E + v x B) - -, dt ‘SP where q, m*,and ‘ s ~are respectively, the electron charge, effective mass, and the momentum relaxation time. Electron velocity v and fields E and B

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

115

include dc and rf components. Electron particle current density is (Krowne and Blakey, 1987) J = qnv,

(172)

with n being the electron density. Particle current density J and n also contain dc and rf components. Decomposition of the variables in dc and rf parts is expressed as (Krowne and Tait, 1989; Krowne, 1987b)

+ EflejW1-yz, B = Bo + Bflejof-Yz,

(173b)

J = Jo + Jflej"r-yz,

(173c)

E = E,

+~ ~ e ' ~ ~ n = no + ndejWt-yz, v = v,

(173a)

- y ~ ,

(173d) (173e)

where o and y are, respectively, the electromagnetic radian frequency and the propagation constant in the z direction. For the simple case under study here with no electric field, current, or voltage dc biases, E, = 0, Jo = 0, and v, = 0. However, there is an applied dc magnetic field B, given by Bo = ~,[cos(+),sin(+), oIT =

BOB,.

( 174)

Inserting the superposition field representation into the equation of motion and the particle current equation, extracting out the zeroth- and first-order perturbations, and dropping all higher-order nonlinear terms yields m*v, 0 = qE0 + WOXBO - -, (175) ZP

J, =

m*(jw

qnov, ,

(176)

+ qvfl x B,,

(177)

Jfi = q(novfi + nfivo) = qnovfi.

(178)

+ t)vfl

= qEfl

The first two equations are trivially satisfied by the chosen dc bias conditions, while the rf current Jfl in the fourth equation is simply related by a dc proportionality constant to vfi. Eliminating vfl from the third equation using the fourth one, and using the plasma and cyclotron

116

CLIFFORD M. KROWNE

frequency definitions

48, -, m*

=

0,

where E is the static semiconductor permittivity, allows a single governing equation to be written (Krowne, 1988a; 1988d): (ju +

t)

Jd

=

cu;Efl

+ m,Jfl x Bo

The dc magnetic field unit vector Bo was defined earlier. The governing equation is recast as

Ed = $Jd

(182)

so that the macroscopic resistivity tensor $ is determined as

p-[

0'0 -t Ti')

0

0 - m, sin 4

0'0 + Ti') m, cos 4

m, sin 4

-!--

-mcCOS4] Em;

0'0

+ ti')

(183)

Notice the inherent asymmetry in the elements of the resistivity tensor. This asymmetry is mathematically caused by the opposite signs of the offdiagonal elements in the 13 and 31 terms and in the 23 and 32 terms. Only by removing the dc magnetic field will the cyclotron frequency m, prefactor go to zero. The diagonal terms are the same with a radian frequency part and a momentum relaxation time zP part due to carrier scattering, which in effect represents the loss of the medium. The requirement of a sourceless field problem and a single 6 x 6 relating i Efi, Dfl, Hd,and Bd leads to permeability constitutive tensor & 1; = Zpo (po = vacuum value), optical activities jjo = jj; = 0, and permittivity determined as follows. Maxwell's curl H equation is aD VXH=-+J, at

where D is the electric displacement vector field. Here D

=

EE,

H = pl'B.

(185a) (185b)

Placing the governing equation and the D, H relations into the curl equation, utilizing the field decomposition, and retaining the first-order

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

117

perturbations gives

V x Hd = (jwe + C-')Ed, leading to the permittivity

For GaAs, r electron valley effective masses are appropriate to use so that (Krowne, 1988c; Tait and Krowne, 1987a, 1987b; Tait and Krowne, 1988) W , = 2.63 x lO"BO(lO3 kG), ( 188)

up= 1.93 x 10'3B,dn(10'7/cm3). Attenuation a arises from the finite value of the momentum relaxation time 7,,in the resistivity tensor. The off-diagonal p elements cause A a and A/3 to be nonzero for finite B,. Here A a = 'a - a- and A/3 = 8" - /I-, where + and - denote the propagation in the forward and reverse directions in the waveguide and 3/ is the phase propagation constant. Relaxation time 7p can be estimated by a single formula (Krowne, 1987b): zp = 3.8048T-0.7361[20.133- 10g(N1)]@~),

(190)

which takes into account ionized impurity scattering (NI = ionized density) and phonon scattering. This equation is based upon experimental and theoretical results for GaAs at lattice temperatures between 77 and 300K (Krowne and Tait, 1988; Krowne, 1988b). For full ionization, no = ND (donor doping) = NI.

XI. CHIRAL AND CHIRAL-FERRITEMEDIA A. Introduction

Chiral media (Jaggard et al., 1990; Jaggard and Engheta, 1988; Jaggard et al., 1991; Engheta and Pelet, 1989; Pelet and Engheta, 1992; Lindell, 1992; Kluskens and Newman, 1991) and ferrite media (Collin, 1960; Neidert, 1992) have been studied over the last 15 and 30 years, respectively, for many applications. Chiral media have been examined as coatings for reducing radar cross-section, for antennas and arrays, for antenna radomes, in waveguides, and for microstrips substrates. Ferrites have seen use in various nonreciprocal devices such as phase shifters, isolators, and circulators.

118

CLIFFORD M. KROWNE

It is our interest here to examine a chiral medium by itself and then a composite medium having both chiral and ferrite properties. The chiral part of the medium by itself will be chosen to have a complex activity coefficient, and this will be shown to suggest, by itself, nonreciprocity in Section XI,B. Of course, by itself, the ferrite part of the medium is nonreciprocal (Krowne, 1993a). The vector Helmholtz equations for the electric field E and magnetic field H with sources are derived in Section X1,C. The sourceless equations are then presented. From these equations the dyadic Green’s functions are found for E and H in reciprocal phase space (Section X1,D). Through integral transformations the direct space dyadic Green’s functions are then found (Section X1,D). In order to find the general characteristic dispersion equation giving the o vs. k behavior, general vector phasor Helmholtzbased equations are derived. The characteristic equations form the basis of determining the modal eigenvalue properties of the sourceless electromagnetic field problem in the doubly anisotropic medium (Section X1,E). From the general dispersion equation, the limiting cases of general chiral and nonreciprocal ferrite media are obtained, and it is shown that the ferrite medium forms the correct dual as the magnetized solid state plasma. Finally, the polatization state of the E field is determined from the solution of the sourceless vector phasor Helmholtz equation. The electric field vector so found is projected onto the desired planar surface for polarization assessment. Diagonalization of the polarization equation for electric field, the polarization ellipse, results in a direct evaluation of major and minor axes. Handedness of the electric field on the plane (REP, LEP: right or left elliptic rotation) follows from a determination of the electric field tip movement as parameterized in terms of time through the phasor angle.

B. Theoretical Foundations The constitutive relations for chiral media are (Lindell, 1992) D = EE + & , H ,

(191a)

B = PH + L E Y

(191b)

+ iacil,

(192a)

t c = ~ a c r ~+ k i 2 9

(192b)

4cl

= acr1

where, for lossless chiral media, acr1 = a c f i =

ac11 . = - aC . l2

-

0,

(193a) (193b)

ELECTROMAGNETICPROPAGATION AND FIELD BEHAVIOR

119

Here i is the imaginary number d- 1. Permittivity E and permeability p are in (193). scalars, as are ac(r,i)(l,2) Ferrite media constitutive relations are (Collin, 1960; Neidert, 1992)

D = EE,

(194a)

B

(194b)

=

FH,

where

p = p ! - ip",

k,,, = kh - ik;,

(196a) (1 96b)

and the dc magnetic field Bo is in the z-direction. The composite medium is hypothesized to have the following constitutive relations : (197a) D = EE + t c l H ,

B

=

ZH

+ &E.

(1 97b)

Maxwell's equations in the frequency domain with electric J and magnetic M volume currents are V

xE

=

- i o B - M,

VxH=ioD+J,

(198a) (198b)

V - B = 0,

(1 98c)

V - D = 0,

(198d)

assuming electric charge neutral problems. In the search for new media which display nonreciprocal properties, it is essential to establish this behavior fundamentally. This is done here by applying the anisotropic reaction theorem (Krowne, 1984b; Kong, 1986) (6, a ) = (6,b ) ,

(199)

or its integral form

In (200), source currents (J,, MJ and (Jb,Mb) produce, respectively, fields (E,, Ha) and (E,, ,Hb), and the tilde over the fields indicates a new medium

120

CLIFFORD M. KROWNE

altered from the original medium with 6 x 6 constitutive tensors:

and

with p^ and p^' being the optical activity 3 x 3 tensors. Reciprocity only occurs if

( b , a ) = (a, b ) ,

(203)

c =e.

(204)

that is, if (199) requires For chiral media, (191) and (192) hold, leading to

2 = &I, p^ = t C I Z , p^' = &ZZ, 4 = pz, (205) defining the component tensors making up 6 in (202). To have reciprocity, (204) imposes &IT

= EZ,

-&-IT

=

&.Z,

-tclZT

=

&Z,

pZT = p1,

(206)

or tCl

(207)

= -&2.

Recalling (192), or

( ycrl .

(208b)

= - a c12 .*

Condition (208b) is the usual one satisfied for lossless chiral media. The existence of aCr1 # 0, acn# 0 suggests a different condition in the chiral media, and (208a) not being fulfilled. Therefore,

0, ~ c #n 0 may make the chiral media nonreciprocal. For ferrite media, (195) and (196) hold, leading to acr1

#

2 = &I, p^ = p^' = 0, To have reciprocity, (194) imposes

F'

= jj,

c; = jj.

(209)

(2 10) (211)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

121

which by (195) requires -ik, = ik, or k, = 0, the nonferrite case. Thus we see that the ferrite medium, as is well known, is nonreciprocal and satisfies the anisotropic reaction theorem (199). A very interesting decomposition by (Bohren, 1974), which constitutes a rotation in phasor field variable space, will be applied to a general chiral medium. Combine Maxwell’s sourceless equations (198a,b) with constitutive relations (191):

V x E = -iopH V

xH

=

ioeE

-

iorc2E,

+ iwrclH.

(212a) (212b)

Multiply these equations by a, and b, , respectively, and add

V x Q = ioCIQ, Q = a,E

(213)

+ blH,

C,Q = -palH

-

(214a)

rc2a1E+ eblE + r C l b l H .

(214b)

Coefficients a, and b l , and C,, can be found by equating field variable coefficients: (215a) alC1= &bl - t c 2 a 1 , blC1 =

rclbl

- Pal 3

(215b)

and taking the ratio of Eqs. (215),

whose solution is

Applying (193b) gives

which may characterize lossy chiral media. For the lossless case, (193a) implies

which agrees with (Lakhtakia et al., (1989). Circulation equation (213) is of special interest because it leads directly to the simple sourceless Helmholtz

122

CLIFFORD M. KROWNE

equation. Take V x of (213):

V x V x Q = ioCIVxQ. Noting that

vx vx

= V ( V * )-

v2

and reapplying (213) yields

V2Q - 0 2 C f Q = 0 . The lossless chiral case by (193a) defines C1in (222) as

c1= +&i. C. Vector Helmholtz Equations

The H-field vector Helmholtz equation is desired by inserting constitutive relations (196) into Maxwell's equation (197):

xE

=

-io/H

VxH

=

io&E + iotClH J .

V

-

i o t C 2 E- M,

(224a)

+

(224b)

Solving for E in (224b) gives

and putting this into (224a) provides the source H-field vector Helmholtz equation, V x V x H + iw(tC2- t C l ) Vx H - 0 2 ~ p o @ / p-o t , l ~ , Z / p O ~ ) H = V

xJ

+ iw&J

-

io&M.

(226)

The E-field vector Helmholtz equation is found by solving for H in (224a),

and placing (227) into (224b), giving the source E-field vector Helmholtz equation, V

x F-'V x E + i o ( t C 2V x F-'E = -V

x

+ iot,,F-'M

- <,,d-'V - ioJ.

x E) - 0 2 & ( I - tCltc2F-'/&)E (228)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

123

Sourceless vector Helmholtz’s equations are found by setting J = 0, M = 0 in (226) and (228). The inverse permeability tensor I-’ is found using (194):

D. Dyadic Green ’s Function Derivation of the H-field dyadic in unbounded composite chiral-ferrite media follows. Start from (226) using (221) and define Uh(r) = V x J

+ iotczJ - ioeM

(230)

so that

-

+

[VV - VzZ- w2epo(F/po-
(23 1) The direct space dyadic for H is defined by

-

du’ Gh(r,r’) Uh(r’).

Putting (232) and Udr)

=

1

V

into (231) gives

du’ Uh(r’)6(r - r’)

(232)

(233)

124

CLIFFORD M. KROWNE

The reciprocal space, Fourier-transformed variable p and Green's function dyadic &(p) are

jym jymS_ S_

Gh(r, r') = (2Z)-'

!-OD

d3pgh(p) eip'(r-r'),

(236a)

d3p eip.(r-r'),

(236b)

--OD

6(r - r') = ( 2 ~ ) ~ ~ gh(P) =

iym Syrnim

d3(r - r')Gh(r - r') e - i P

'

(r-r')

.

(236c)

--OD

The translation invariance assumed in the use of r - r' comes from the unbounded nature of the problem although it is uniformly anisotropic. Using the operator properties on exp(ip r)

-

vv

V = ip,

-v2=p2,

=

-pp,

v x ~ = i p x z

(237)

and inserting (236a,b) into (235) gives

s_ jYmjm

d3P[-PP + P2Z -

02wo(F40

-tCltC24PON)

--OD

+ io(tC2- tcl)ip x I] eip.@-")- ghb) =

S_ S_ jm -m

(238)

d3pzeip.(r-r').

-

Because of the common exp[ip (r - r')] coefficient in (238), ghb) = [-PP + P'Z - 0

2~~o
The direct space dyadic is found from (239) by doing the inverse operation implied by (239) and applying the transform (236a) on g(p). Derivation of the E-field dyadic follows the H-field steps, starting from (238) and defining U,(r)

= -V

xF-'M

+ iotclF-'M

- ioJ,

so that [-v

x F-'v x z + 0 2 & ( 1 - &l~c2F-'/&)z + io(-tC2V x F-' + t,,F-'V x Z)] - GJr, r') = Z6(r

- r').

