Computational Methods
for Electromagnetic Inverse Scattering
Computational Methods
for Electromagnetic Inverse Scattering
Xudong Chen
National University of Singapore
This edition first published 2018
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10
9
8
7
6
5
4
3
2
1
To Lin, Yuexin, Yide, and my parents.
vii
Contents
Foreword xiii
Preface xv
1
1.1
1.2
1.3
1.4
Introduction 1
Introduction to Electromagnetic Inverse Scattering Problems 1
Forward Scattering Problems 2
Properties of Inverse Scattering Problems 3
Scope of the Book 6
References 9
2
Fundamentals of Electromagnetic Wave Theory 13
2.1
2.1.1
2.1.2
2.1.3
2.1.4
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.8.1
2.8.2
2.8.3
2.8.4
2.9
2.9.1
2.9.2
2.10
Maxwell’s Equations 13
Representations in Differential Form 13
Time-Harmonic Forms 14
Boundary Conditions 15
Constitutive Relations 16
General Description of a Scattering Problem 16
Duality Principle 18
Radiation in Free Space 18
Volume Integral Equations for Dielectric Scatterers 20
Surface Integral Equations for Perfectly Conducting Scatterers
Two-Dimensional Scattering Problems 22
Scattering by Small Scatterers 24
Three-Dimensional Case 24
Two-Dimensional Case 27
Scattering by a Collection of Small Scatterers 28
Degrees of Freedom 28
Scattering by Extended Scatterers 29
Nonmagnetic Dielectric Scatterers 29
Perfectly Electrically Conducting Scatterers 31
Far-Field Approximation 32
21
viii
Contents
2.11
2.12
Reciprocity 34
Huygens’ Principle and Extinction Theorem
References 39
3
Time-Reversal Imaging 41
3.1
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
3.2
3.2.1
3.2.2
3.2.3
3.3
Time-Reversal Imaging for Active Sources 41
Explanation Based on Geometrical Optics 41
Implementation Steps 43
Fundamental Theory 45
Analysis of Resolution 48
Vectorial Wave 49
Time-Reversal Imaging for Passive Sources 53
Imaging by an Iterative Time-Reversal Process 54
Imaging by the DORT Method 55
Numerical Simulations 56
Discussions 62
References 64
4
Inverse Scattering Problems of Small Scatterers 67
4.1
4.2
4.2.1
4.2.2
Forward Problem: Foldy–Lax Equation 68
Uniqueness Theorem for the Inverse Problem 69
Inverse Source Problem 70
Inverse Scattering Problem 71
Locating Positions 72
Retrieving Scattering Strength 72
Numerical Methods 73
Multiple Signal Classification Imaging 73
Noniterative Retrieval of Scattering Strength 77
Inversion of a Vector Wave Equation 79
Forward Problem 79
Multiple Signal Classification Imaging 82
Nondegenerate Case 82
Degenerate Case 83
Noniterative Retrieval of Scattering Strength Tensors 88
Subspace Imaging Algorithm with Enhanced Resolution 90
Discussions 97
References 99
4.3
4.3.1
4.3.2
4.4
4.4.1
4.4.2
4.4.3
4.4.4
4.5
5
Linear Sampling Method 103
5.1
5.2
5.2.1
5.2.2
Outline of the Linear Sampling Method 104
Physical Interpretation 106
Source Distribution 106
Multipole Radiation 108
35
Contents
5.3
5.3.1
5.3.2
5.3.3
5.3.4
5.4
5.5
Multipole-Based Linear Sampling Method 109
Description of the Algorithm 109
Choice of the Number of Multipoles 110
Comparison with Tikhonov Regularization 113
Numerical Examples 114
Factorization Method 116
Discussions 118
References 119
6
Reconstructing Dielectric Scatterers 123
6.1
6.1.1
6.1.2
6.1.3
6.2
6.2.1
6.2.2
6.2.3
6.2.4
6.2.5
6.3
6.3.1
6.3.2
6.3.3
6.3.4
6.4
6.4.1
6.4.2
6.4.3
6.4.4
6.5
Introduction 124
Uniqueness, Stability, and Nonlinearity 124
Formulation of the Forward Problem 126
Optimization Approach to the Inverse Problem 127
Noniterative Inversion Methods 129
Born Approximation Inversion Method 130
Rytov Approximation Inversion Method 130
Extended Born Approximation Inversion Method 131
Back-Propagation Scheme 133
Numerical Examples 134
Full-Wave Iterative Inversion Methods 139
Distorted Born Iterative Method 139
Contrast Source Inversion Method 142
Contrast Source Extended Born Method 144
Other Iterative Models 146
Subspace-Based Optimization Method (SOM) 149
Gs-SOM 149
Twofold SOM 161
New Fast Fourier Transform SOM 164
SOM for the Vector Wave 169
Discussions 171
References 174
7
Reconstructing Perfect Electric Conductors 183
7.1
7.1.1
7.1.2
7.2
7.3
7.3.1
7.3.2
7.4
Introduction 183
Formulation of the Forward Problem 183
Uniqueness and Stability 184
Inversion Models Requiring Prior Information 185
Inversion Models Without Prior Information 186
Transverse-Magnetic Case 187
Transverse-Electric Case 192
Mixture of PEC and Dielectric Scatterers 196
ix
x
Contents
7.5
Discussions 202
References 203
8
Inversion for Phaseless Data 207
8.1
8.2
8.2.1
8.2.2
8.2.3
8.3
8.3.1
8.3.2
8.3.3
8.3.4
8.3.5
8.4
8.5
Introduction 207
Reconstructing Point-Like Scatterers by Subspace Methods 209
Converting a Nonlinear Problem to a Linear One 210
Rank of the Multistatic Response Matrix 212
MUSIC Localization and Noniterative Retrieval 213
Reconstructing Point-Like Scatterers by Compressive Sensing 214
Introduction to Compressive Sensing 214
Solving Phase-Available Inverse Problems by CS 215
Solving Phaseless Inverse Problems by CS 216
Applicability of CS 218
Numerical Examples 219
Reconstructing Extended Dielectric Scatterers 220
Discussions 223
References 224
9
Inversion with an Inhomogeneous Background
Medium 227
9.1
9.2
9.3
9.4
9.5
9.5.1
9.5.2
9.6
Introduction 227
Integral Equation Approach via Numerical Green’s Function 229
Differential Equation Approach 235
Homogeneous Background Approach 240
Examples of Three-Dimensional Problems 243
Confocal Laser Scanning Microscope 246
Near-Field Scanning Microwave Impedance Microscopy 249
Discussions 252
References 254
10
Resolution of Computational Imaging 257
10.1
10.2
10.2.1
10.2.2
10.3
10.4
10.5
10.6
Diffraction-Limited Imaging System 257
Computational Imaging 261
Inverse Source Problem 261
Inverse Scattering Problem 262
Cramér–Rao Bound 264
Resolution under the Born Approximation 268
Discussions 272
Summary 277
References 278
Contents
Appendices
A.1
A.2
A.3
A.3.1
A.3.2
A.3.3
A.4
A.4.1
A.4.2
A.4.3
A.4.4
A.5
281
Ill-Posed Problems 281
Regularization Theory 282
Regularization Schemes 283
Spectral Cutoff 284
Tikhonov Regularization 285
Iterative Regularization 285
Regularization Parameter Selection Methods 286
Discrepancy Principle 287
Generalized Cross Validation 287
L-Curve Method 287
Trial and Error 288
Discussions 288
B
Least Squares 291
B.1
B.1.1
B.1.2
B.2
Geometric Interpretation of Least Squares
Real Space 291
Complex Space 292
Gradient of Squared Residuals 292
C
Conjugate Gradient Method
A
Ill-Posed Problems and Regularization
291
C.1
C.1.1
C.1.2
C.2
295
Solving General Minimization Problems 295
Real Space 295
Complex Space 296
Solving Linear Equation Systems 296
D
Matrix-Vector Product by the FFT Procedure
D.1
D.2
One-Dimensional Case 299
Two-Dimensional Case 300
Appendix References 301
Index 303
299
xi
xiii
Foreword
I am thankful to Dr. Xudong Chen for asking me to write a Foreword to his
book on Computational Methods for Electromagnetic Inverse Scattering. This
book comes at an opportune time as the field of inverse scattering has been
studied for several decades now. I feel that this field is about to enter a new era,
just as the field of artificial intelligence has evolved in the last three decades.