(240)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

The reciprocal space dyadic ge(p), ge(P)

=

jymjymIm

d3(r - r')Ge(r - r') e-iP' (r-r')

--OD

is found using operator properties

VXF-' =ipxp-l,

~ x ~ - ' ~ x ~ = - p x ~ - ' p x ~ ,

yielding

x p-'p x z + OZ&(l- (cl(c2p-1/&)I - o(-
ge(p) = [-p

The direct space dyadic is then determined by using the same form of transform as (236a). Equation (244) may be compared to results in (Lakhtakia et af., 1989). E. Dispersion Relations

Dispersion relations for the propagation vector k versus frequency o can be obtained from the general vector phasor electric field equation derivable from the vector Helmholtz equation or from the general vector phasor magnetic field equation derivable from the vector Helmholtz equation. We start with the H-field relation because it is somewhat simpler than the E-field relation for completeness, and go on to the rest of the analysis. Define the H-field phasor as H = ~ ~ i 9 u f .H = ~ ~ ~ - i k ' r (245) P

9

where H appears in (226), and thus write the real time-dependent magnetic field as H(r, t ) = Re[Hp] = R ~ [ H ,ei(of-k r) I. (246) '

Using the Vx property in (237) repeatedly, the H-field vector Helmholtz equation with (245) inserted yields k X k X Ho

+ o(tCz- &i)k X Ho - oZWo(F/Po - T c i Tcz/(Wo))Ho = 0. (247)

The triple cross-product may be also rewritten using the operator properties

V2 = -k2

V. = -ik., as

k X k X H,

= -k(k

*

Ho)

+ k2Ho.

(248) (249)

126

CLIFFORD M. KROWNE

Now put H, into rectangular coordinates for discussion's sake, Ho = Hx2 + H y j

+ H,?,

(250)

and place this into (247), separating out the individual component equations. The three component equations will determine the eigenvectors, and the determinant of the coefficient component matrix Mh will determine the eigenvalues, thereby yielding the o vs. k dispersion diagram in phase space. For the electric field, the following E-field vector Helmholtz equation is obtained: -k x F - ' k x E,

+ w(-tC,F-'k

- w'E(I - (c.~c2F-1/~)Eg where E, E replaces H, H in (245) and (246). Returning to the H-field,

x

+ tc2k xF-')E,

= 0,

127

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

(-kj + k’ - a)

(-k,k, (-k,k, + bk, + ic) (-k; ( - U z -

bk,)

-

bk, - ic)

+ k’ - a) (-k,k, + bk,)

(-kj - k2 - a) (-k,k, - ic) (-k,k, + ic) (-k; + k2 - a) - kx kz

- kY kz

(-k,kz + bk,) =

(-k,kz - bk,)

0.

+

(4; k’ - a f )

- kx kz (-k,kz - bk,) (4;+ k2 - a ’ )

=

0. (259)

- k2 sin 8 cos 0 cos 4 -k’sinBcosBsin+

(k2sin’ B

(PIE,

- n2)

-iE1Em

0

ieriim (P,E,

-

n’ cos2 0)

n’ sin 0 cos 8

-

= 0.

(261)

a’)

0

n2 sin 0 cos 0 (E,

- n’ sin2 8)

= 0.

(263)

128

CLIFFORD M. KROWNE

Here,

Em

p, = p / p o ,

E, =

(264)

= km/po.

The form of (263) is exactly the same as for a solid-state magnetoplasma (Baynham and Boardman, 1971) and so constitutes an exact dual of the magnetized ferrite component. It reduces to, upon expansion,

~

,

= pl sin2 e = -(p:

+n~ , n~’ + A,

= 0,

(265)

+ plIcos2 8,

- &)

(266a)

sin2e - plpll(l

+ cos2e),

(266c)

1, = P I l ( d - P 3 , PI

(266b)

= ~ r ~ r , PII = ~ r ,

-

(266d)

PH = E r k m .

There are two eigenmode solutions,

The two signs implied in n2 correspond to forward and backward waves. Consider next the limiting case where ferrite behavior is dropped and the chiral part of the medium maintained. Then

(-k:

+ k2 - a)

(-k,k, + bk,) (-k,k, - bk,)

(-k,k, - bkJ (-k,k, + bk,) (-ky” + k2 - a) (-k,k, - bk,) (-k,k, + bk,) (4:+ k2 - a)

= 0. (269)

-k2 sin e cos 8 cos4 + bk sin e sin4 - k2 sin 8 cos 0 sin 4

- bk sin 0 cos 4

(k2sin2e - a ’ )

=

0.

(270)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

k2cos28- a

-bkcos8

-k2sin8cos8

bk cos 8

k2 - u

-bksin 8

-k2sin8cos8

bksin8

k2sin28- a

=

0.

129

(271)

(272) (273a) (273b) (273c) (273d) There are two eigenmode solutions for k2, with the two signs in k2 being forward and backward waves. The general composite medium-case characteristic equations is found from (257) by employing transformation (260) and expanding the determinant (253), giving Alk6 + A2ks + A3k4 + A4k3 + A,k2

+ A6k + A7 = 0,

(274)

0,

(275a)

A’ = 0,

(275b)

A1 =

A3 = (sin’ 8 cos’ 4 - l)[a sin’ 8 - a’(sin20 sin’ 4 - 1) - b2 sin28 cos’ 41 - a sin’ 8 cos’ 4 - sin4 8 cos 4 sin ~ ( C O 4 Ssin 4[b2 - a ‘ ] - ic) - ic sin’ 8 cos 4 sin 4 - sin’ 8 cos’ 8 cos 4[(b2 - a) cos 4 - ic sin 41, (27%) A4

A, = aa’(2 - sin’ 8) + (a2 - c’) sin’ 8 - b2(asin’ 8 + a’ cos’

A7

(275d)

= 0,

e),

(275e)

= -2ica’b COS 8,

(275f)

= a’(c2 - a’).

(27%)

There are four different eigenmodes for k as implied by (274), and these modes each correspond to a different eigenvector electric field to be derived in Section XI,F in order to find the polarization.

130

CLIFFORD M. KROWNE

F. Electric Field Polarization

Electric field components are found from the matrix equation (276)

MeEo= 0,

which is like (252) used for the H, field. Equation (276) is derived from the sourceless E-field vector Helmholtz equation (228). The elements of Me are meil =

melz =

k;ph - ( P ’ + i4tC1 - tczIkh)kz - w2&+ ~

~ t ~ ~ t ~(2774 z ~ ’ ,

-kxkyph + ikhk; + kxp‘ + o(tcl - tc2)p’kz+ io2tc1tC2kh, (277b)

+ o(phtc2 - p ’ t ~ l ) + ~ yi w < c l k h , me21 = -ikhk; - kXkyph+ 4 t C 2 - tCl)p’kz+ iw2tCltc2kh, me13

(277c)

= -ikhkykz

meZ2= p’k;

(277d)

+ phk: + iw(tc2- tcl)khkz- 0 2 e - w Z t c 1 t c 2 p ‘ ,

(277e)

+ ikhkx + iwtclkhky+ o(p’tc1- p6Cc2Wxf me31 = -kxkzp‘ + ikhkykz - iwtc2khkx- w(p‘tc2- p6tC,>ky, me23

me32

= -kykzcc’

= -kykzp’

me33= (k:

(277f) (277g)

- ikhkxkz - iotc2khky+ o ( p ’ t c 2 - phtcl)kx, (277h)

+ ki)p’ - 0 2 e - w2
(2773)

Here the primes on the variables indicate an inversion generating the elements of p-’ : ph = p i 1 ,

p’ = p / ( p 2

- ki),

kh

=

- k m / ( p 2 - kZ,). (278)

To find the electric field polarization, a plane upon which we project the field must be chosen. Let the plane be defined by the normal to its surface i t .Then the real electric field projection is written as & =

€,tat

+ E W Y ’ p+ &nn,iY,

(279)

k - r.

(280)

where

0 = cot

& = Re[E, ei6],

-

Transforming & in the c system to & in the r‘ system is done by &“rt

&nr,

=

[En,,

&nyl

-u. . = i i * % j ; A,=cosO

=

(281)

f

E, = [&,*

EYl

&)IT,

i=x,y,z, j=x’,y’,z’,

(282a,b) (282c)

with 8, being the direction cosine angles between the xith and xj’th axes, i = x, y, z and j = x’, y’, z ’ .

ELECTROMAGNETICPROPAGATION AND FIELD BEHAVIOR

131

The polarization ellipse, describing the most general type of polarization found after projecting onto the h‘ plane, is found by defining

Eo = E l

+ iE2

(283)

and by using this in (280), (Elicos 0 - E2isin 0 ) G i .

& =

(284)

i

Following a procedure similar to that used by Chen (1983), we extract and then eliminate 0 functions. The 2’ projection is & 2‘ = & m l ,

-

pCxcos 0 - p, sin 0, &wt = SCycos 0 - 8, sin 0, &m,

(285a)

=

8cx = El, AXXI + Ely + ElZAZd 8, = EaAXxI+ GYAyxI + E2ZA7y 4 x 1

scy

= EIXAV, + E1yAyy’

Psy

= ElrAxyl

(285b) Y

(286a)

3

(286b)

+ ElZAZYY’,

+ EzyAyyt + E2zAry1,

(286c) (286d)

Multiplying (285a) by 8, and (285b) by ,3,/ R11

-

€mt

Rll

=

R12Eny#= cos 0, DSY

8, s c x

(288a)

- s,scy

s,

R12=

(287)

8,8cx - ss ,;

(288b)

Similarly, Rzl€ M r - R22&w3 = sin 0, R2l = R22 =

BCY

s,scx

- ssxscy ’

8,

s s y flu - asxDcy‘

(289) (290a) (290b)

Parameterization in terms of 0 can be eliminated by adding squares of (287) and (288) to obtain the polarization ellipse equation

+

R,&LI + R2GmgEw( R3&iyt= 1, R1 = R:l R3 = R:,

+ Ri1, + R&,

R2 = -2(RiiR12 + R21R22)-

(291) (292a) (292b) (292c)

132

CLIFFORD M. KROWNE

To obtain the major and minor axes of the polarization ellipse (291), it should be diagonalized by a coordinate transformation

[ :I [ =

cos6 sin6 -sin6 cosd)

[ :I

(293) 9

where the rotation in (293) is about z, z’ by angle 6, right-hand rule counterclockwise. Following the prescription

for real field E ,

[:I

[

=

cos6 -sin6 sin6 cosd]

EL”

&&”

PI

p2

[ ]I:

-+--1=o, PI = R , cos2 6

(295)

+ R2 cos 6 sin 6 + R , sin26,

(296a)

+ R , cos26,

(296b)

P2 = R , sin2 6 - R , cos 6 sin 6

cos26

(294)

=f

RI - R3 [(R, - R,)’ + Rz]”2’

(297)

G . Conclusions

Electromagnetic properties of chiral and nonreciprocal composite chiralferrite media have been delineated in this section. Exploring new media for use in microwave and millimeter-wave devices is of great interest today. This is especially true in view of the trend toward miniaturization, including the current emphasis on monolithic microwave circuits using Si, GaAs, and other substrates operating in the 1-100 GHz frequency spectrum. The trend toward examining alternatives to the current workhorse for nonreciprocal applications, ferrites, will continue in terms of exploring new materials as in this section and new active circuit replacements. Further examination of ferrite-related materials will occur in Sections XI1 and XIII. XII. VECTORVARIATIONAL AND WEIGHTED RESIDUAL FINITE ELEMENT PROCEDURES FOR HIGHLY ANISOTROPIC MEDIA A . Introduction

The solution of modern electromagnetic problems involving waveguide discontinuities, irregular waveguiding, antenna, scattering, and resonator

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

133

and coupling structures necessitates the use of extremely general computer simulators. These simulators must have very flexible geometry drawing features. To treat anisotropic media, especially those which are lossy, the simulator must go well beyond the capability of isotropic material description. The purpose of this paper is to develop theoretical approaches applicable to the numerical simulation of structures which contain the most general lossy, unsymmetric, anisotropic media. We therefore focus here on approaches which are compatible with finite-difference or finite-element solutions of electromagnetic problems. Much work concerned with the above problem has been done, and a completely exhaustive survey would be impractical, so only a few papers will be mentioned by way of background and review and as necessary to set the context of the present work. Works concerned with the overall development or utilization of the finite-element method are Daly (1971), Koshiba et a/. (1985), English (1971a, 1971b, 1971c), Rahman and Davies (1984), Burnett (1987), Sadiku (1992), Silvester and Ferrari (1990), and Chari and Silvester (1980), to name a few. Treatments implementing procedures to assure uniqueness applying tangential (or edge) elements besides more traditional approaches (normal vector components, solenoidality condition-divergenceless constraints) and to study the nature of uniqueness include Cendes (1991), Cendes and Lee (1988), Bossavit and Mayergoyz (1989), Lee et al. (1991a, 1991b), Lynch and Paulsen (1991) and Paulsen and Lynch (1991), Bardi and Biro (1991a, 1991b), and Svedin (1989). In the area of lossy media, waveguide applications such as McAulay (1977a, 1977b) may be cited. For the area of anisotropy, applications and theoretical extensions, works such as Konrad (1976), Kang and Chen (1984), and Svedin (1990) for chiral media may be noted. Many past works have employed variational methods. Among them we mention McAulay (1977b), Chen and Kiang (1980), English (197 la), Cendes (1991), and Cairo and Kahan (1965). Studies which have focused on properties of operators which affect the formulation of electromagnetic problems and having bearing on the ability to address lossy media, reduction of media symmetry, anisotropy, increase in the number of tensors required or change in their properties include the doubly anisotropic Hermitian tensors of Konrad (1976), the self-adjoint formulas of Cvetkovic and Davies (1984), the non-self-adjoint operators of Chen and Lien (1980), the self-adjoint formulation based upon nonself-adjoint operators of Jeng and Wexler (1978), and the principle-of-leastaction approach of Morishita and Kumagi (1977).

134

CLIFFORD M. KROWNE

These papers specifically address such issues as dual tensor character (i.e., tensor permittivity and permeability), Hermitian tensors, nonHermitian tensors, self-adjoint operators, non-self-adjoint operators, and definitions of integral inner product. We note here in particular that self-adjointness requires boundary condition field constraints through surface integrals in 3-D problems which we wish to avoid here. We will instead let them develop naturally from the derivation directly of variational functional expressions, in the variational approaches presented. The added requirement of self-adjointness of adjoint operators makes further restrictions upon media, which are known not to hold when lossy or unsymmetric media are examined. When the operators for increasingly complex media become non-self-adjoint, but only adjoint, the boundary condition surface integrals still occur. It is possible (Chen and Lien, 1980) even with non-self-adjoint operators to construct formulation procedures for such media as dual tensor media. The penalty paid for such an approach is the need to define adjoint constitutive tensors for the permittivity or permeability (or both), and adjoint E and H fields. We wish to avoid that need here. With the addition of tensor constitutive optical activities, the field operators are considerably changed and they fail to possess adjointness. In terms of E-field operators, this failure occurs because of the single-curl multitensor parts of the operators. Such results are found both in the variational and weighted residual approaches when having a full super-tensor constitutive relationship (Krowne, 1984b) for the media. Construction of energy-like expressions for various classes of anisotropic media and the use of the weighted residual technique for those cases which don’t obey Hermitian properties and have full super-tensors (the 6 x 6 tensor is filled, and is arbitrary and completely anisotropic) is undertaken. From these energy-like expressions are derived energy-like functionals which are shown to have volumetric and surface parts. Volumetric parts contain electromagnetic governing equations of the media, and surface parts provide constraints on field components on interfacial boundaries. Media examined here include gyroelectric, gyromagnetic, combined gyroelectric and gyromagnetic, and complex media displaying gyroelectric, gyromagnetic, and optical activities. Energy-like expressions are constructed here (Krowne, 1994) from three component vectors (that is, complete vectors) in contrast to some earlier work which used single components (especially convenient to use in simple uniform waveguide structures) but consistent with more recent 3-D work employing complete vectors (e.g., Cendes and Lee, 1988) and older work modeling inhomogeneously loaded guiding structures (e.g., English, 197la, 1971b).

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

135

B. Vector Helmholtz Sourceless Equation

Maxwell's curl equations for harmonic conditions with exp(iot) assumed time dependence are V x E = -ioB, (298) V x H = iwD.

(299)

The most general constitutive relationships for linear media are (Krowne, 1984a, 1984b)

+ fiH, D = E^E+ p^H.

B

=

C'E

(300)

(301)

Here 8, fi, p^, p^' are, respectively, the 3 x 3 permittivity, permeability, and optical activity tensors. Placing (300) and (301) into (298) and (299) eliminates the electric D and magnetic B displacement fields, giving V x E = -io(p^'E V x H = io(2E

+ fiH),

+ CH).

From (302) we can determine H in terms of the E field only, yielding i H = -fi-'V w

x E - fi-'b'E.

(304)

Placing both (304) and its curl into (299) using (301) gives the sourceless vector E field Helmholtz equation V x (fi-'V x E)

+ iw[V x (fi-'C'E)

-

@fi-'V x El = w2(E^- Cfi-'Cf)E.

(305)

Any single-energy functional must have its variation satisfy this governing equation (valid for describing the time harmonic electromagnetic fields within the general medium). This governing equation includes gyroelectric, gyromagnetic, and optical activity effects. For no optical activity, (305) becomes V x (fi-'V x E) = 02E^E, (306) which is considerably simpler than the original equation. Adding back p^' gives V x (fi-'V x E) + ioV x (,E'bfE) = 02E^E, (307) or adding back p^ gives V x (fi-'V x E) - iwbfi-'V x E = oZEIE.

(308)

136

CLIFFORD M. KROWNE

C. Gyroelectric-Gyromagnetic Variational Analysis

The procedure to follow is nonunique in the sense that more than one expression for energy can be postulated and used to determine an energylike functional suitable for variational analysis. Let us consider the energy expression G=B*.H-D*E* and define the energy functional as

For p^

=

0, 6' = 0, the constitutive relationships become B = fiH, D = EIE.

Equations (302) and (303) reduce to VxE

=

-ioB

xH

=

iwEIE,

V

=

-iw$H,

and the H formula (304) becomes

x E. The energy functional F can be written down as F(E) =

ssi

[v x E* -fi-'V x E

-

-

0 2 E * EIE] dC2

This energy-like functional F can be minimized by setting E

=

E,

+ /3t

and determining the first variation of F(E). Here E , is the exact solution of the vector E field Helmholtz equation (306), p is a number, and t is an arbitrary vector function. Putting (317) into (316) and expanding to second order in the perturbation gives

sss 111

F ( E ) = F(E,)

+ /3

+ /3*

[v x

[ V ~ t * * f i - ' V x E-, w ~ { * * E I E , ] ~ C ~

-

-

EZ fi-'V x ( - 0 2 E : 2 4 dC2

+ O(p2),

(318)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

137

where the zeroth-order term is given by

The first variation of F(E) with respect to

a* is found to be

making the normalized first variation

Similarly, the first variation with respect to

is

Inspection of (3 18) produces the result for the normalized first variation, bplF(E) =

1s

[V x

<* - b - ' V

-

x E, - o'<* EE,] dQ,

(323a)

The arbitrary perturbational vector field <* can be factored out of the (323a) integrand expression employing the vector identity for the divergence of a cross-product, V - ( a x b ) = ( V x a ) * b- a - ( V x b ) .

(324)

Identify

a = (*;

b = @-'V x E,,

(325)

so that

+

fi

<* x ( i - ' V x E,) - d S , (326)

138

CLIFFORD M. KROWNE

where the surface integral is over the closed surface S and the divergence theorem has been employed. Placing (326) into (323a) gives the final form for the variation: s8*F(E) =

111

<* - [V x ( i - ' V

+ (# <* x ( p - ' V

x E,) - w2E^E,]d n

-

x E,) dS.

(327)

The Hermitian conjugate of the /3 variation result, using the transpose properties of vectors and tensors, is

+

fi

<* x (jt-'v x E,) - dS.

Since the vector Helmholtz equation (306) holds for the exact field solution E, , the first integral in (327) drops out, leaving

which may be cast as 8,,F(E)

=

-io

# r*

x He * dS

(330)

using the curl E, expression. Setting the first /3* variation of F(E) equal to zero to satisfy the energy-like functional minimization requirements produces

Q

<* x He - dS

=

0.

(331)

For Hermitian properties of the anisotropic medium,

fi

= $,

(332a)

E^

= it,

(332b)

and the right-hand side of (328) is equal to the right-hand side of (327). Thus, setting the first /3 variation of F(E) equal to zero again produces the same surface integral result (331) while now satisfying (306).

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

139

D. Isotropic Gyroelectric and Gyromagnetic and Optically Active Variational Analysis

Consider the case where electric and magnetic anisotropies reduce to the isotropic situation, but the material is still optically active. The simplified constitutive relationships (300) and (301) look like B

=

p‘E + pH,

(333)

D

=

EE + pH

(334)

after removing individual tensor properties. Examine the new energy-like expression G=-B.H-D.E

(335)

and note the scalar form of (304) for the vector H field: i 1 H = -V x E - - p’E. P

UP

(336)

From (333)-(336), the energy-like expression G can be expressed as 1 i G =TVXE*VXE --(p U P

- p’)VxE.E-

UP

(337)

and the energy-like functional written as

Using the perturbational relation E

in (338) gives

=

E,

+ 8s

(3 17)

140

CLIFFORD M. KROWNE

where the zeroth-order term is given by

i

- -( p - p‘)V x E,

UP

*

E, -

Now refer back to the vector identity (324) for the divergence of a cross-product and assign

a

= (,

b = p-’V

so that we find

1sj

V

1

x ( - v x E, dQ

=

x E,,

1s1 -

( V

x

(341) 1

V

x E, d M

+ $l(xbVxEe-dS.

(342)

Again using the vector identity (324) but choosing a = (,

b

=

(i/(20p))(p - p’)V x E,,

(343)

we find

- p’)E,

*

dS.

(344)

Putting (342) and (344) into the energy-like functional expression F(E) in (339) yields F(E) = F(E,) + 7 0 2pIss(-[Vx~VxEe-02 -

9 0

iss(-

+2

[L(p P

$l ( x - v

o2

1

P

1

1

- p ’ ) ~ x E +, v x - ( p - p’)E, d M P

x E, . d S -

- p’)E;dS. 0

The first variation of F(E) with respect to p is

(345)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

141

making the normalized first variation

Inspection of (345) gives the result for the normalized first variation,

-i w

111 r - [1

+2 w2

#I

P

1 ( x -v P

( p - p’)v x E,

-

x E, dS

1

+ v x -1 ( p - p’)E, P

dC2

-

( p - p’)E, dS.

-

(348)

By requiring the difference between the optical activity parameters divided by the permeability to be spatially invariant, the second integral term in (348) becomes

and this combined with the first volume integral generates

2 w2

111r . [

iw

vx

Clearly this form is obtained if both optical activity parameters and the permeability are spatially uniform. The sum within the brackets is zero since the vector Helmholtz equation in (305) can be applied. Consequently, the normalized variation becomes w

- p’)E;dS. (351)

Utilizing the electric field curl relation (298), and setting the first variation of F(E) equal to zero in order to obey the energy-like functional minimization requirements, admits 2#I<x!B*dS

-iw

(352)

142

CLIFFORD M. KROWNE

E. Complex An is0 tropic Variational A nalysis

In general, the most complex media have lossy properties and are entirely anisotropic in some or all of the four descriptive constitutive tensors found in the relations (300) and (301). We choose an energy-like expression as follows, for reasons which will become apparent later in the development: 1

1

G = T V x E * . i - ' V X E - -(bi-'V 0

-

x E . E * - V x E * -b-'p^'E)

0

(El

-

bF-'b')E.

E*.

(353)

Again use the perturbational relation, E

=

Ee + Bt,

(3 17)

in (353) to obtain an expansion up to first order in /3 and /3*:

-

I

1

-0( b ~ - %x E , - < *- v x t* . F - ' ~ ' E , )- ( ~ l - b j i - ~ p ) ~ ~,. t * (354)

where the zeroth-order energy-like expression G(E,) is given by G(E,)

1

= TV

w

i

x E: . i - ' V x E,

- - (&-'V 0

x E, - E:

-V

x E: - L-'b'E,)

- (2 - bk-'b')E,

- E:. (355)

Examining (338), we write the energy-like functional as

F and incorporating (3 17) gives

=

jjs

GdQ,

(356)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

143

Expansion of the right-hand side of (357), employing (354), gives F(E) = F(E,)

+ p*

iii

[$Vx<*~fi~'VxEe

1

- - ( b f i - ' V ~ E , - < * - VX(*.fi-'$'E,) W

-

(E

- &"b')E,

1

- <*

dS1,

(358)

keeping only the p* terms. F(E,) is found from

The normalized first variation of the energy-like functional is determined from

Inspection of (358) gives, for 8,*F(E),

1

-

- (bfi -'V W

x E, - <* - V x <* fi-'$'E,) *

The Hermitian conjugate of the /? variation is by inspection of (354), using vector and tensor transpose properties,

-

- r t - t - l -t

(it - p p

p )(* * E ,

144

CLIFFORD M. KROWNE

Referring to vector identity (324), with j-',(or p-'jj') in the variational expression, we can write

V x (* * $-'V x E, = Vx

(*

r* - V x (p-'V

j-'i'E, = (*

x E,)

+V

*

[(* x

(p-'V

x E,)], (362)

V x (@-'/.?'E,) + V * [(* x (b-'@'E,)].

(363) Putting (362) and (363) into the variational expression (360) allows us to factor out inside of the integrals either in vector or cross-product form. The result is *

r*

&F(E)

=

11

(*

*

[-$V x

-'V x E,

i a

- - (pp

V x E,

-

V x p -'fi'Ee)

- (El - bj-'b')E,

Notice that the terms within the brackets sum to zero due to the sourceless vgctor E field Helmholtz equation (305). Therefore, the normalized variation reduces to 1

F-'V

X

E, * dS

+

Utilizing the electric field curl relation (298), and setting the first p* variation of F(E) equal to zero in order to obey the energy-like functional minimization requirements, produces

For Hermitian properties of the permittivity and permeability tensors according to (332), p =jt, (332a)

El = El+ and the optical activity tensors related by jj' = jjt

(332b)

(367) cause the right-hand side of (361) to equal the right-hand side of (360). Therefore, setting the first /?variation of F(E) equal to zero again upholds (305) and produces the same surface integral result (366). Equations (332a), (332b), and (367) imply a lossless anisotropic medium having a completely filled 6 x 6 superconstitutive tensor describing the material. Some chiral

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

145

media can be characterized by this tensor description. There are two extremely different methods of generalizing the medium further which will be mentioned. One method is to use an auxiliary vector function to artificially reintroduce some of the lost symmetry into the variational energy-like functional (Krowne and Salvino, 1993). The other method is to apply the weighted residual process directly to the governing equation field operator (s). The weighted residual process will be covered in the next section.

F. Weighted Residual Analysis of Non-Hermitian 6 x 6 Tensor Medium In the last section, the first-order variational requirements were met by invoking (332) for Hermitian behavior of the permittivity and permeability, while the optical activities were related by a Hermitian symmetry constraint (367). When the medium has no symmetry and lossy characteristics, a much more general method, the weighted residual method, can be applied to the original equations. Consider the operator equation LE = g, where L is a linear operator which may contain derivative or integral terms. For fully three-dimensional problems, E is a three-component vector, namely, E = [Ex Ey &IT, (369) where the unknown field components and the known function g are expanded as N

Exe =

C Exeiai(xt Y , z), i=

(370a)

1

N

Eye =

C Eyeiai(X,Y, z),

(370b)

i= 1

N

Eze =

C Ezeiai(X,Y , z), i= 1

ge =

C geiai(X,y,2). i= 1

(3704

N

(371)

Here the element shape functions ai(x,y , z) are themselves made up of polynomial expansions obtained from a Pascal tetrahedron.