To recount the history of artificial intelligence briefly, it started out as a field
in computer science to emulate human intelligence with computers. However,
to emulate human intelligence with the computers of three decades ago was a
tall order. Very quickly, the field evolved to a less ambitious goal of developing
expert systems to replace humans. Expert systems found applications in many
machines that can perform quasi-intelligent menial tasks for humans. When
the field of artificial neural networks was conceived, it again aroused much
excitement in the computer science community: it portended great potential
for machines to emulate the inner workings of the human brain. However, the
excitement period subsided gradually, as many of the algorithms were too slow,
and it was too difficult and time consuming to train neural networks of high
complexity. Nevertheless, neural networks re-emerged later in the new field of
machine learning. This was especially significant when machines were trained
to beat humans in a game as complicated as the ancient oriental board game go
in Japanese, or weiqi (weichi in Wade–Giles phonetics) in Chinese.
Three main reasons precipitate this breakthrough in artificial intelligence:
(1) Computers have become at least 10 million times faster in the last three
decades. (2) Computer memories are a lot cheaper compared to three decades
ago, due to the compounding effect of Moore’s Law. (3) Algorithms for information propagation through neural nets have become cleverer and faster.
Inverse scattering is facing the same juncture at this point as it shares many
similar features with artificial intelligence; for instance, one of the bottle-necks
of the inverse scattering algorithm is its computational cost or labor. But after
several decades, computer technologies have grown a lot more powerful and
cheaper. The clever use of modern computer technologies in massively parallel
computations, the use of a priori data in inverse scattering and imaging, and
xiv
Foreword
the development of compressive sensing knowledge can be the game changers
in this field. Moreover, the dogged pursuit of more efficient inverse scattering
algorithms by many researchers makes the time ripe for this field to undergo a
major revolution, as has been witnessed in the field of artificial intelligence.
Another reason that this field has become very interesting is that it is a
field that is highly inter-disciplinary, drawing upon knowledge from mathematics, wave physics, and signal processing, as well as computer science.
The confluence of various forms of knowledge and their judicious synergy are
important to stimulate the next generation of technology that can follow from
inverse scattering: for instance, in various forms of imaging, detection, and
identification applications. This book will become an excellent resource for
researchers and students who wish to learn the relevant knowledge needed
for studying inverse scattering and related topics. Dr. Chen has started from
the fundamentals of electromagnetic scattering theory and guides the readers
slowly into the advanced form of scattering and inverse scattering theory. He
also gives comprehensive coverage of the major inverse scattering techniques,
plus pertinent signal processing methods. It is pleasing to see that both perfect
electric conductor inversion and dielectric object inversion are discussed,
as well as the complicated case when the background is inhomogeneous.
Small-scatterer inversion is discussed alongside with large-scatterer inversion. The issue of phaseless imaging (or reconstruction) as well as imaging
with phase information have been discussed. Phase imaging has been done
at microwave frequency but is becoming increasingly popular at optical
frequency as optical measurements become more precise. The manner the
book is organized makes this knowledge accessible to researchers who are
not in mainstream electromagnetic physics. Also, topics are added to ease the
learning of computational mathematics and signal processing.
In summary, Dr. Chen should be lauded for spending the effort to write this
book, which will become an important resource for researchers and students
in this field.
September 2017
Weng Cho Chew
Purdue University
xv
Preface
This book is dedicated to presenting computational methods for solving
electromagnetic inverse scattering problems. The intended audience includes
graduate students and researchers in electrical engineering and physical
sciences who are interested in inverse scattering and related imaging or who
may encounter this subject in their work. Researchers in applied mathematics
might also find the book useful.
There are two main reasons that motivated me to write this monograph.
First, despite the fact that a rapidly expanding number of research articles on
inverse scattering have been published, thanks to its wide range of real-world
applications as well as the availability of powerful and cheaper computational
resources, few research textbooks have been written on the subject. In particular, there has not yet been a book dedicated to solving electromagnetic inverse
scattering problems without making linearization approximations. The lack of
a suitable reference book has been an inconvenience for many researchers who
are either in this area or are interested in entering into this subject. Second,
although progress in the research into inverse scattering would not be possible
without the confluence of various forms of knowledge, researchers in the engineering community usually have little knowledge on the theories and tools that
have been developed in the applied mathematical community. Although there
are excellent textbooks on the topic in applied mathematics, these books are
usually inaccessible to engineering readers due to a lack of sufficient training in
mathematics.
Based on my research experiences in the subject during 2006–2016, I
wrote this monograph, keeping in mind these two concerns. The book mainly
addresses inverting exact wave equations, without making linearization
approximations, which results in a highly nonlinear problem. The book is
written in such a way that it presents the following features:
1) Most of the major inversion algorithms are reviewed and, in particular, their
strengths and weakness are discussed, as well as their relationships to other
algorithms.
xvi
Preface
2) Important mathematical concepts, such as existence, uniqueness, and stability, are introduced. A general introduction to ill-posed problems and regularization is provided in the Appendix. Some inversion algorithms that
prevail in the applied mathematical community are also introduced, such as
the well-established linear sampling method. All these mathematical topics
are presented in a way accessible to engineering readers.
3) The book is highly oriented to the practical implementation of algorithms.
The details of solving the forward problem and the implementation steps of
individual inversion algorithms are presented such that readers can practice
them without a long learning curve. Along the same pragmatic direction,
several important tools are provided in Appendices.
To summarize, the book presents inverse scattering for an engineering audience in a well-balanced way; that is, emphasizing pragmatism of computational
methods but still with the right formal rigor.
Keeping in mind that the research into the inverse problem requires a deep
or fairly good understanding of the corresponding forward problem, I always
hesitate to directly apply a general optimization method to a high-dimensional
nonlinear problem, where the original forward problem is iteratively evaluated.
I am convinced that insights and intuitions, no matter whether they are mathematical, physical, or engineering, potentially help us to solve the problem in a
more efficient and elegant way. In inverse scattering problems, induced source
plays an essential role. The analysis of induced source, such as its degrees of
freedom, multipole expansion, Fourier series, and expansion with respect to
singular vectors, provides deep insights into solving inverse scattering problems, which is demonstrated throughout this book.