146

CLIFFORD M . KROWNE

The a i ( x , y , z) are known and are found from the following procedure (Burnett, 1987; Sadiku, 1992; Silvester and Ferrari, 1990; Chari and Silvester, 1980). Consider any 3-D function f that we desire to calculate. The entire solution space is divided into domains called elements. Here we are treating tetrahedra. There are four vertices for each tetrahedron. The number of nodes j which become the locations (xi ,yi ,zi) where fi is considered unknown and a solution desired must equal or be greater than the number of vertices. The greater the degree n of polynomial expansion used in constructing q ( x , y, z), the greater the potential numerically determined accuracy of the final solution. For a third-order polynomial, ai

+ axx + ayy + azz + a 3 x z + a,xy + axzxz+ + ay,yz + a2z2 + ax3x3+ a h x ’ y + as,xzz + u,2xy2 + axz2xz2+ ay3y + ayzzy2z + a y t y z 2 + az3z3,

= a,

ay2y2

3

(372)

where the coefficients are different for each i = 1 - N . An nth-order polynomial will have 1 N = -(n + l)(n + 2)(n + 3) (373) 6 terms, and a corresponding number of nodes N in the tetrahedron. If n = 1, then ai = a, + axx + a,y + azz. (374) We require that the function f in an element f, be represented by the interpolation polynomial construction such that at each node, there being N, f = f, = fi, i = 1 - N. This results in N linear equations for N a coefficients to be found. When n = 1 , N = 4, and only four nodes are needed, actually just the vertices, and the four shape functions can be expressed as

a1 =

-

1 6v

(375a)

a2 =

-

1 6v

(375b)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

147

(37%)

(375d)

where the determinant of the system of equations is six times the volume of the tetrahedron v:

d, =

1 x1

Y1

z1

1

x2

Y2

22

1

x3

Y3

23

1

x4

Y4

z4

=

6 ~ .

(376)

Shape functions obey two simple properties:

x, y, z are the global coordinates of the solution problem region. In finite element solutions, it is often convenient, although not necessary, to define element local coordinates specific to each element. For very high-order polynomial approximations, the shape functions become defined by very large determinants in the (x, y , z) global system, and that, coupled with the existence of integration identities in the local coordinate system, suggests the use of a local coordinate system in the process of obtaining a finiteelement solution. For the tetrahedral element, the local coordinates are &, i = 1, ...,4, the ith local coordinate measured as the perpendicular distance from the point (x, y , z) to the planar side opposite the vertex i divided by the total perpendicular distance between the ith vertex and the opposite side. Local coordinates obey 4

c ti

i=l

=

1.

(379)

148

CLIFFORD M. KROWNE

Integrations which result from volume or surface integrals in the global coordinate system can be done in the (tl,t2,t3,t4)system by noting that dC2

sss

=

s: I:-"so

dx dy dz

= 6~ d < 1 d < 2

(380)

d<3,

1-b-h

f(x, y, z) dx dy dz

= 6v

f ( r l , (2

3

(3) d
f

(381)

The tetrahedral coordinates are related to the global Cartesian coordinates by

x

=

Y

=
z =



+ ( 2 x 2 + t 3 x 3 + <4X4f + t2Y2 +

+ (222

t3Y3 t3z3

+ 44Y4f

+ <424*

These relations lead to the derivative formulas

The inverse transformation from global to local coordinates is

(383a) (383b) (383c)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

149

and a-,. =

a.I - a4 ,

a=x,y,z,

i=l,2,3,

and leads to derivative relations

Each nodal point may be labeled by a 4-set of integers ( i , j , k , m), called associated with the local coordinate system. This set is chosen so that they are integers which make (tl,t 2 t, 3 t4) , be filled by ones or zeros and Pijkm,

where n is the polynomial order. The summations contained in (370) and (371) are changed to be consistent with the local coordinate system of indexing points: N

n

c-c

i=l

n

n

c

c

i=O j = O ; i + j s n k=O;i+jsn;rn=n-i-j

,

where the index i on ai goes to ijkn and we can write

p,(<) =

[Lrz(nt-s), r! s = o L 1,

r>O

,

r

E

(i, j , k , m ) .

(390)

r=O

Go back to (368). If we consider Maxwell’s equations (at least the two curl equations) collapsed into a single field vector form to be acted upon by operator L , then the sourceless vector Helmholtz equation (305) is appropriate to define L, and it can be stated as L =

v x ( ~ - ‘ v x )+ io[vx($-’p”’)

-

~ $ - ‘ V X-] 02(i- p”b-’p”’). (391)

Function g in (368) becomes g = 0.

(392)

150

CLIFFORD M.KROWNE

Let us evaluate the first term of the operator on E: L I E = V x (,i-’Vx)E.

(393)

Define the permeability inverse tensor as pt

+ pi2[ -

a’Ex

a”, a22 -

a”, s] 4axay +

1-

a2Ez

a’Ex - pi2 a2Ex -

ayaz

p-I

(394)

a2EY

-

[

=

a’Ex

-

a2EY

-

pi3[

a2Ex

a’Ez

[=-$I’

a2Ez] - P I 3, a’Ey

F] a2Ey

(396)

-

a2Ez

axay

a2Ez

3 1 - pi][ i j j -~

a’Ey

(398)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

15 1

Let us evaluate the second term of the operator on E : L2E = V x (i-’p^’)E,

(399)

Define the product of the permeability inverse tensor and the optical activity tensor as A

=

i-’cl.

(400)

Then (399) can be expanded as

Qy

=

The other first order curl term in L operates on E such that

Define the product of the optical activity tensor and the permeability inverse tensor as

E

=ci-1

(406)

152

CLIFFORD M. KROWNE

Now expand (405) according to standard curl properties:

The last term in L operates on E such that L4E

=

(El

-

@-'F')E.

(409)

Define the shifted tensor permittivity as the permittivity minus the product of the optical activity tensor times the inverse permeability tensor times the other optical activity tensor: 2' = El - cb-lcf* (4 10) Next write (409) as L4E = [SX

sy &I,

(41 1)

where Sx = &;,Ex+ €i2Ey + &;3Ez,

+ &i2Ey + &i3EZ,

S, = &;]Ex S,

=

&;,Ex+ &;2Ey+ &;3Ez.

(4 12a) (412b) (412c)

In the weighted residual method, a residual quantity equal to the difference between the operator equation with the approximate numerical field solution and the operator equation with the exact field solution is found. The inner product of weighting functions and the residual is calculated and required to be zero: (413)

Rs = LEapprox - LEexactr ( w m , R S= )

iii

wmR:dC2

=

0.

(4 14)

Refer to (368), which has the second term LE,,,,, in (413) equal to zero by (392), so that (414) can be recast

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

153

Since there are four terms in L , (Wm,LE,)

=

(Wm,LIE,) -

+ io(wm,L2E,)

io(w,,L,E,) - 02(w,,L4E,).

(4 16)

The weighting (or testing) functions will be chosen to exist only over the same subdomain, that is, the element, as the shape functions used to expand the field components in (370). We will require as many weighting functions as unknowns in the problem. Examining (370), there are N unknown coefficients for each E field component, corresponding to each of the N nodes in the tetrahedron. Because there are three vector components, there are a total of 3N unknowns requiring 3 N weighting functions. Each weighting function is placed into (415) and demanded to satisfy it. This results in a total of 3N equations in 3N unknowns and the ExeirEye;,Ezei unknown coefficients of the shape functions can then be determined. Substitute into (416) the operator expressions for Li,i = 1 , 2 , 3 , 4 : (wm, LE,)

= (Wm,

“G Ty TI)

+ io(Wm, [Qx

Qy

Qzl)

-

i o ( w m , [ R R xRy R,I)

-

W2(wm,[ s x

sy

(4 17)

~ Z I ) .

Since the inner product of the weighting function and the residual must be zero by (415), each component of (417) must be zero. This produces three separate equations, each having N unknowns, generating the required total of 3N equations: ( w , , T, + iwQ, - i o R , - 02S,)

0,

(4 18a)

+ iwQy - ioRy - w2Sy)= 0, (w,,,, T, + ioQ, - i o R , - w2Sz)= 0.

(418b)

=

( w , , Ty

(418c)

A popular approach is to apply the Galerkin method of setting the

expansion and test (weight) functions to be the same: wi=ai,

i=l,

..., N.

(4 19)

Each (418) inner product may be expanded. For example, (418a) becomes ( w , , T,) + i o ( w , , Q,) - io(w, ,R,) - w2(w, ,S,)

= 0.

(420)

154

CLIFFORD M. KROWNE

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

155

The third inner product is

The last inner product is

The remaining inner products in (418b) and (418c) may be similarly evaluated to that done for (418a) and are given in the following equations. Next evaluate the inner products in (418b):

156

CLIFFORD M. KROWNE

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

157

158

CLIFFORD M. KROWNE

G . Conclusion This section has covered generalizations of energy-like functionals used in variational approaches as varying degrees of symmetry removal have occurred in highly anisotropic media. The media considered here have, in essence, required the most complete linear media description in terms of tensors. Addition of optical activities and loss of Hermitian behavior have been treated. The media can include, but are not limited to, gyroelectric, gyromagnetic, gyroelectromagnetic, gyroelectrochiral, gyromagnetochiral, and gyroelectromagnetochiral. The media involved are generally lossy and nonuniform. Normally accepted practices in recent finite-element work apply here in regard to apportioning some of the interfacial boundary conditions to the volumetric integrals, some to the surface integrals (natural BCs), and some as constraints imposed directly on the fields making use of various Green’s theorems. For the most unsymmetric media in developing the weighted residual method, conversion of some of the volume integrals to surface integrals is done by applying integration by parts directly.

XIII. FERRITEMEDIA A. Introduction

Recently, the investigation of thin gyromagnetic films for integrated circuit applications has become of great interest in order to obtain some of the nonreciprocal functions required. The development of such films necessitates new technology intiatives for producing these films. Garnet, spinel, and hexagonal ferrites are all of interest. Which ones will become the most suitable for future practical device fabrication is not entirely apparent now. Hexagonal ferrites appear to provide a way to remove the external magnetic field required in circulators. This is because of a self-biasing field characteristic of hexagonal ferrites. Ferrites are basically magnetic ceramics. The essential composition for them is Fe,O, . Distinguishing atomic elements for garnet, spinel, and hexagonal ferrites are respectively Y, Mg or Mn or Ni or Zn or combinations, and Ba. A number of theoretical techniques have been used to analyze circulators for planar-type structures or integrated circuit applications, which is to be our focus in this section. These include Green’s function methods in 2-D real space (Bosma, 1964; Neidert and Phillips, 1993), finite-element methods (Helszajn and Lynch, 1992; Lyon and Helszajn, 1982), contour integration methods (Miyoshi et al., 1977), and analytical techniques (Helszajn and James, 1978).

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

159

In order to show how a relatively fast finite-element simulator may be developed to treat arbitrary-geometry circulators, we derive here a 2-D formulation which may be used to model triangular, rounded triangular, hexagonal, distorted hexagonal, and many other geometries. Some of the material presented here is somewhat different than that found in the literature, and may be considered new.

B. Governing Helmholtz Wave Equation We repeat here the sourceless Maxwell equations (298) and (299) for convenience: V x E = -ioB, (298)

V xH

=

ioD.

(299)

These two equations are valid within the ferrite region (it may be a semiconductor region if we try to use the magnetoplasma effect) and the immediately surrounding dielectric region. The constitutive relationships (311) and (312) are given by

B = @H,

(31 1)

D

(312)

=

EIE.

In the ferrite region, we will assume that the dielectric tensor reduces to a scalar: &

(433)

= E.

The general expression in matrix notation for the curl of an arbitrary vector field is

(434) For the 2-D problem we are constructing, it is sufficient to drop a dimension by setting

a

-=

az

Curl A then expands as

VxA=-zl’ ax A,

0.

(435)

‘l+-l

A,

a a

ay A ,

izl,

(436)

160

CLIFFORD M. KROWNE

or

a

V x A = --(Azj

- Ay?)

ax

+ -a( A z 2 aY

- A,?)

To be more consistent with notation in the circulator literature, we will set K = k, in (295): y iK (438)

0

0 Po

BY (3111, p

iK

pH,

0

0 Po

+ iuHy

Using the expression for curl E found in (437), A = E, and the B result just given, the curl equation becomes -aEz i - - y +aEz ay ax

A

(2 T)? =

)

pHx + iKHy - i o ( - iKH, + pHy ,

(440)

PO H,

which can be written out in terms of the three component equations:

aE. = - i o ( p H , + iKHy), aY --aEZ ax

- - io(- iKH,

(!z 5) -

= - i o p oH,

+ pH,),

.

Following similar steps, the curl H equation becomes

(441 a) (441 b) (441 c)

ELECTROMAGNETICPROPAGATION AND FIELD BEHAVIOR

161

and again listing the component equations, (443a) (443b)

(22)

= io&Ez.

-

(443c)

If we consider the cases where there is at least one ground plane in the real device, then (435) implies E,(z)

=

S'

0 dz

+ E,(z~),

(444)

and the ground plane forms the Dirichlet tangential boundary condition in the third coordinate direction, E,(zo)

=

(445)

0,

if the conductor is assumed perfect. This leads to EJz) = 0

(446)

and the dropping of this field component in the analysis. A similar null condition holds for E,(z)

=

(447)

0.

Examination of (441c) in light of (446) and (447) gives (448)

Hz(z) = 0.