Supplementary materials, such as the MATLAB m-files used to generate
many of the examples and figures, can be found on my personal website.
These materials help readers make rapid progress in learning the subject and
comparing the various solution methods.
I am indebted to my Ph.D. supervisor Professor Jin Au Kong who taught
me electromagnetic wave theory and to my Masters supervisors Professor
Guangzheng Ni and Professor Shiyou Yang who introduced me to the field of
optimization and taught me the importance of physical insight. Their passion
and enthusiasm in teaching greatly influenced my view on education. I am
very grateful to Professor Weng Cho Chew who was so generous in writing the
Foreword to the book and provided me with valuable suggestions on my writing. The depth and width of his knowledge, as well as his interest in learning
whenever and wherever possible, have deeply impressed and influenced me. I
would like to thank my close collaborators Dr. Dominique Lesselier, Professor
Colin Sheppard, Professor Lixin Ran, and Professor Zhi-Xun Shen, together
with whom I worked on various inverse problems and imaging projects. I
appreciate my friendship with many mathematicians; in particular, Professor
Preface
Gunther Uhlmann, Professor Jun Zou, Professor Hongkai Zhao, Professor
Jenn-Nan Wang, and Professor Gen Nakamura, who have helped me in various
ways, taught me mathematics, and influenced my style of research.
I have been very fortunate to work with brilliant Ph.D. students and postdoctoral fellows on this subject, in particular, Yu Zhong, Krishna Agarwal, Li Pan,
Xiuzhu Ye, Rencheng Song, Rui Chen, and Zhun Wei. Dr. Zhong and Dr. Agarwal, my first two Ph.D. students, started working on inverse scattering almost
at the same time as I did. I cherish the time and effort we spent together in
embarking on a new journey in inverse scattering. Special thanks go to Dr. Wei
and Dr. Chen who generated many of the figures and provided a lot of editorial
assistance to the book. I would also like to thank Dr. Maokun Li, who read most
of chapters and provided many suggestions for improvements.
Finally, I am deeply grateful to my wife, Lin, my children, Yuexin and Yide,
and my parents, for their tremendous support, patience and love during this
project.
September 2017, Singapore
Xudong Chen
xvii
1
1
Introduction
The purpose of this chapter is to provide an overview of the book. First, the
concept of electromagnetic inverse scattering problems (ISPs) is introduced,
which is followed by their scientific and real-world applications. Second, we
address the forward scattering problem, also known as the direct problem.
Third, the fundamental properties of electromagnetic ISPs, including the
existence, uniqueness, and stability of the solution, are presented. The inherent
nonlinearity of ISPs is emphasized and the classification of ISPs is discussed.
Finally, the scope of the book is specified. The topics covered by the remaining
chapters are overviewed, which is followed by extension of the methods
presented in the book to other areas. Other related topics that are not covered
by the book are briefly mentioned.
1.1 Introduction to Electromagnetic Inverse
Scattering Problems
The electromagnetic scattering problem deals with determining the scattered
field generated by a given scatterer when it is illuminated by incoming electromagnetic waves. This is also called the forward or direct problem. The opposite
of the forward problem is called the inverse problem. Electromagnetic inverse
scattering is concerned with determining the nature of an unknown scatterer,
such as its shape, position, and material, from knowledge about measured scattered fields.
Figure 1.1 shows a schematic diagram of inverse scattering problems. An
unknown scatterer is located in the domain D, referred to as the domain of
interest (DOI), and is illuminated by incoming waves generated by transmitters
labelled Tx1, Tx2, …. For each illumination, the scattered fields are measured
by an array of receivers labelled Rx1, Rx2, …. The goal of the inverse scattering
problem is to determine the shape, position, and material of the scatterer from
the measured scattered fields.
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
2
Electromagnetic Inverse Scattering
T×1
Ei, Hi
Figure 1.1 Schematic diagram of inverse
scattering problems.
R×1
S
Es, Hs
T×2
D
R×2
T×3
R×3
Using electromagnetic waves to probe obscured or remote regions, the
imaging techniques based on electromagnetic ISPs are suitable for a wide
range of applications. For example, in nondestructive evaluation (NDE), the
ISP has been applied to detection of possible cracks in civil and industrial
structures [1–4]. In geography, this is used in remote detection of subsurface
inclusions, such as detecting unexploded ordnance and mines [5, 6]. In the oil
industry, it is used for oil and gas exploration [7]. In medicine, it is used for
the detection of the early stages of breast cancer [8–12]. In security checks, it
is applied to concealed weapon detection [13]. It can also be used for material
characterization, such as the determination of constituents and evaluation
of porosity [14]. Some real-world applications of inverse scattering in the
microwave range can be found in chapter 10 of [15]. In physical science,
the interpretation of Rutherford’s gold foil experiment, which discovered the
atomic nucleus, is also an inverse scattering problem.
From this short and incomplete list, it is apparent that the scope of electromagnetic ISP is extensive and its applications are diverse and important. Nevertheless, compared with its increasing importance, research in inverse scattering
technique is still in the nascent stage. The purpose of this book is to introduce several computational methods for solving electromagnetic ISPs. Before
discussing the inverse problem, we have to give the rudiments of the corresponding forward problem, which is the topic of the next section.
1.2 Forward Scattering Problems
Electromagnetic scattering theory is based on Maxwell’s equations. Maxwell’s
equations are four partial differential equations that describe the electric and
magnetic fields arising from distributions of electric charges and currents.
Electromagnetic scattering occurs when scatterers are illuminated by a
Introduction
radiation source. The perturbation field due to the presence of scatterers is
referred to as the scattered field; that is, the scattered field is the difference
between the fields with and without the scatterers. Since the scattering problem is formulated in an unbounded domain, the boundary condition at infinity
is called the radiation boundary condition, which requires the scattered field
to be a local plane wave that propagates outward.
Broadly speaking, scatterers can be categorized into two types: Penetrable
and impenetrable scatterers. For penetrable scatterers, the wave field is not zero
inside the scatterers and satisfies the wave equation that depends on the constitutive parameters of the scatterer. At the interface between a penetrable scatterer and the background medium, continuity of certain components of electric
and magnetic fields should be satisfied. For an impenetrable scatterer, the wave
field is zero inside the scatterer and the total field satisfies a certain boundary
condition, such as the Dirichlet (or sound-soft [16]) boundary condition, Neumann (or sound-hard [16]) boundary condition, or the impedance boundary
condition. In this book, scatterers made of nonmagnetic dielectric material are
penetrable scatterers, and scatterers made of perfect electric conductors (PEC)
are chosen for impenetrable scatterers. In solving ISPs, the values of permittivity of dielectric scatterers have to be reconstructed, whereas the boundary of
PEC scatterers has to be determined. In the applied mathematical community,
scattering problems involving penetrable and impenetrable scatterers are often
referred to as the medium and obstacle problem, respectively [16, 17].
This book deals with time-harmonic waves; that is, monochromatic
waves. We do not specify any particular frequency range; for example, radio
frequency, microwave, millimeter wave, or optical wave. Instead, we are
interested in expressing dimensions and positions in terms of wavelength.
The mathematical methods, both theoretical and numerical ones, used to
investigate the forward and inverse scattering problems depend heavily on the
operating frequency of the wave. For scatterers whose dimensions are much
larger than the wavelength, the mathematical methods used to study their
scattering phenomena are very different from those used for scatterers whose
dimensions are much smaller than, or comparable to, the wavelength. This
book is primarily concerned with the forward and inverse scattering problems
associated with the scatterers whose dimensions are much smaller than, or
comparable to, the wavelength.
The theories, formulations, and computational methods for the (forward)
scattering problem are provided in Chapter 2.
1.3 Properties of Inverse Scattering Problems
Following the definition by Hadamard [18], a problem is well posed if its
solution exists, is unique, and depends continuously on data. If one of these
conditions is not satisfied, the problem is ill- or improperly posed. It is obvious
3
4
Electromagnetic Inverse Scattering
that the first two properties, that is, the existence and uniqueness, should be
discussed when the data is noise-free. Otherwise, for example, for a given set
of measurement data that are contaminated with noise (such as measurement
error and background noise), if there is no candidate acting as an input to
the problem that produces an output exactly matching the measured data,
then the solution to the problem does not exist. The last property, referred to
as continuity (or stability), essentially means that a small perturbation of the
data results in a small perturbation of the solution. Mathematical techniques
known as regularization methods have been developed to construct a stable
approximate solution of an ill-posed problem. More details on ill-posedness
and regularization can be found in Appendix A.
For electromagnetic inverse scattering problems, we will address the following questions: the existence, uniqueness, and stability of the solution, the inherent nonlinearity, and classifications.
For electromagnetic inverse scattering problems, the question about existence is trivially confirmative since the measured scattering data for an inverse
scattering problem must be generated by a certain scatterer and obviously this
scatterer is an automatic solution to the inverse scattering problem. Turning to
the question of uniqueness, [19] and section 7.1 of [16] proved the uniqueness
theorem under certain conditions for dielectric and PEC scatterers, respectively. The conclusion for dielectric scatterers is that, under certain conditions,
for a fixed wavenumber and all directions of incidence and all polarizations of
the electric field, the knowledge of the electric far field pattern for all angles
uniquely determines permittivity. The conclusions for PEC scatterers are that
(1), for a fixed wavenumber, the electric far field patterns for all incidence direction and all polarizations uniquely determine the PEC scatterer; and (2), for
one fixed incidence direction and polarization, the electric far field pattern
for all wavenumbers contained in some interval uniquely determines the PEC
scatterer.
It is important to note that this book concentrates mainly on computational
methods that solve inverse scattering problems with a unique solution. Inverse
scattering problems that do not have a unique solution are not considered in
this book. In fact, the conditions of non-uniqueness are rather stringent, and
thus in practice such inverse scattering problems are not often encountered.
For example, for anisotropic scatterers, if the permittivity and permeability are
allowed to be zero or infinite, then it is possible to have infinite solutions to the
inverse scattering problem. One of the applications of such non-uniqueness is
invisibility and cloaking, and the idea of designing such kinds of anisotropic
scatterers is referred to as transformation optics [20, 21]. In addition, many
inverse scattering methods cannot work reliably when a penetrable isotropic
scatterer does not scatter off a certain incidence wave for certain wavenumbers. In the mathematical community, such a wavenumber is referred to as the
transmission eigenvalue of the interior transmission problem [22]. This book
Introduction
considers only scattering problems where all discrete physical and numerical
resonance frequencies are avoided [23].
Next, we turn to the question of stability. Inverse scattering problems involving dielectric or PEC scatterers cannot be stably solved. In fact, even if the
amount of data collected is sufficient to guarantee uniqueness, the unknown
parameters (either the boundaries for PEC scatterers or the values of permittivity for dielectric scatterers) do not usually depend on the measured data in
a stable way (mathematically referred to as continuous). An obvious question
to ask is how large the error of the solution could be in the worst case if the
error in the measured data is at most 𝜏. For an ill-posed problem, the error in
solution could be arbitrarily large, which means instability. In order to recover
some kind of stability, we need to restrict the space of admissible unknowns
by assuming that they satisfy a priori conditions, such as some kind of smoothness, sparseness, or non-negative constraint. With this a priori information, it is
possible to prove that the unknowns depend in a continuous way on the measured data. Determining the modulus of this continuity is referred to as the
stability estimate (section 2.2 of [24]). For electromagnetic inverse scattering
problems, it has been proven that the stability is of a logarithmic type [25, 26].
Roughly speaking, if the error in the measured data is at most 𝜏, then the error
of solution in the worst case is on the order of | ln 𝜏|−s (where 0 < s < 1). By
the L’Hôpital’s rule, as 𝜏 approaches zero, we see that a small error in measured
data leads to a much larger error in the solution.
In addition to instability, the second main difficulty of inverse scattering problems is the fact that inverse problems are nonlinear, even if the corresponding
forward problems are linear ones. The inverse scattering problem deals with
the relationship between scattered field and scatterer’s parameter, whereas the
forward scattering problem deals with the relationship between the incident
and scattered fields. The nonlinearity of the inverse problem is obvious due to
the fact that the scattered field will not be doubled when the scatterer’s permittivity is doubled. The nonlinearity is due to the multiple scattering effect that
physically exists. In addition, the nonlinearity is not a convex one. The intrinsic
nonlinearity of the inverse scattering problem makes the development of effective algorithms a difficult task because a solution procedure can get trapped in
false solutions that are in fact different from the exact one.
Since multiple scattering effects physically exist, any imaging algorithm
that ignores multiple scattering effects will cause an error, which is hard or
impossible to remove using simple post-processing methods. Thus, numerical
reconstruction that takes multiple scattering effects into account is expected
to be one of the main research directions for the inverse problem community
in the near future. Usually, the nonlinear problem is solved by casting it into
an optimization problem, where the mismatch between the measured and
predicted data is minimized by adjusting the unknowns that are used for
prediction. The bottleneck of reconstruction algorithms that take multiple
5
6
Electromagnetic Inverse Scattering
scattering effects into account lies in the high computational cost for solving
the associated forward problem conducted at each iteration of the optimization process. However, the enormous increase in computing power of modern
computers and the development of powerful inversion algorithms are expected
to make it possible to reconstruct objects within the framework of multiple
scattering in many real-world applications in the near future.
Electromagnetic inverse scattering problems can be classified by different
criteria. Depending on the material properties of scatterers, the ISPs can be
classified into penetrable and impenetrable ones. Depending on the size of
scatterers, in comparison with the wavelength, the ISPs can be classified into
two types involving small scatterers (also known as point-like scatterers) and
extended scatterers. Depending on the property of background medium, classification can be homogenous and inhomogeneous background ISPs. In terms
of availability of phase information of measured scattered fields, classification
can be ISPs with phaseless data or phase-available data. From the angle of
inversion algorithms, classification can be linear and nonlinear, iterative and
noniterative, or quantitative and qualitative. When multiple scattering is taken
into account, the ISP is inherently nonlinear, and consequently any inversion
algorithm that directly solves the nonlinear problem is called a full-wave
nonlinear algorithm. Under some conditions, such as scatterers being weak in
scattering, the nonlinear problem can be well approximated by a linear one,
and the accompanying algorithms are linear ones. Certain inversion algorithms
are able to provide accurate or good approximate solutions with one or only
a few steps of manipulation, whereas some algorithms provide final reconstruction results by iteratively minimizing the mismatch between measured
and predicted data. In practical applications, the objectives of ISPs can be very
different. If the values of permittivity are needed, then quantitative inversion
algorithms should be adopted. If only approximate information on the shape,
position, and size of scatterer is needed, then qualitative inversion algorithms
suffice.