Magnetic horizontal fields can be found from (441) by multiplying the first equation by - iK and the second by ,LA and subtracting: -iK-aE, aY

2 + p-aE, = io(p - K ~ ) H ~ . ax

(449)

Solving for Hy in terms of partial derivatives of E, , Hy =

and its partial derivative

ax

(450)

162

CLIFFORD M. KROWNE

Multiplying the first equation by ip and the second equation by subtracting, ip-aEz aY

aEz = a ( p2 -

K-

K

ax

2

)Hx,

-K

and (452)

Inserting (451) and (454) into (443c), we obtain

or, in a slightly reduced and more familiar form, a’Ez

a2Ez

+ k&EZ = 0 , -&T+p

(456)

with the definitions k,2f = U2W,ff9 /.12 - K 2

Peff =

-*

P

Form (456) for the governing Helmholtz equation in rectangular coordinates agreees with the earlier result provided in cylindrical coordinates (Bosma, 1964). C. Finite-Element 2 - 0 Equations

We would like to solve the 2-D boundary value problem for a circulator which has the input microstrip line imposing the Dirichlet boundary condition Ez = Ezo (459) and the other two ferrite puck-microstrip line interfaces imposing some sort of self-consistent boundary condition based on the microstrip wave impedance. This will be discussed later in Section XII1,D. The remaining ferrite-dielectric interfaces will be assumed to provide perfect magnetic walls. That is, Ht = 0, (460) For a horizontal boundary, this condition becomes H,

=

0.

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

163

Referring to (453) for the H, field component,

or

aE,

-

aY

K aEz + 1. -= 0.

P ax

(463)

If we note that the y-direction is the normal direction, (463) can be generalized for arbitrary boundary contour orientation:

The same result is found for a vertically oriented boundary contour employing Hy in (450). This form agrees with Lyon and Helszajn (1982). Consider the general 2-D boundary value problem which has the form

Looking back to (456), identify

W,Y)

=

Ez,

(468)

f(X,Y)

=

0,

(469)

where the last condition on the internal load forces f says that we are not internally driving the device. The exact field solution U(x, y) in (465) can be approximated by n

P(x,.Y)

=

c aj4j(X9Y)

(470)

j = 1

if we restrict the 3-D expressions of (370) to 2-D. The tilde indicates this is an approximate field; superscript e indicates that the field is being examined within the eth element; 4;(x, y) are the two-dimensional shape functions; and there are n nodes per element (we will use N to denote the total number of nodes later in the assembly procedure).

164

CLIFFORD M. KROWNE

The residual R in 2-D space is seen to be, by inspecting (465),

Multiplying by the ith shape function, the residual weighted equation is

1

R(x,y)&(x, y ) dxdy = 0,

i

=

1,2, ..., n,

(472)

integrated over the area of a single parent element. Putting in the explicit expression for R yields

Integration by parts can be performed once to reduce the derivative order on the unknown in the first two integrals;

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

165

Placing these results into (473) gives

- f(x,y)4f(x,y)]dxdy =

0,

i

=

1,2, ..., n.

(475)

Let us define the x and y flux components: (476a) (476b) The weighted residual equation can now be expressed in the compact form

i

=

1 , 2 , 2 , ...,n.

(477)

Define the 2-D vector u as

-

f;4fi + 5"'Y@ IY = (?,'a + f,'3)4f =

=

547,

(478)

with the flux vector defined as

5

=

5;a

+ 5;j.

(479)

166

CLIFFORD M. KROWNE

The first integral in (477) can be recast as

480)

Utilizing (480) in (477) produces

=

=

i ie

s

f4f dx dy f4f dx dy

?'4f - ds

+e

-

$e

& ?:

ds.

e

(48 1) Substituting into this formula the flux definitions gives the following form, which demonstrates the effect of the integration by parts on the symmetry, if any, of the individual parts of the numerical finite-element equation:

Notice that we have used the unknown expansion series. The kernel on the left-hand side of (482) is a 2 x 2 matrix and multiplies the unknown node values aj of the unknown variable. The right-hand side for a boundary value problem must be known. The first double integral represents the internal forcing function, which is known. The second line integral is known once we specify the normal flux value along the boundary contour. It is possible, if we don't have numerical values available for the flux on part of the boundary where we need it, to develop an equation describing the flux properties, insert it into the line integral, and move it to the left-hand side of the equation so that the unknowns may be self-consistently solved along with the rest of the problem. The part of the line integral which is

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

167

acceptable for the rest of the flux specified boundary remains on the righthand side of the equation. The result is a splitting up of the line integral. In preparation for matrix notation, define

(484) Using (483) and (484) in (482), n

1 K;aj=ee,

;=

i = 1 , 2,..., n ,

1

or

Keae = Fe. It is apparent from (483) that the kernel is symmetric in indices i and j .

D. Interfacial Boundary Conditions The standard continuity relations for normal and tangential field components across a boundary interface hold for the problem at hand, and we won’t attempt to summarize all of them or repeat such derivations. For example, in the ferrite anisotropic medium from one element to the next, the normal B, and D, are continuous, and the tangential Ht and Et are also continuous. These continuities are broken for D, and Ht , respectively, for surface charge and current. At this time, interelement surface charges and currents will be neglected as they don’t form part of the physical problem. Between two elements going from a ferrite to a dielectric medium, the same comments may be applied to the continuity properties. A derivation of the flux continuity behavior across an interface will be given, as it is extremely illuminating to show how this information may be obtained from the governing equation (456), and it highlights the importance of these Neumann-type boundary conditions. Then, special attention to interelement boundary conditions from the ferrite to the dielectric medium will be treated in some detail, as we will need this information to make available a Neumann boundary condition based upon the lack of specific knowledge of an explicitly fixed flux value. This need arises in the spatial transition in going from the ferrite medium inside the puck (whatever its shape) to the microstrip line (remember that there are three lines for the simple type of multiport circulator we are considering).

168

CLIFFORD M. KROWNE

The governing equation (456) without the spatial dependence written out looks like (Burnett, 1988)

Following the arguments surrounding (480), and noting the use of the 2-D operator

v,

=

a

a,. +y ay

-2 ax

in the derivation there, write (487) as

-v,

*

T(X, Y )

+ P W X , Y ) = f(x, A.

Perform a 2-D surface integration on this equation:

which, by the divergence theorem, becomes -

$I

T(X, y )

- ds +

ss

BU(x, y ) dA =

f(x, v)dA.

Let the small area we are integrating across the interface have area

S = tL, where t is the thickness of the rectangle perpendicularly crossing the interface and L the length parallel to the interface. Now consider the limit in which t + 0, lim 1-0

[$I

T(X, y )

- ds] + l i i

1

pU(x, y ) dA = lim t-to

The second integral is lim t-0

11

ss

f(x, y ) dA

.

(492)

pU(x,y) dA = limp(au)U(au) t-0

=

(493)

0,

where we have accounted for the possibility of jump discontinuities in p and U by inserting the averages for them and extracted them from the integral. Thus, (492) transforms into the form lim t-to

[-

-

~ ( xy ,) ds] =

f'-"o

11

f(x, y ) dA.

(494)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

169

If we do not allow forcing functions at the interface,

Noting that the path integrations normal to the interface are infinitesimals measured by the product of t and a constant, (495) may be stated as L

+ tlim tc, + t : 2 +~lim tc, -0 t-0

= 0,

(496)

which reduces to 2 + rn2 = 0.

1

tnl

(497)

Here the superscripts indicate the regions 1 and 2, which could be ferrite regions, a ferrite and a dielectric, or two dielectrics. Subscripts indicate the outward normals in each region. These normals obey the simple relation

62

-61,

(498)

1 2 rnl = t n l -

(499)

=

which leads us to rewrite (497) as

Now let us develop equations which express the interfacial boundary condition properties of crossing two different ferrite materials. Focus on normal B conditions, because these will provide nontrivial information on flux behavior, whereas D normal conditions can at most provide scale ratio information for scalar permittivity change from region to region. In fact, the continuity of the normal B field will be shown to result in equations such as (499). Recall (439) in abbreviated form as

(-

pHx + iKHy

B=

i?c:;Hy).

Consider first a vertical interface between region 1 on the left and region 2 on the right. Normal field continuity is then stated as B: = B,'.

(501)

From (500),this may be restated in going from ferrite region 1 to ferrite region 2 as p l H x l + iKIHyl= p2HX2 + k H Y 2 .

(502)

170

CLIFFORD M.KROWNE

Substituting in the H field expressions in (453) and (450),

The cross terms involving p and

K

will exactly cancel out:

aEZ1 - - -a E z 2

a~

aY

(504) *

Using flux notation, this becomes Tyl

=

(505)

Ty2,

and recalling that y is in the tangential direction, Ttl

=

(506)

Tr2.

Consider next a horizontal interface between region 1 on the bottom and region 2 on the top. Normal field continuity is stated as B,! = By".

(507)

Using (500), the equation may be recast -iK1~,1

+ PlHYl

=

-K2Hx2

+ P2Hy2.

(508)

Again substituting in the H field expressions,

Again the terms cancel, resulting in

and noting that x i s in the tangential direction, flux notation gives the same form (506).

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

171

When the dc biasing z-directed magnetic field is turned off, the offdiagonal terms in permeability tensor (438) go to zero. This is because K = 0. Relations (502) and (508) become PIHXI

=

P2Hx2,

(511a)

PUlHYl

=

P2Hy2,

(511b)

which possess the same scalar ratio form as the dielectric behavior on the displacement field continuity. In the microstrip line we will make assumptions as necessary to obtain a tractable flux boundary condition constraint equation. Assume single dominant mode operation and write a unique wave impedance

=

zoJ-p d r &dr

-20

6’

(512)

where 2, is the free space wave impedance (377 a).Relative permeability is set to unity for the dielectric loaded microstrip. From (453) and (450), the H fields are

The total H field is

H

=

H,a

+ Hyj.

(514)

The measure of the squared magnitude of this field is

H.H*

=

HxH,* + HyHy*.

In terms of (513) components,

Define the wave impedance for the two cases of interest as

(515)

172

CLIFFORD M. KROWNE

to select out the power flow directions normal to the interface. Replace the partial derivatives by the flux definitions, obtaining

where the second equalities occurs because of tangential electric field continuity. These equations write the region 2 fluxes in terms of the z electric-field component in region 1 . We will perform the splitting up of the forcing function flux line integral which occurs in (482):

fe

?& :f

ds =

feE

f,"cpf ds

+ feI

?,& ' ds.

(521)

The first line integral on the right-hand side of (521) is the explicitly applied flux boundary condition. It may be thought of as having a numerically provided flux value from data or computed explicitly. The second integral has flux determined from unknown information and must go on the kernel side when reinserted into (482)-(484). The closed line integral notation is kept because in total generality, either one of them may be closed and the other become zero. Thus, the individual elements of the kernel matrix and components of the forcing vector are changed in (486) to be

F,

f4fdx dy

=

- *eE ?,"&

ds.

(523)

e

The line integral in (522) will now be evaluated for the ferrite puck-microstrip line interface. Consider a vertical boundary. Then (519) applies, and (524)

The flux in (519) can be found using (470), giving tx2=

ipo

-Ezl 2;:

=

ipo

-

Z;: j = 1

ajq!$(x,y ) .

(525)

173

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

Inserting this formula into (524) yields

so that the additional kernel term is

Label a four-node parent element counterclockwise from the bottom right corner, 1 , to the last node 4 at the bottom left corner. This implies a C o linear shape function for a quadrilateral parent element. Isoparametric mapping will be used. The 12 side is vertical along ( in the local parent coordinate system, is located at q = -1 in the local parent element coordinate system, and is the side along the interface between the ferrite on the left and the dielectric on the right. The other sides of the element will not contribute to the assembled kernel, so we won't need to discuss them. The differential length is

ds

=

ds=

m, (gdt

(528)

+ &dq) ax + ($dt

(529)

or

ds

= ~ ( J ,dIt

+ J21 &I2 + (J12 dT + Jz&I2

9

(530)

where the Jacobian expressing the transformation between local and global coordinates,

l& &I has been used in (530). For the 12 side, dq = 0 and (529) simplifies to dcf,

ds =

or

dS

=

Jrd(,

Jr =

m.

(532)

(533)

In this last form for ds, the Jacobian for differential mapping is given.

174

CLIFFORD M. KROWNE

The integration in (527) is from -1 to +1 in (:

1

1

K$"p = @

z:

Q$(<, -l)$((,

-l)Jr(<, -1)dt.

(534)

-1

The shape functions in the element are

44(L a) = *(I - m 1 - a), &(t,a) = + O(1 - a),

(535a) (535b)

4;(L a) = *
(535c) (535d)

To evaluate the Jacobian, we note that for an isoparametric coordinate transformation, the mapping functions are chosen to be the same as the shape functions: n=4

x

=

c x ; m , a),

k=

1

Therefore, referring to (531),

Note also by (535) that

This information can be used to find Jflcr,

1 2

- 1) = - ( x i

- xf),

.If&, - 1)

1 =

(y,' - y?),

and

JF((, -1)

=

1 2

-.\/(xz" - X f y

+ (yz' - y ; ) 2 = L2'

where L is the length of the 12 side we are integrating along.

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

175

From (534)' and utilizing (541), K,?"P

(542)

= --

This is a symmetric matrix. Evaluate the elements of this kernel matrix:

The form of the kernel matrix is 1 1 0 0

K = - ipwL [32;

01 01 0 0

1.

(547)

0 0 0 0

Let us return to the magnetic wall boundary condition (464): aE, an

aE, p at

. K

-+1--=0.

(464)

176

CLIFFORD M. KROWNE

Consider a vertical wall. Then, noting that n^ and ? constitute a right-handed coordinate system, h=f, t^= -9, (548) for a right side vertical boundary on an element.