1.4 Scope of the Book
The main purpose of this book is to introduce computational methods for solving electromagnetic inverse scattering problems. The remaining chapters cover
the following topics regarding forward and inverse electromagnetic scattering
problems.
Chapter 2 gives the rudiments of the theory behind the forward scattering
problem. The fundamentals of electromagnetic wave theory are reviewed, and
equations are developed for scattering problems involving dielectric scatterers
and perfect electric conductors. The chapter aims to provide readers with some
quick practice in solving forward scattering problems.
Introduction
Chapters 3–5 present qualitative inversion algorithms, which include
time-reversal imaging, the multiple signal classification (MUSIC) method, and
the linear sampling method. These inversion algorithms do not provide the
values of permittivity of scatterers, but instead provide indicators that show
the possibility of the existence of scatterers at particular spatial points from
which the size, shape, and position of scatterers can be inferred. In particular,
Chapter 4 shows that the MUSIC theoretically reaches an unlimited resolution
for point-like scatterers in absence of noise and, once the position is determined, the scattering strengths of point-like scatterers can be noniteratively
retrieved.
Chapters 6–9 address quantitative inversion algorithms. The focus of these
chapters is full-wave nonlinear ISPs where the multiple scattering effect is
taken into account. Chapter 6 deals with reconstructing dielectric scatterers
where both iterative and noniterative algorithms are introduced. Chapter 7
addresses the ISPs involving PEC scatterers, either PEC scatterers alone or a
mixture of PEC scatterer and dielectric scatterers. Chapter 8 covers inversion
algorithms for phaseless data. Chapter 9 deals with inverse scattering problems
that have an inhomogeneous background medium, which is general enough
to model most real-world applications. In these chapters, although an optimization approach is used to solve full-wave nonlinear ISPs, the main focus
is not optimization algorithms, but instead the application of mathematical
and physical insights to the development of objective functions with reduced
nonlinearity.
Chapter 10 discusses the resolution of an image that is obtained by solving
inverse scattering problems. An important conclusion is that the classical resolution widely used in traditional optical microscopy is not applicable to inverse
scattering problems.
Four Appendices are provided at the end of the book. Appendix A is a
short introduction to the mathematical theories of inverse problems, mainly
ill-posedness and regularization. The text is oriented to engineering readers.
This topic has been deliberately chosen for placement in the Appendices, since
otherwise it might disrupt the flow of engineering readers if it was placed in
the main body. The remaining three appendices are pragmatic and concise.
Appendices B and C outline two widely used optimization methods; that is,
the linear least squares method and the conjugate gradient method. Appendix
D presents the fast implementation of the convolution-type matrix-vector
product. We mention in passing that, although not provided in the appendix,
matrix analysis is important for analyzing and implementing inversion algorithms throughout the book. Readers are referred to [27, 28] for information
about matrix analysis.
It is worth stressing that the concept of induced source is used throughout
the book. Induced source plays an important role in solving electromagnetic
inverse scattering problems since it provides a bridge to the inverse source
7
8
Electromagnetic Inverse Scattering
problem, a linear problem. The analysis of induced source, such as its degrees
of freedom, multipole expansion, Fourier series, and expansion with respect
to singular vectors, provides deep insights into solving inverse scattering problems. In fact, inverse scattering problems can be viewed as inverse source problems subject to constraints imposed by the material properties of a scatterer
when it is under multiple illumination sources.
For all presented inversion algorithms, numerical examples are provided
to test their performances. In addition to synthetic data, some lab-controlled
experimental data are used to test some of the proposed inversion algorithms.
Whenever possible, the last section of each chapter lists relevant references
that test inversion algorithms with lab-controlled experimental data or even
real-world field data.
The computational methods introduced in this book, used for solving
electromagnetic ISPs, can be extended to solve many other inverse problems,
such as acoustic ISPs [29], seismic exploration [30], inversion of elastic
wave equation [31], quantum ISPs (chapter 20 of [32]), inverse transport
equation [33, 34], optical diffusion tomography [35], and electric impedance
tomography [36].
Since electromagnetic ISP is a multidisciplinary topic, it is impossible to cover
all relevant topics in a single volume. I have chosen the topics in electromagnetic inverse scattering problems from the perspective of my own research
experiences. This inevitably means that certain areas of electromagnetic ISP are
either given only cursory attention or completely ignored. First, I’d like to mention some books that primarily focus on electromagnetic ISPs. Theories and
numerics on fast linear methods, such as diffraction tomography, are extensively discussed in [37–39]. Instrumentation and apparatuses for microwave
imaging, as well as layouts of transmitters/receivers, can be found in [15, 38].
The inverse source problem is addressed in great detail in chapter 5 of [37].
Theories and examples for one-dimensional electromagnetic ISPs are given in
[40–42] and section 9.2 of [43]; Various optimization methods for solving electromagnetic ISPs are presented in [44]. The test on lab-controlled experimental
data is presented in chapter 9 of [39]. Rigorous mathematical treatments of
the electromagnetic ISP, which is nevertheless less pragmatic, can be found
in [16]. This book focuses on full-wave nonlinear inversion methods and is
complementary to the aforementioned books. Next, I’d like to mention some
aspects of electromagnetic ISPs that are not covered by this book. Time domain
inversion algorithms can be found in [45–49]. Hybrid imaging modalities, such
photoacoustic or thermoacoustic tomography, can be found in [50–53]. Inverse
problems for scattering by periodic structures can be found in [54]. Electromagnetic ISPs can be tackled by the Bayesian approach [55] or along the same
lines, the “combination of states of information” approach [56]. Machine learning approaches, such as support vector machines and neural networks, have
been applied to solve electromagnetic ISPs [57, 58].
Introduction
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13
2
Fundamentals of Electromagnetic Wave Theory
Before discussing the inverse scattering problem we have to give the rudiments
of the theory behind the corresponding forward problem. The electromagnetic
scattering problem is based on Maxwell’s equations. The purpose of this
chapter is to review the fundamentals of electromagnetic wave theory. In
addition, equations are developed for scattering problems involving dielectric
scatterers and perfect electric conductors (PEC). The presentation is intended
as a review of the aforementioned concepts rather than an introduction, and
an in-depth treatment can be found in references [1–7]. The presentation
materials in this chapter are highly oriented to the following two aspects:
(1) numerical implementation of forward problem solvers; and (2) theories
and formulas that can be directly used in later chapters. The details of linear
equation systems for some types of forward problems are provided, which
should facilitate readers’ quick practice and understanding of the numerics of
the forward problem.