Refer back to (470) for the expansion for E,. n

E;(X,Y ) = C aj+j(x,Y ) j= 1

Substituting this expansion into (549), we obtain a direct constraint equation on the element boundary side:

Using flux notation, (549) and (551) can be recast:

r,

K

- i-5, P

=

0,

K

(553) j= 1

Condition (552) can be used to evaluate the second internal integral found in (522). We will try to follow the steps beginning at (524) employed for the microstrip boundary condition:

and the additional kernel term is

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

177

The third equality in (554) came from the flux definition in (476),

using ( S O ) , and aJx, y ) = - 1 in (466). Along the 12 side, the kernel is

where (533) has been used for the differential mapping Jacobian. It has already been evaluated in (541). Therefore,

Let us evaluate this kernel:

0

2

- - 1 -.- ,K L 8P

(558)

(559)

178

CLIFFORD M. KROWNE

The form of the kernel is

which is similar to the unsymmetric part seen by Lyon and Helszajn (1982) but done there for a triangular element. Now treat the horizontal magnetic wall condition where the outside of the element is on the top and i = j ,

-

1

t=x.

(565)

From (464) using the preceding prescription,

To obtain a direct constraint equation on the element boundary side, we invoke (550) and place it into (566):

Using flux notation, (566) and (567) become

r,

J=

1

+ i -Kr y

=

0,

(568)

“I

=

0.

(569)

P

[ r x j + i - r y j aj P

Condition (568) can be used to evaluate the second internal integral found in (522). We will try to follow the steps beginning at (524) employed for the

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

179

microstrip boundary condition.

and the additional kernel term is

which has an opposite sign to that in formula (555).

E. Assembly Process We will discuss here a few issues related to the assembly of the parent element kernels for the entire problem. In this discussion, we treat the essential constraint boundary condition on the electric field, namely

Ez = Ezo,

(459)

which is a Dirichlet boundary condition on the input microstrip line side of the circulator. Equation (459) can be discretized on the global coordinate system: m = nodes along the contour of input line. (572) Ezm = Ezo, Because the nodal finite-element method associates the unknown variable value at the node with the solution unknowns u j , f o r j = all the nodes in the global ordering system, Ezm = a,,

m

=

nodes along the contour of input line.

(573)

Looking at (572) and (573), we immediately see that a, = Ezo,

m = nodes along the contour of input line.

(574)

The relevant equation to examine here is the global form of (486) after assembly,

Ka = F,

(575)

180

CLIFFORD M. KROWNE

where a is the global unknown vector and F the known forcing function vector. Equation (575) is explicitly N

c Kijaj=F,,

j = 1

i = 1 , 2,..., N.

(576)

The sum in (576) can be broken up into two pieces. Therefore, N

c

j = 1; j # m

Kijaj +

c K"aj

j=m

= F,,

i = 1,2, ..., N.

(577)

i = 1,2, ...,N.

(578)

Moving the second sum to the right-hand side, N

1 K U. . aJ .= 4 - j =1 Kiiaj, j=l;j#m m

where the second sum is over all m on the contour of the input line. Let us rewrite the second sum in (578) to reflect this directly. N

c

j = 1;jfm

&aj=&-

cKimam, m

i = 1 , 2,..., N.

(579)

To retain the old sum over N j values, redefine the kernel as

Thus, N

c Kj>aj

=

j = 1

4

-

c Kimam, m

i = 1,2, ..., N.

(581)

Equation (574) eliminates Munknowns along the contour of the input line. Consequently, we can eliminate exactly Mequations from (581), noting that M < N. Choosing these equations to be at i = m', N

c Khtjaj

=

j= I

c K,,,a,,

4

-

-

c Kimam,

m' IM.

m

(582)

Utilizing (582) in (581), N

c K;aj

j= 1

Note from (574) that

=

4

m

i IN ; i z m'.

(583)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

18 1

This equation can be written in summation form as N

c KLojaj

=

E,,

=

FA,,

m' I M ,

(585)

j = 1

KAtj =

j = m', j # m'.

1, 0,

The Dirichlet constraint (585) can be combined with (583) to give the final assembled kernel: N

c K!.a. IJ

J

= F! I ,

j = 1

F;! =

:,[

=

'S2,

' . * ¶

Kimam,

N, i

(587) f

m, i

5

N, (588)

i=msM.

Assembly for the elements into a global kernel using global notation is relatively straightforward to state. It reflects the fact that each node represents a continuity of the unknown variable U (or E, in our specific case here). This translates into the rules K lJ. . = Ke. IJ ' (589)

c e

where the e sum is over all elements sharing the ith node for the i = j self terms, and the e sum is over all elements sharing the 0th side for i # j cross terms; and

e =tee,

(590)

e

where the e sum is over all elements sharing the ith node. F. Determination of Circuit Parameters

The input reflection coefficients for the three ports of the circulator can be calculated from

Here the line impedance may be taken as the fundamental mode of microstrip operation, and

where the electric and magnetic fields are averaged values of the solution across the port contours. Ht is along the port contour.

182

CLIFFORD M. KROWNE

The off-diagonal s port parameters can be found in a manner like that used by Miyoshi et al. (1977) and restated more briefly by Lyon and Helszajn (1982). We reproduce below only the formulation part of the circuit parameter derivation for the stripline type of configuration from Miyoshi et al. (1977). Introduce the Green’s function which satisfies the boundary condition along contour C : .KaG aG J----= 0. (593) p at an The rf voltage at a point in the circuit is

9 P

V

= jwpeffd

G(-i,,)dt.

(594)

C

Here 2d is the thickness of the planar circuit. Expand the Green’s function in terms of the complex eigenfunctions 4, which derive from the eigenvalue problem defined by (v; + w,2&p,ffMa= 0, in D, (595a) (595b) (59%) The rf voltage in the circuit, using eigenfunctions, is given by

To calculate the circuit parameters of the equivalent multiport, we define approximately the rf voltage on a port and the total current flowing into a port, respectively, as

K=-

‘I

v(ti)dt,,

qw,

Ij

=

3‘

( - 2i,(tj)] dt.

(597)

A;.

Substituting (597) into (596),

where rn is the number of ports coupling to the planar circuit, a circulator in our case. Thus, the 0 t h element of the impedance matrix of the equivalent

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

183

multiport becomes

The impedance matrix is not symmetric because the eigenfunctions complex-valued. However, if the circuit is lossless,

z.. 1J = -z*.* 1J

are (600)

G . Conclusions The theoretical underpinnings for solving irregular-geometry circulators loaded with ferrite material have been provided here. We had posed a boundary value problem, but it is possible to have approached it as an eigenvalue problem, with the loading ferrite material acting as a resonator. Then, after eigenmode determination, proper superposition of the eigenmodes would be necessary to satisfy the boundary conditions. The eigenvalue problem kernel is somewhat different than that derived here for the boundary value kernel, but the essentials are the same. The finite-element method, which formed the basis for this section, is a method well-suited to problems possessing irregular geometries and inhomogeneous and anisotropic materials. Although we have worked from higher-order governing equation(s) guided by a physical insight arising from electromagnetic, quantum-mechanical, acoustic, and mechanical types of problems, it is possible to take a more mathematical approach using lowerorder equations and deriving the finite-element formulation, making sure to maintain rigorous element-to-element continuity conditions. Regardless of the finite-element approach used by the researcher, this technique is an extremely powerful method for solving complex electromagnetic problems. Accurate numerical results are a consequence of using this approach. XIV. NUMERICALLY CALCULATED RESULTS FOR GUIDED-WAVE STRUCTURES A . Semi-infinite Planar, Rectangular, Circular, and Planar Microstrip and Slotline Structures

Here we discuss a number of important numerical results obtained in unbounded planar, rectangular, planar Kerr geometry, cylindrical coaxial, and bounded planar layered structures. Figure 1 shows the dispersion diagram

184

CLIFFORD M. KROWNE

a FIGURE1 . Exact and finite-element (open circles) dispersion spectra of a dielectricsemiconductor single interface for (solid line) P, = 80pm, P2 = 100pm and dashed line) PI = 320pm, P2 = 5Opm. (From Krowne, 1993b; 0 1993 IEE.)

region

region 1

'/z

Bx

*{ 1

{1

-a-

semiconductor slab

-

~ . '

B=eoB,

FIGURE 2. The H-plane loading geometry. (From Krowne, 1993(b); 0 1993 IEE.)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

1 85

for TM modes obtained using the finite-element method (Mohsenian et al., 1987) for a two-layered structure, infinite in the y-direction. Normalized o versus /3 is calculated for an n-doped GaAs sample with a carrier concen~ , to a plasma frequency of tration of N~ = 2.1 x 1015~ m - equivalent 1013rad/s, and a cyclotron frequency of 1OI2rad/s corresponding to a B, dc magnetic field of 3,810G parallel to the interface. Dielectric and semiconductor layer thicknesses were on the order of 100pm. Focus in that work and related earlier research by those authors was on millimeter- and submillimeter-wave frequencies. The Fig. 1 results lead to suggestions for developing isolators and circulators with low attenuation in the 358 to 41 1 GHz and 377 to 396 GHz bands. Modal field patterns are given also for x , z , and y E and H field components. Inhomogeneous loading of a rectangular waveguide with a semiconductor slab in the presence of a magnetic field applied parallel to the surface and perpendicular to the propagation direction is shown in Fig. 2 (Godshalk and Rosenbaum, 1984). For asymmetric slab loading with respect to the top and bottom guide surfaces, the field displacement effect will occur, causing more fields to exist in the non-slab region for one propagation direction than the reverse direction. The mechanism for this unequal distribution of power can be analyzed in terms of the coupling of higher-order modes such as the TEll and TM,, with the dominant TE,, mode via the off-diagonal permittivity tensor elements (Arnold and Rosenbaum, 1971). An important qualitative requirement to assure nonreciprocal behavior is that Wet>

1,

(601)

or that the product of the phenomenological carrier collision time due to scatterers in the semiconductor and the cyclotron electron frequency exceed unity. Figure 3 shows the attenuation constant versus carrier concentration for a 250pm thick Si slab at 92 GHz at 77 K. Room-temperature calculations and measurements were also done. Analysis of parallel-plate waveguide modes enabled construction of the modes of a rectangular waveguide loaded with semiconductor and B, applied normal to the propagation direction (Sorrentino, 1976). Backward waves were numerically seen for an X-band waveguide by finding indexes of refraction. Figure 4 gives the first six modes (those with the lowest attenuation) forf = 9 GHz, (T = 1.8 C2-' m-', C ~ / W E = 0.719, thickness to guide-height ratio 0.5. Real and imaginary parts of the normalized phase and attenuation constants against RB, are plotted. Also, a comparison to an earlier Si-loaded waveguide (Arnold and Rosenbaum, 1971) was made demonstrating considerably better agreement with experiment using a few modes of the present theory versus the earlier approach.

186

CLIFFORD M. KROWNE -

looor

carrier concentration, ~ r n - ~

FIGURE3. Attenuation vs. carrier concentration for a 250pm thick slab of Si. (From Krowne, 1993b; 0 1993 IEE.)

1 X band waveguide

0.755

-

0.505 0.421

0.485 0.303

0.155

X

u

c

C

v

v

aJ

-E

LL

0.303

0.485 0

.

5

w

o -0.155 -0.421 -0.505

-0,7 -1 0

0.5

RBo(Qm)

1

0

0.5

1

RBo(Qm)

FIGURE4. Normalized phase constants and attenuation constants of the first six modes vs. RE,, for b , / b = 0.5. (From Krowne, 1993b; 0 1993 IEE.)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

N

187

IY

plasma

FIGURE5 . Orientations of field vector E, propagation vector (From Krowne, 1993b; 0 1993 IEE.)

r, and dc magnetic

field B.

Reflection-beam isolators (Kanda and May, 1973) using the Kerr transverse magneto-optic effect (plane of polarization of the wave in the plane of incidence, which is perpendicular to a dc magnetic field) at submillimeter wavelengths (infrared, 337 pm) were studied (Fig. 5). A semiconductor slab (InSb) covered with a dielectric layer acting as a matching transformer is operated at room temperature. Figure 6 provides reflection loss against incident angle for an HCN laser in B,, = 15 kG. The nonreciprocal results seen are due to the combined effects of carrier collisions which are required t o cause an elliptic motion of electrons (and similar electric field polarization) in a solid state plasma without a dc magnetic field, and the dc magnetic field. A reflection beam isolator at millimeter wavelengths at 94 GHz was examined (Kanda and May, 1975a) using n-type 1.44Q.cm GaAs, N,, = 7.1 x 1014/cm3, mobility p = 6.1103 cm2/V.s, lattice dielectric constant E = 12.53, electron effective mass m* = 0.067 m,. The experimental setup is shown in Fig. 7. Although theory predicted over 40 dB isolation with 5 dB insertion loss, observed values were 12 dB with 11 dB insertion loss. Better results were found at submillimeter wavelengths (Kanda and May, 1973). Use of a longitudinal dc magnetic field (Faraday configuration) in a cylindrical system is illustrated by a 35 GHz isolator using a coaxial InSb sample (Figs. 8 and 9) with carrier concentration N D = (4.8-21.2) x 1013/cm3, mobility p = (4-4.8) x lo5 cm2/V-s, and ambient temperature 75 K (McLeod and May, 1971). Figure 10 shows how the isolator works by the field exclusion effect, there being more microwave or millimeter-wave absorption for left-hand circular polarization than for right-hand circular polarization (also obtained by switching either the direction of propagation or the dc magnetic field). Figure 10a shows the low-loss direction of

188

CLIFFORD M. KROWNE

t

incident angle, deg

FIGURE 6 . Theoretical and experimental reflection loss of InSb at 337,um as a function of incident angle. Geometry of the isolator is shown in Fig. 7, with I = 250,um and B = 15 kG. (From Krowne, 1993b; 0 1993 IEE.)

propagation. The effective dielectric constant in the InSb rod had a large real part, causing field displacement or exclusion. Figure 10b shows field reversal: The fields are still largely excluded, but in Fig. 1Oc with B, chosen to make the real part of the left effective circularly polarized dielectric constant nearly zero, there is a great increase of field density in the rod and a peak in attenuation. Figure 11 gives the calculated and measured attentuation versus magnetic field for a guide loaded with an InSb rod. The mode theory of a coaxial loaded circular guide with longitudinally applied B, field has been studied for rotationally symmetric TM modes (Miteva and Ivanov, 1989; 1988). Interesting TM,, results are shown in Fig. 12 for p = rl/ro = 0.2, parameterized in terms of V / K ,~where K~ is the relative dielectric constant and q = (a/we,)pB,/[(l +jot)2+ (pB0)2].Notice that backward waves are predicted when the slope is below zero. Discussion is given regarding designs of phase shifters and cutoff switches for the Vand W-bands (70-92 GHz). One simple but repeatedly noticed criterion

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

i

isolator

I

7

94GHz I

klystron

-

power supply

I

lkHz

modulator

~

A

I

94 GHz

precision attenuator

189

1

pyramidal horn

pyramidal

E'

i

quarter wave matching plate stycast (E I 15)

o

k

lock - in amplifier

I

t crystal detector

4 .4 m m

GaAs

FIGURE7. Experimental setup used to measure reflection from GaAs at 94GHz. (From Krowne, 1993b; 0 1993 IEE.)

throughout the literature for nonreciprocal operation of magnetoplasma devices is that 0,> 0, (602) where we see that the cyclotron frequency should exceed the rf frequency. The background wave phenomenon has been further commented upon by a ,circular

waveguide,

:e

RG 96lU waveguide

FIOURE 8. Cross-sectional view of the InSb isolator. (From Krowne, 1993b; 0 1993 IEE.)