2.1 Maxwell’s Equations
2.1.1
Representations in Differential Form
Maxwell’s equations govern electromagnetic phenomena. In three-dimensional
vector notation, Maxwell’s equations are
𝜕
(2.1)
∇ × H(r, t) = D(r, t) + J(r, t)
𝜕t
𝜕
∇ × E(r, t) = − B(r, t)
(2.2)
𝜕t
∇ ⋅ D(r, t) = 𝜌(r, t)
(2.3)
∇ ⋅ B(r, t) = 0
(2.4)
where E, D, B, H, J, and 𝜌 are real-valued functions of space and time. E is the
electric field strength, D is the electric flux density (also known as the electric
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
14
Electromagnetic Inverse Scattering
displacement), B is the magnetic flux density, H is the magnetic field strength,
J is the electric current density, and 𝜌 is the electric charge density. The units
for these physical parameters are as follows: volt/m for E, coulomb/m2 for D,
weber/m2 for B, ampere/m for H, ampere/m2 for J, and coulomb/m3 for 𝜌.
Equation (2.1) is Ampere’s law, (2.2) is Faraday’s law, (2.3) is Gauss’ law for
electric fields, and (2.4) is Gauss’ law for magnetic fields. The electric current
density J and the electric charge density 𝜌 are governed by the continuity law,
𝜕
𝜌(r, t)
(2.5)
𝜕t
which describes the transport of electric charge that is a conserved quantity;
that is, the divergence of current J from an infinitesimal volume surrounding r
is equal to the decreasing of the electric charge density 𝜌 with time t.
∇ ⋅ J(r, t) = −
2.1.2
Time-Harmonic Forms
For electromagnetic waves of a particular frequency in the steady state, the
fields are time-harmonic and are known as monochromatic waves or continuous waves. For time-harmonic fields, it is mathematically convenient to use
their complex representations, which are also known as phasors. For an angular
frequency 𝜔, the complex representation E(r) of the time-domain field E(r, t)
is defined to satisfy
E(r, t) = [E(r)e−i𝜔t ]
(2.6)
where [⋅] denotes the real part of a complex quantity. The complex-valued
E(r) is a function of space only and independent of time. Note that e−i𝜔t
denotes the convention of time dependence adopted in this book. Under
this time convention, the plane wave E(x) = ŷ E0 eikx propagates in the +x
direction, which can be seen from the fact that when time t increases, x has to
increase so that the time-domain expression E(r, t) = ŷ E0 cos(kx − 𝜔t) retains
a fixed phase.
Under complex representation, it is easy to verify that Maxwell’s equations
read
∇ × H(r) = −i𝜔D(r) + J(r)
(2.7)
∇ × E(r) = i𝜔B(r)
(2.8)
∇ ⋅ D(r) = 𝜌(r)
(2.9)
∇ ⋅ B(r) = 0
(2.10)
where the time derivative is replaced by the factor −i𝜔. The continuity law (2.5)
becomes
∇ ⋅ J(r) = i𝜔𝜌(r)
(2.11)
Fundamentals of Electromagnetic Wave Theory
2.1.3
Boundary Conditions
The time-harmonic Maxwell’s equations (2.7)–(2.10) are differential forms
applied locally at each point in space. By means of the divergence theorem and
Stokes’ theorem, they can be cast in integral form. At the boundary of two
different media, the integral-form Maxwell’s equations can be used to deduce
the boundary conditions. Consider two media that are shown in Fig. 2.1, with
the normal n̂ of the boundary pointing from medium 1 into medium 2. For
an infinitesimal closed contour C with the long arms on either side of the
boundary, as shown in Fig. 2.1, the application of Stokes’ theorem to (2.7) and
(2.8) gives the following boundary conditions, in the limit of zero length of the
short arms of C,
n̂ × (H2 − H1 ) = Js
(2.12)
n̂ × (E2 − E1 ) = 0
(2.13)
where the subscripts 1 and 2 denote fields in medium 1 and 2, respectively, and
Js is the surface current density with the unit ampere/m. For an infinitesimal
pillbox V straddling the boundary, as shown in Fig. 2.1, the application of the
divergence theorem to (2.9) and (2.10) gives the following boundary conditions,
in the limit of zero height of the pillbox,
n̂ ⋅ (D2 − D1 ) = 𝜌s
(2.14)
n̂ ⋅ (B2 − B1 ) = 0
(2.15)
where 𝜌s is the surface charge density, with the unit coulomb/m2 .
It is important to note that the boundary conditions (2.12) and (2.13) automatically imply (2.14) and (2.15), respectively. This is because such a dependence relationship roots from the corresponding differential forms of Maxwell’s
equations. Indeed, the divergence of Ampere’s law (2.7), together with the vector identity ∇ ⋅ ∇ × H = 0 and the continuity law (2.11), implies Gauss’ law for
Figure 2.1 Schematic
diagram of a boundary
between two different
media. The infinitesimal
loop and pillbox are used
to derive boundary
conditions.
2
1
C
n
E2, B2
D2, H2
ρs, Js
V
E1, B1
D1, H1
15
16
Electromagnetic Inverse Scattering
electric fields (2.9). Similarly, Faraday’s law (2.8) automatically implies Gauss’
law for magnetic fields (2.10). Thus, it suffices to apply only two boundary conditions (2.12) and (2.13) when solving electromagnetic wave problems.
2.1.4
Constitutive Relations
The electric and magnetic flux densities are related to the electric and magnetic
field strengths via constitutive relations. The constitutive relations characterize
the medium that is under study. The simplest constitutive relations are for a
linear and isotropic medium,
D = 𝜖E
(2.16)
B = 𝜇H
(2.17)
where the scalars 𝜖 and 𝜇 are the permittivity and permeability of the medium,
respectively. In free space, 𝜖 = 𝜖0 ≈ 8.85 × 10−12 farad/m and 𝜇 = 𝜇0 = 4𝜋 ×
10−7 henry/m. For a material, if its permeability is equal to 𝜇0 , then it is called
nonmagnetic. A more complex constitutive relation is for a linear anisotropic
medium, where 𝜖 and 𝜇 are 3 × 3 matrices. Other complex constitutive relations
are beyond the scope of this book.
It is worth mentioning the conducting medium, which is characterized by
J = 𝜎E, where 𝜎 denotes the conductivity. The right-hand side of (2.7) can be
rewritten as −i𝜔(𝜖 + i𝜎∕𝜔)E. Thus, we can define 𝜖 + i𝜎∕𝜔 as the complex permittivity and simply write it as complex-valued 𝜖, without arousing ambiguity.
We often define relative permittivity 𝜖r = 𝜖∕𝜖0 , which is consequently also complex valued. Thus, both insulating material (𝜎 = 0) and conducting material
(𝜎 ≠ 0) can be represented by relative permittivity. Since relative permittivity is
often referred to as the dielectric constant, we use the term “dielectric material”
to describe both insulating and conducting materials.
For a perfect electric conductor (PEC), 𝜎 approaches infinity and thus PEC
can be treated as a special dielectric material.
2.2 General Description of a Scattering Problem
This section and the next three sections describe the electromagnetic scattering
problem involving dielectric scatterers. The topic of perfect electric conductor will be discussed in Section 2.6. Consider a dielectric scatterer, as shown
in Fig. 1.1, that is located otherwise in a homogenous background medium.
For simplicity, let the homogenous background medium be free space (or more
realistically, air). The scatterer is illuminated by a primary source, also known as
the active source, located somewhere outside the scatterer. The fields E and H
can be decomposed into two parts due to the principle of linear superposition.
The first part is incident fields, Ei and Hi , generated by the primary source in the
Fundamentals of Electromagnetic Wave Theory
absence of the scatterer. The second part is scattered fields, Es and Hs , which
are due to the presence of the scatterer. The fields in the immediate vicinity of
the scatterer satisfy Maxwell’s equations in the presence of the scatterer,
∇ × H = −i𝜔𝜖E
(2.18)
∇ × E = i𝜔𝜇H
(2.19)
The incident fields in the immediate vicinity of the scatterer satisfies Maxwell’s
equations in the absence of the scatterer,
∇ × Hi = −i𝜔𝜖0 Ei
i
∇ × E = i𝜔𝜇0 H
i
(2.20)
i
s
(2.21)
i
s
Since E = E + E and H = H + H , we find that (2.18)–(2.21) lead to
∇ × Hs = −i𝜔𝜖0 Es − i𝜔(𝜖 − 𝜖0 )E
(2.22)
∇ × E = i𝜔𝜇0 H − [−i𝜔(𝜇 − 𝜇0 )H]
(2.23)
s
s
By comparing (2.22) and (2.7), we find the last term in (2.22) plays the role of
electric current density,
J = −i𝜔(𝜖 − 𝜖0 )E
(2.24)
The last term in (2.23) is mathematically equivalent to the magnetic current
density,
K = −i𝜔(𝜇 − 𝜇0 )H
(2.25)
Note that K does not physically exist, but instead is a concept of mathematical
equivalence. The introduction of the magnetic current density makes Maxwell’s
equation symmetric, which will be discussed in the next section. The electric
current density (2.24) and the magnetic current density (2.25) are referred to
as secondary source, induced source, or passive source. Equations (2.22) and
(2.23) state that the scattered fields can be regarded as the radiated fields by
secondary sources in free space. The task of finding radiation fields in free space
will be presented in Section 2.4.
Since the scatterer is distributed in a finite region, radiating in an unbounded
space, boundary conditions must be imposed at infinity to ensure a unique solution. Such boundary conditions are called the radiation boundary conditions
and require outgoing scattered fields. In a three-dimensional problem, where
r = r̂r is the conventional spherical coordinate variable, radiation conditions
take the form
lim r(Es + r̂ × 𝜂0 Hs ) = 0
r→∞
(2.26)
(2.27)
lim r(H − r̂ × E ∕𝜂0 ) = 0
√𝜇
0
, with the unit ohm, is defined as the intrinsic impedance of
where 𝜂0 =
𝜖
s
s
r→∞
0
free space.
17
18
Electromagnetic Inverse Scattering
2.3 Duality Principle
Before formulating the radiation fields in free space by an electric current or
magnetic current, it is worth introducing the duality principle. After the introduction of the mathematical-equivalence concept of magnetic current density, Maxwell’s equations exhibit almost prefect symmetry, and consequently
any equation describing electric or magnetic fields and sources can be used
to directly arrive at a dual equation describing the complementary fields and
sources. For example, under the following replacements,
E → H,
H → −E,
J → K,
K → −J,
𝜖 → 𝜇,
𝜇→𝜖
(2.28)
Ampere’s law and Faraday’s law are a dual pair and one can be derived from the
other,
∇ × H = −i𝜔𝜖E + J
(2.29)
∇ × E = i𝜔𝜇H − K
(2.30)
The constitutive relations (2.16) and (2.17) are also a dual pair and one can be
derived from the other by this replacement. Since the replacements listed in
(2.28) are sufficient within the scope of this book, other replacements, such as
charge density, will not be discussed.
2.4 Radiation in Free Space
As presented in Section 2.2, scattered fields can be regarded as the radiated
fields in free space by induced electric and magnetic currents. This section is
devoted to the task of finding radiation fields in free space.
Due to the linear superposition principle, it is sufficient to separately consider
the radiating fields due to electric current and magnetic current. First, we consider the case where only electric current exists in free space. From Maxwell’s
equations,
∇ × H(r) = −i𝜔𝜖0 E(r) + J(r)
(2.31)
∇ × E(r) = i𝜔𝜇0 H(r)
(2.32)
we eliminate H, by taking the curl of (2.32) and then using (2.31), and obtain
the vector wave equation,
(2.33)
∇ × ∇ × E(r) − k02 E(r) = i𝜔𝜇0 J(r),
√
where k0 = 𝜔 𝜇0 𝜖0 is the wavenumber in free space. The unit of k0 is rad/m,
and it can be proven that k0 = 2𝜋∕𝜆0 , where 𝜆0 is the wavelength in free space.
For an arbitrary electric current distribution J(r), it is convenient to calculate
Fundamentals of Electromagnetic Wave Theory
its radiation fields by using the concept of the dyadic Green’s function G(r, r′ ),
which is defined to satisfy
∇ × ∇ × G(r, r′ ) − k02 G(r, r′ ) = I𝛿(r − r′ )
(2.34)
where I is a unit dyad that can be represented by a unit diagonal matrix. The
operation of I on any vector yields the vector itself. Roughly speaking, the
dyadic Green’s function can be treated as a vector version impulse response
function. Through the superposition principle for linear problems, the radiation fields by the current distribution J(r) can be formulated as the convolution
of the dyadic Green’s function with J(r),
E(r) = i𝜔𝜇0
∫∫∫
G(r, r′ ) ⋅ J(r′ )dr′
The explicit expression of the dyadic Green’s function is given by
[
]
1
′
G(r, r ) = I + 2 ∇∇ g(r, r′ )
k0
(2.35)
(2.36)
where the scalar Green’s function g(r, r′ ) satisfies the following differential
equation,
(∇2 + k02 )g(r, r′ ) = −𝛿(r − r′ )
(2.37)
The scalar Green’s function can be treated as an impulse response function, and
its explicit expression is given by
eik0 |r−r |
4𝜋|r − r′ |
′
g(r, r′ ) =
(2.38)
The uniqueness of the solution to (2.37) is ensured by the radiation condition,
also known as the Sommerfeld radiation condition,
[
]
𝜕g(r, r′ )
′
lim r
(2.39)
− ik0 g(r, r ) = 0
r→∞
𝜕r
which represents an outgoing wave.
For convenience to use in later chapters, the closed-form elements of the
3 × 3 matrix G(r, r′ ) are provided here,
)
[(
i
1
𝛿u,𝑣
1+
−
Gu𝑣 (r, r′ ) =g(r, r′ )
k0 R k02 R2
(
)]
Ru R𝑣
3i
3
1+
(2.40)
− 2
−
R
k0 R k02 R2
where u, 𝑣 = 1, 2, 3, R = r − r′ and R = |R|.
19
20
Electromagnetic Inverse Scattering
Once the radiated electric field is obtained from (2.35), the magnetic field is
consequently deduced from (2.32),
H(r) = ∇ ×
∫∫∫
=
∫∫∫
=
∫∫∫
G(r, r′ ) ⋅ J(r′ )dr′
(2.41)
∇ × [Ig(r, r′ ) ⋅ J(r′ )]dr′
[∇g(r, r′ ) × I] ⋅ J(r′ )dr′
where the vector identity ∇ × ∇f = 0 for an arbitrary scalar function f is used.