190

CLIFFORD M. KROWNE

perfect electric conductor

FIGURE9. Schematic of the plasma-loadedcylindrical waveguide and the coordinate system. (From Krowne, 1993b; 0 1993 IEE.)

group of investigators (Gibson et al., 1991; Sloan et al., 1991) who studied the azimuthally magnetized coaxial guide to obtain and plot power density distribution against radius showing regions of negative power flow, and cases when total power flow exceeds (d/3/do > 0) zero or is below zero (d/3/do < 0) (Sloan et al., 1991). In later work they emphasized the double root nature of the dispersion diagram (Gibson et al., 1991). A paper which focused on slow wave propagation with a slow wave circuit replacement in mind for M-type TWT applications found a nonreciprocity of 60 dB and 6.5 dB insertion loss and formed a precursor for the later azimuthally magnetized papers just discussed (Obunai and Hakamada, 1984). Interestingly, a discussion on a parallel-plate analogue to an azimuthally magnetized solid plasma coaxial guide demonstrated the existence of a Goubau-line surface wave mode influenced by the outer conductor, an image-guide mode affected by the inner conductor, and a mode more nearly tied directly to the plasma-air interface. Figure 13 shows a suspended single slot line structure (Krowne et al., 1998a). Propagation is either in the z direction, out of the paper ("+" or forward direction), or into the paper ("-" or reverse direction). The suspended single slot line rests on two substrates, a dielectric with relative permittivity Ed = 12.5 and the semiconductor with E , = E d . Regions of thickness hl and h4 are air (relative permittivity = E , ~= 1, relative permeability p r = 1). Slot width w = 1 .O mm, and the other geometric dimensions are b = 2.35 mm, hl = h4 = 2.1 mm, and h2 = h3 = 0.25 mm. Thickness h3 is varied. Numerical results are first presented for

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

-21

191

radius, mm

FIGURE10. Power density curves in the loaded section of guide calculated from 0.5Re (E x H*) * u, at the plane z = 0. Frequency f = 35 GHz, n = 4.8 x 10" ca~riers/cm', mobility = 4 x 10' cmZ/V.s, InSb rod radius = 0.764 mm, guide radius = 2.36 mm. (a) Dc magnetic field B, = +0.1 Wb/m2. (b) B, = -0.2 Wb/m2. (c) Bo = -0.23 Wb/mz. (d) B, = -0.33Wb/m2. (From Krowne, 1993b; 0 1993 IEE.)

4 = O", with w, = 3.14 x 10l2rad/s (Bo = 12 kG), w p = 6.28 x 10l2rad/s (no = 10l6cm3), and tp= s (room-temperature operation). Only the dominant mode is calculated and discussed here. This mode is odd with respect to E, and even with respect to E x . In the Bo 0 limit, the regular dominant slot line mode is obtained. Figure 14a shows CY (dB/mm) (with CY ='YC or CY-)versus frequency f (GHz) between 45 and 85 GHz. At the a peaks, ACY/CY = 15%. The normalized phase propagation constant p = /3/fl0 with /3 = 8' or /3-, Po = free space value, is shown in Fig. 14b. Results show that AD increases with decreasing h3 since h3 reduction places the magnetoplasma layer in closer proximity to the slot. For h3 = 0, A/3//3 = 1.3% across the 45 through 80 GHz frequency range. For f = 75 GHz, Figs 15a and 15b show CY and /3 under varying 4. Notice the monotonic decrease of ACYand A p with increasing 4, and 'YC > a- and > p-. For the choice of parameters selected, A C YApy ~ 01-, p', and +

a'

a-

192

CLIFFORD M. KROWNE

\/\

measured attenuation propagation against magnetic field

I-\

'

45 -

40 -

,,\

I I

measured attenuation, propagat ion with,,' magnetic ,, field

I

calculated mode two

I

calculated mode one I

l

o

Y

iu

11

12

13

14

magnetic field, k G

FIGURE11. Calculated (solid line) and measured (dashed line) attenuation of the 35-GHz signal. Calculations are for propagation against the magnetic field for the guide loaded with a rod of InSb with n = 2.12 x loi4 carriers/cm3, mobility = 4.8 x ID5cmZ/V.s, rod radius = 0.51 mm, rod length = 8.65 mm, and guide radius = 2.36 mm. (From Krowne, 1993b; 0 1993 IEE.)

are all roughly constant up to a 15" inclination angle. The attenuation constant 'a hardly varies (compared to a-)up to = 60". As approaches go", both A a and AD approach zero. An explanation for such a limiting characteristic is the creation of cyclotron orbits which are perpendicular to the B, direction and parallel to the planar interfaces.

+

+

normalised guide radius p o r o K

FIGURE12. Normalized phase characteristics of azimuthally magnetized millimeter-wave solid-plasma circular guide for p = 0.2. (From Krowne, 199313; 0 1993 IEE.)

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

193

FIGURE13. Cross-section of a single suspended slot line structure. Magnetic field bias B, is at inclination angle 9 to the planar surfaces. (From Krowne ef ul., 1988a; 0 1988 IEEE.)

Four other planar structures in Fig. 16 were examined for their dominant mode behavior with C$ = O", two slot and two microstrip. The even mode of the suspended coupled slot structure was studied in Fig. 16a for the same geometric and physical parameters as in Fig. 14, except h, = Omm and w = w1 = 0.2 mm (f= 75 GHz). The last structure analyzed was the dielectric-covered single microstrip over a doped semiconductor layer of thickness h2 = 0.25 mm and an undoped substrate of thickness hl = 2.1 mm ( =A4). It is shown in Fig. 16d. Here the cover has h, = 0.1 mm, w1 = 0.2 mm, and &d = E , ~= E , ~= 12.5. Figures 17a and 17b give a and j? versus B, up to nearly 35 kG at f = 50 GHz with cop = 6.28 x 10" rad/s (no = 10'4/cm3). The even-mode results are parameterized in terms of tp = (1, 5 , 10) x lo-', s. At low B,, a increases, an effect not seen beyond 5 kG. As tpapproaches infinity, a approaches 0 (not shown), as expected for no carrier scattering. The quantity A a first increases then decreases with 1.4 dB/mm and occurs for 3 < Bo < 7 kG with rising Bo Maximum A a tp = s. This corresponds to a A d a = 47% at 5 kG. One notices that these curves are extremely nonlinear for Bo < 20 kG and tP # s. Highly nonlinear behavior is mimicked in the versus Bo curves for B, < 20 kG and tp2_ 10-13s (Fig. 17b). Beyond 20 kG, A p is approximately constant and behaves linearly. The attentuation dependence of a on tp at B, = 20kG is presented in Fig. 18a for different no =

.

=

194

CLIFFORD M. KROWNE

E . 3.0

frequency f ,

GHz

FIGURE 14. (a) Dispersion behavior of 01’ and 01- for the odd mode of single suspended slot line. Bo = 12 kG, 6 = O”, no = lOi6/cm’, 7r,= 0.1 ps, w = 1.0 mm, h, = h, = 2.1 mm, h, = h, = 0.25 mm, E, = ed = 12.5. (b) Dispersion behavior of j’ and j-.Parameters are the same as in Fig. 14a. (From Krowne et al., 1988a; 0 1988 IEEE.)

1014cm3.As tp approaches 10-14s, both a and A a decrease. Also, as expected, a and A a tend to decrease as tpgets large (beyond tp = lo-” s for no = 1014cm3). However, A d a = 130% near tp = 6.2 x lO-”s. When no increases, the absolute a maxima move to the left (or smaller tp). Figure 18b presents the corresponding versus tpdependence. A a increases with increasing n o . For no = 1014cm3, ha E constant up to tp= 2.5 x lO-”s, beyond which it monotonically increases to A/?//? G 11Yo.

a

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

195

I

0

20

60

40

80

100

inclination angle of rnagnetlc bias field $ . deg

FIGURE15. (a) Variation of'a and a- with inclination angle $J.(b) Variation of 8' and b with inclination angle 6. (From Krowne et al., 1988a; 0 1988 IEEE.)

a

Figures 19a and 19b provide a and against h2 for B, = 10 kG, s, and w1 = 0.2 mm with parameterized h3. rp = The last figure, Fig. 20, shows the phase and attenuation constants against frequency for the three fundamental modes appearing in a multilayered three strip asymmetrical structure (Mesa et al., 1992). The multilayered medium is composed of layers, the first a ferrite, the second a dielectric, the third a semiconductor, and the fourth air. A dc magnetic field was applied parallel to the interfaces, normal to the strips.

196

lc

CLIFFORD M. KROWNE

T

i 7

il

+ b FIGURE16. Cross-sections of planar structures possessing a semiconductor(s) under dc magnetic field basis B, and inclination angle t$ = 0". The gyroelectric effect is studied in (a) suspended coupled slot line, (b) sandwiched suspended single slot line, (c) suspended single microstrip, and (d) covered single microstrip. (From Krowne el al., 1988a; 0 1988 IEEE.)

B. Conclusions

The future directions of research for employing the magnetoplasma effect in solid-state loaded guiding structures will of necessity involve using new semiconductor crystals, compounds, and heterostructures. Some of the

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

E

-

E m

197

Tp 1.ops

U

c

0

c

C

8

I

10

15

20

25

30

35

magnitude magnetic field B O , k G

FIGURE 17. (a) Variation of a+ and a- with magnetic field Bo for the even mode of the covered single microstrip structure (Fig. 16d). Parameters same as in Fig. 14, except that here h, = 0.25 mm, h, = 0.10 mm, ed = eS1= eSZ= 12.5, no = 10i4/cm3, and f = 50 GHz. Results are parameterized in T ~ (b) . Variation of 8' and 8- with magnetic field Bo for the Fig. 17a case. (From Krowne el al., 1988a; 0 1988 IEEE.)

classical and quantum aspects of carrier transport and magnetic field effects have been reviewed in previous sections (Sections 11, IV, V) and applied to the numerical analysis provided here. Since some analytical methods can be a guide in electromagnetic studies, one simple variational approach was presented for the propagation constant in an anisotropically loaded waveguide (Section VII). Then a summary was provided of a finite-element method used to solve a layered magnetoplasma

198

CLIFFORD M. KROWNE

8 c

+ C

1.0“0

C

U

C

0

c

4

0.5-

C a,

c c

o

I

/’

n

/

OI

--7-----

a-

10

momentum relaxation time

T

~ 0 ,.

01~~

FIGURE18. (a) Variation of a+ and 01- against T~ for the Fig. 17a case with B,, = 20 kG. Parameterized curves for no = 10” and 10’4/cm3. @) Variation of 8’ andg- against T~ for the Fig. 13a case. no = lo’’, loi3,and 10i4/cm3.(From Krowne et al., 1988a; 0 1988 IEEE.)

problem at millimeter and submillimeter wavelengths. The finite element is a very general method and is readily adaptable to many irregular geometries and anisotropic materials. It is an approach whose use and full potential is just beginning to be realized (Section VIII). Because planar problems are of such great importance in hybrid and monolithic circuits, special emphasis was placed on the construction and solution of slot and microstrip problems found in enclosures using the

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

199

E E m

. U

d

bIm

3.5-

c

0

c

C

e

3.4-

C

!?

c

;3 . 3 a

g Q

2 3 10

. 102

103

1004

layer thickness h 2 ,&rn

FIGURE 19. (a) Variation of a+ and a- against h, for the Fig. 17a case. Here B, = 10 kG and w = 0.2mm with h, = 0.1 or 0.5 mm. (b) Variation of 8' and 8- against h, for the Fig. 19a case. (from Krowne el al., 1988a; 0 1988 IEEE.)

spectral domain approach (Section IX). This approach is considered a quasi-analytical method in comparison to the finite element technique. Normal-mode fields (part A), the transformation operator P (part B), the dyadic impedance Green's function (part C), strip surface currents (part D), the dyadic admittance Green's function (part E), slot surface fields (part F), the anisotropic determinantal equation (part G ) , and the magnetoplasma permittivity tensor (Section X) were all covered.

~

200

CLIFFORD M. KROWNE

3.7,

------I2

3.6

1.8

E 1.6 E m

.

2 3.5 \

m

U

tl

3.4

1.4

3.3

1.2 I

L

7

9

12 15 frequency, GHz

18

20

FIGURE20. Modal propagation parameters for the three fundamental modes in a transmission line with a gyrotropic-dielectric-semiconductor-air composite medium and three asymmetric strips above the second layer. h, = 200pm, E , = 12.9, H, = (1,000Oers.) a,, n = 1014cm-3, 4 = lO-"s, 4xM, = 1,600G, h, = 100pm, E, = 12.6, h, = 100pm, c3 = 12.3, u3 = 0.01 (Qrnm)-', h, = 1 mm, e4 = 1, c1 = 50pm, w , = 100pm, c2 = 275prn, w 2= 150pm, c3 = 600pm, w1 = 200pm. (From Krowne, 1993b; 0 1993 IEE.)

Finally, numerical results were presented in this section for important infinite planar, rectangular, circular or coaxial, and enclosed layered waveguiding structures demonstrating some aspects of the magnetoplasma effect,

CONCLUSION XV. OVERALL This chapter in the Advances in Imaging and Electron Physics series has covered a number of extremely important and interesting areas from both an electronics point of view and a physics point of view. We have examined gyroelectric behavior due to the magnetoplasma effect. The electron transport physics and the electromagnetic properties in various waveguiding structures were discussed. Various theoretical approaches for treating anisotropic problems have been presented, including variational, spectral domain, and finite-element methods. Many fascinating subjects in the area of anisotropy may be investigated which could have an immense impact upon the future technologies being developed for monolithic integrated circuits. We have tried to touch upon some of those areas in this survey which may significantly modify the future of microwave and millimeter-wave circuits. Included in this survey have

ELECTROMAGNETIC PROPAGATION AND FIELD BEHAVIOR

20 1

been the gyroelectric effect (Section X), the optical activity effect and the optical activity-gyromagnetic combined effect (Section XI), and the gyromagnetic effect (Section XIII). Studying the most general anisotropic linear media, often referred to as bianisotropic media, has been done by presenting two very different theoretical approaches which are very effective, the spectral domain method (Section IX) and the finite-element method (Section XII). The spectral domain method was used here to treat waveguiding problems of uniform layers but arbitrarily placed conducting metalizations, essentially making the approach a 1-D Fourier transform analysis. The finite-element method has been developed in 3-D real space. It does not require any domain transformations in the usual sense. The only transformations, or more appropriately, mappings, which are required to derive the method and implement it, involve mappings from the local real space of the parent element to the individual elements in global space.

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Index

A

Bismuth, properties, 82 Boundary value kernels, 175-179, 183

Adaptive contrast enhancement, 34-36 Adaptive extremum sharpening filter, 44-48 Adaptive quantile filter, 14-16 Anisotropic determinantal equation, 1 12- 1 14 Anisotropic media chiral and chiral-ferrite media, I 17- 132 computer simulations, 133 electromagnetic properties and field behavior, 80-95, 114-1 17, 132-158, 200 ferrite media, 158-183 guided-wave structures, 183-200 planar guiding structures, 95- 132 Anisotropic reaction theorem, 119- 120 Anisotropy, adoptive extremum sharpening filter, 44-45 Assembly process, ferrite media, 179- I8 1

Canberra distance, 35 Cayley-Hamiltonian theorem, 102, 103 Center-weight median filtering, 17 Chiral-ferrite media, 117, 132 constitutive relations, 1 19 dispersion relations, 126 dyadic Green’s function, 123 Chiral media, 117, 132 constitutive relations, 118-1 19 dispersion relations, 125- 129 electric field polarization, 130- 132 vector Helmholtz equations, 122- 125 Circuit parameters, ferrite media, 181- 183 Composite enhancement filter, 16- 17 Computer simulations, anisotropic media, 133 Conductivity tensor, 83-85, 90-93 Constitutive relations chiral media, 118-119 gyroelectric-gyromagnetic variational analysis, 136 linear media, 135 Constitutive tensor, 114 Contextual region, 36 Contour chain processing, binary image enhancement, 70-72 Contrast enhancement adaptive contrast enhancement, 34-36 extremum sharpening, 32-34

B Background subtraction, 21 gray-value tracking, 24 linear regression, 22-24 Band structure, anisotropy, effect on conductivity tensor, 90-93 Bessel function, 109 Binary image enhancement, 64-75 contour chain processing, 70-72 distance transform, 72-75 polynomial filtering, 68-70 rank-selection filters, 65-68 215

216

INDEX

inverse contrast ratio mapping, 30-3 1 local range stretching, 27-30 Contrast stretching, 4-6 Convolution kernel, 55-58 Coupled slot, surface field, 1 10- 1 1 I Cyclotron effective mass, 8 1 Cyclotron frequency, 81 Cyclotron mobility, 87 Cyclotron resonance, 89

D Degree of local spatial order, 45 Detail enhancement, 38-52 adaptive extremum sharpening filter, 44-48 mathematical morphology, 48-52 multistage one-dimensional filter, 39-44 Digital image processing, see Image enhancement Discretization noise, distance transform to smooth, 72-75 Dispersion relations, chiral media, 125- 129 Distance transform, binary image enhancement, 72-75 Drude model, I14 Dyadic Green’s function, 103-106, 108, 1I2

E Electric field polarization, chiral media, 130- 132 Electromagnetic properties, highly anisotropic media, 80-200 Energy bands, anisotropic, 85-88 Enhancement, see Image enhancement k? tensor, I14 Extremum sharpening filter, image enhancement, 32-34, 44-48

F Ferrite media, 117, 132 assembly process, 179- 181 chiral-ferrite media, 117, 119, 123, 126, 132 circuit parameters, 181- I83 constitutive relations, 119, 120 finite-element 2-D equations, 162- 167 Helmholtz wave equation, 159- 162 Field behavior, electromagnetic, highly anisotropic media, 80-200 Filters adaptive extremum sharpening, 44-48 adaptive quantile, 14- 16 center-weight median, 17

composite enhancement, 16- 17 extremum sharpening, 32-34, 44-48 iterative noise peak elimination, 17- 18 linear, 55-62 low-pass, 21 majority, 65-68 max/min-median, 40-42 morphological, 49-52, 75 multistage one-dimensional, 39-44 non-adaptive rank-selection, 66-68 non-linear mean, 18 polynomial, 68-70 quadratic, 68 rank-order, 14-18 recursive, 21 -22, 57 soft morphological, 49-52 zonal, 25 Finite-elemen t method anisotropic media, 132-134, 145-157, 183 2-D equations, ferrite media, 162- 167 propagating structures, 94-95

G Gallium arsenide, properties, 82, I 17 Generalized soft dilation, 52 Generalized soft erosion, 52 Germanium, properties, 82 Gray-scale transformations, histogram equalization, 6-9 Gray value, 4-5, 9, 19, 20, 27, 38 Gray-value function, 6, 29, 62-63 Gray-value transformations, 4-9 Green’s functions, dyadic admittance, 108, 112 chiral-femte media, 123- 125 impedance, 103-106, 108 Guided-wave structures, 183-200 Gyroelectric-gyromagnetic variational analysis, 136- I38 isotropic case, 139-141

H Helmholtz wave equation, ferrite media, 159-162 Histogram equalization, 6-9, 28-30 Histogram stretching, 27-28 Hybrid multistage filter, 40

I Iconic maps, 53-55 Image enhancement, 1-3

217

INDEX binary image enhancement, 64-75 contour chain processing, 70-72 distance transform, 72-75 polynomial filtering, 68-70 rank-selection filters, 65-68 detail enhancement. 38-52 adaptive extremum sharpening filter, 44-48 mathematical morphology, 48-52 multistage one-dimensional filter, 39-44 gray-scale transformations, 4-9 histogram equalization, 6-9 line pattern enhancement, 52-64 iconic maps, 53-55 linear filters, 55-62 top-hat transformation, 63-64 topographical approach, 62-63 local contrast enhancement, 26-38 adaptive contrast enhancement, 34-36 extremum sharpening, 32-34 inverse contrast ratio mapping, 30-31 local range stretching, 27-30 pyramidal image model, 36-38 shading compensation, 20-26 background extraction, 22-24 rank-order statistics, 25-26 weighted unsharp masking, 24-25 uniformity enhancement adaptive quantile filter, 14-16 center-weight median filter, 17 composite enhancement filter, 16- 17 iterative noise peak elimination filter, 17-18 Image smoothing, see Smoothing Indium stibnite, properties, 82 Input reflection coefficient, 181 Inverse contrast ratio mapping, 30-31 Inverse mobility tensor, 87 Irregular-geometry circulator, 183 Iterative noise peak elimination filter, 17-18

K Kleincordon equation, 88

L Linear filters, 21, 55-62 Linear media, constitutive relations, 135 Linear minimum square error technique, 10 Line pattern enhancement iconic maps, 53-55 linear filter, 55-62

top-hat transformation, 63-64 topographical approach, 62-63 LMMSE, see Linear minimum square error technique Local anisotropy measure, 53-55 Local contrast enhancement adaptive contrast enhancement, 34-36 extremum sharpening, 32-34 inverse contrast ratio mapping, 30-3 1 local range stretching, 27-30 Local degree of anisotropy, 44 Local range stretching, 27-30 Local spatial order, 45-48 Local variation coefficient, 15

M Magnetic fields, quantum phenomena, 88 Magnetoplasma physics, 80, 81 anisotropic energy bands, 85-88 band structure anisotropy effect, 90-93 chiral-ferrite media, 117, 132 constitutive relations, I 19 dispersion relations, 126 dyadic Green’s function, 123 chiral media, 117, 132 constitutive relations, 118-1 19 dispersion relations, 125-129 electric field polarization, 130- 132 vector Helmholtz equations, 122- 125 conductivity tensor, 83-85, 90-93 femte media, 117, 132 assembly process, 179- 18 I chiral-ferrite media, 117, 119, 123. 126, 132 circuit parameters, 181-183 constitutive relations, 119, 120 finite-element 2-D equations, 162- 167 Helmholtz wave equation, 159- 162 finite-element method, 132-134, 145-157, 183 2-D equations, 162-167 propagating structures, 94-95 guided wave structures, 183-200 permittivity tensor, 114-1 17 planar guiding structures, 95-132 anisotropic determinantal equation, 112-114 dyadic Green’s functions, 103- 106, 108, 112 normal mode field, 95-101 slot surface fields, 108- 11 1 strip surface currents, 106-108

218

INDEX

transformation operator matrix, 102- 103 propagation constant, variational formula, 93 quantum phenomena, 88-90 semiconductors, 81-83 variational analysis complex media, 142-145 gyroelectric-gyromagnetic, 136- 138, 139-141 propagation constant, 93 Majority filter, 65-68 Mapping region, 36 Mathematical morphology image enhancement, 48-52 top-hat transformation, 63-64 Max/min-median filter, 40-42 Median multistage filter, 40 Microwave conductivity tensor, 83-86 Modulation transfer function, 26 Morphological filter, image enhancement, 75, 49-52 Multistage one-dimensional filter, 39-44

N Neglected residual technique, anisotropic media, 132-134, 145-157 Noise additive, 10- 13 binary images, 66-68, 72-75 discretization, 72-75 LMMSE, 11-13 multiplicative, 13-14 Non-adaptive rank-selection filter, 66-68 Non-linear gray-value transformation, 4-6 Non-linear local contrast enhancement, 27-36 Non-linear mean filter, 18 Normal mode field, 95-101

0 Optical activity tensor, 135

P Particle current density, 115 Permeability tensor, 135 Permittivity tensor, 114-1 17, 135 Planar guiding structures, 95-132 anisotropic determinantal equation, 112-1 14 dyadic Green’s functions admittance, 108, 112 impedance, 103-106, 108 normal mode field, 95-101 slot surface fields, 108-1 11

strip surface currents, 106-108 transformation operator matrix, 102- 103 Plasma frequency, 8 1 Polarization ellipse, 131- 132 Polaron, theory of, 89 Polynomial filter, binary image enhancement, 68-70 Prewitt operator, 34 Propagation, electromagnetic, highly anisotropic media, 80-200 Propagation constant, variational formula, 93 Pyramidal image model, 36-38

Q Quadratic filter, 68 Quantum phenomena, magnetoplasma physics, 88-90

R Radar, smoothing of SAR speckle noise, 13-14 Rank-order filtering, 14-18 Rank-selection filter, binary image enhancement, 65-68 Recursive filter, 21 -22, 57 Remote sensing image contrast enhancement, 28-29 detail enhancement, 45, 46 extremum sharpening, 33,46 max/min-median filter, 41 smoothing of SAR speckle noise, 13-14 top-hat transformation, 64

S SAR, see Synthetic aperture radar Self-adjointness, 134 Semiconductors, 81-83, 86, 89, 116 Shading compensation, 20-26 background extraction gray-value tracking, 24 linear regression, 22-24 rank-order statistics, 25-26 weighted unsharp masking, 24-25 Silicon, properties, 82 Single slot, surface field, 108-109 Smoothing, 9-19 with additive noise or texture, 10-13 with multiplicative noise or texture, 13-14 rank-order filtering, 14-18 adaptive quantile filter, 14-16 center-weighted median filter, 17

INDEX composite enhancement filter, 16- 17 iterative noise peak elimination filter, 17-18 streak suppression, 19 Soft erosion, 49 Soft morphological filter, 49-52 Sourceless Maxwell equations, 135, 159 Speckle noise, 13-14 Static semiconductor permittivity, 116 Streak suppression, 19 Strip surface currents, 106- 108 Synthetic aperture radar detail enhancement, 45,46 extremum sharpening, 33, 46 max/min-median filter, 41 smoothing of speckle noise, 13-14 top-hat transformation, 64

T Tellurium, properties, 82 Toboggan algorithm, 34 Top-hat transformation, mathematical morphology, 63-64 Transformation operator, 102- 103

with additive noise or texture, 10-13 with multiplicative noise or texture, 13-14 nonlinear mean filter, 18 rank-order filtering, 14-18 adaptive quantile filter, 14-16 center-weight median filter, 17 compositc enhancement filter, 16- 17 iterative noise peak elimination filter, 17-18 streak suppression, 19 Unsharp masking, weighted, 24-25

V Variational analysis complex media, 142-145 gyroelectric-gyromagnetic, 136- 138 isotropic gyroelectric and gyromagnetic, 139-141 propagation constant, 93 Vector Helmholtz equation chiral media, 122-123, 126, 130 sourceless, 135, IS9

W Weighted median filter, 17

U Uniformity enhancement, 9- 19

219

Z Zonal filtering, 25