Note that r and r′ are different coordinate systems, so that the ∇× operator
can be moved inside the integral. For convenience to use in later chapters, the
closed-form elements of the 3 × 3 matrix ∇ × G(r, r′ ), equivalently ∇g(r, r′ ) × I,
is provided here,
0
⎡
⎢ (z − z′ )
′
∇ × G(r, r ) = ⎢
−(y − y′ )
⎢
⎣
−(z − z′ ) (y − y′ ) ⎤
(
)
ik0
0
−(x − x′ ) ⎥
1
′
)
−
g(r,
r
⎥
(x − x′ )
0
R
R2
⎥
⎦
(2.42)
Next, we consider the radiation fields when only the magnetic current exists
in free space. The answer can be found by duality relationships. The dual
equations to (2.35) and (2.41) are, respectively,
H(r) = i𝜔𝜖0
E(r) =
∫∫∫
∫∫∫
G(r, r′ ) ⋅ K(r′ )dr′
[−∇ × G(r, r′ )] ⋅ K(r′ )dr′
(2.43)
(2.44)
Now we have found the expressions of the radiation fields in free space by electric and magnetic currents.
2.5 Volume Integral Equations for Dielectric Scatterers
From the relationship between the scattered fields and induced currents, (2.22)
and (2.23), we are able to directly arrive at the following expressions of the scattered fields by using the concept of dyadic Green’s function that was introduced
in the previous section 2.4,
E(r) − Ei (r) = i𝜔𝜇0
+
∫∫∫
∫∫∫
G(r, r′ ) ⋅ J(r′ )dr′
[−∇ × G(r, r′ )] ⋅ K(r′ )dr′
(2.45)
Fundamentals of Electromagnetic Wave Theory
H(r) − Hi (r) = i𝜔𝜖0
+
∫∫∫
∫∫∫
G(r, r′ ) ⋅ K(r′ )dr′
[∇ × G(r, r′ )] ⋅ J(r′ )dr′
(2.46)
It is important to note that the right-hand sides cannot be calculated yet since
the induced electric current density (2.24) and magnetic current density (2.25)
depend on the electric field E and magnetic field H, respectively, which are the
unknowns to be determined, however. Instead, we should treat (2.45) and (2.46)
as volume integral equations for the unknowns E(r) and H(r) where r is inside
scatterer. Equations (2.45) and (2.46) are referred to as the electric field integral
equation (EFIE) and the magetnic field integral equation (MFIE), respectively.
In the special case in which the scatterer is nonmagnetic, that is, 𝜇 = 𝜇0 , then
the induced magnetic current vanishes, leaving EFIE and MFIE as
E(r) − Ei (r) = i𝜔𝜇0
H(r) − Hi (r) =
G(r, r′ ) ⋅ J(r′ )dr′
(2.47)
[∇ × G(r, r′ )] ⋅ J(r′ )dr′
(2.48)
∫∫∫
∫∫∫
respectively. It is obvious that the EFIE (2.47) involves only E as the unknown
since J = −i𝜔(𝜖 − 𝜖0 )E. By the relationship of E and H that is shown in (2.18),
we have J = (1 − 1∕𝜖r )∇ × H so that the MFIE (2.48) can be rewritten such that
it involves only H as the unknown.
2.6 Surface Integral Equations for Perfectly
Conducting Scatterers
After presenting the scattering problem for dielectric scatterers, we turn to
the scattering problem for scatterers made of perfectly electrically conducting
materials. Consider a PEC scatterer, occupying a space D in free space, with
the closed boundary 𝜕D that is illuminated by a primary source. Since electromagnetic waves cannot penetrate PEC materials, the electric and magnetic
fields inside the PEC scatterer are identically zero. Consequently, the boundary
conditions (2.12) and (2.13) reduce to
n̂ × H = Js
(2.49)
n̂ × E = 0
(2.50)
where the normal direction n̂ points from the PEC scatterer into free space.
Thus, the surface electric current with density Js is induced on the PEC boundary, and there is no induced surface magnetic current.
21
22
Electromagnetic Inverse Scattering
Following the same procedure presented in Section 2.5, we can derive the
following surface integral equations from (2.49) and (2.50),
{
}
[∇ × G(r, r′ )] ⋅ Js (r′ )dr′ = Js
(2.51)
n̂ × Hi +
∫ ∫𝜕D+
{
}
n̂ × Ei + i𝜔𝜇0
G(r, r′ ) ⋅ Js (r′ )dr′ = 0
(2.52)
∫ ∫𝜕D
where the 𝜕D+ denotes an surface at an infinitesimal distance outside of the
scatterer surface 𝜕D. Equations (2.52) and (2.51) are referred to as the surface
EFIE and surface MFIE, respectively. The uniqueness of the solution to surface
EFIE and surface MFIE is discussed in detail in chapter 6 of [4]. We mention
in passing that for infinitesimally thin open PEC scatterers, the original boundary condition (2.12) should be used since either side of PEC scatterers is not
necessarily zero.
2.7 Two-Dimensional Scattering Problems
The electromagnetic radiation and scattering problems discussed in
Sections 2.4–2.6 are general three-dimensional problems. A special type
of three-dimensional problem is those with all physical parameters invariant
in one direction. Throughout this text, we let the direction of invariance be
in the z direction, and we use the term “cylinder” to denote structures whose
parameters are invariant along the z-axis. All physical parameters are functions
of x and y. Thus, this special physically three-dimensional problem is referred
to as a mathematically two-dimensional problem.
All conclusions drawn in Sections 2.4–2.6 can be directly, or after minor
modifications, applied to two-dimensional problems. Since 𝜕∕𝜕z = 0, the ∇
becomes 𝜕∕𝜕xx̂ + 𝜕∕𝜕ŷy, which can be written as ∇t in shorthand, where
the subscript t means “transverse.” Since the source is also invariant along
the z-axis, the integration of the three-dimensional scalar Green’s function
(2.38) along the whole z-axis leads to the two-dimensional scalar Green’s
function,
i
(2.53)
g(r, r′ ) = H0(1) (k0 |r − r′ |)
4
where H0(1) (⋅) is the zeroth order Hankel function of the first kind.
It is usually convenient to decompose the fields into transverse electric
(TE) and transverse magnetic (TM) parts. Whether a field is a TE or TM case
depends on whether an electric or magnetic field is transverse to a chosen
reference. The reference chosen here is the z-axis, and consequently the TE
case means that the electric field is transverse to the z-axis, that is, the z
Fundamentals of Electromagnetic Wave Theory
component of the electric field is absent, whereas the TM case means that the
magnetic field is transverse to the z-axis.
Before solving scattering problems, it is important to first discuss radiation
problems. The z component of the electric current density J generates the TM
component, and the transverse components of J generate the TE components.
Consequently, (2.35) is decomposed into
g(r, r′ )̂zẑ ⋅ J(r′ )dr′
]
[
1
I 2 + 2 ∇t ∇t g(r, r′ ) ⋅ [I 2 ⋅ J(r′ )]dr′
ETE (r) = i𝜔𝜇0
∫∫
k0
ETM (r) = i𝜔𝜇0
∫∫
(2.54)
(2.55)
and (2.41) is decomposed into
HTM (r) =
HTE (r) =
∫∫
∫∫
∇t g(r, r′ ) × [̂zẑ ⋅ J(r′ )]dr′
(2.56)
∇t g(r, r′ ) × [I 2 ⋅ J(r′ )]dr′
(2.57)
We observe that, for the TM case, the electric field only has the z component,
whereas the magnetic field only has the transverse component. For the TE case,
the magnetic field only has the z component, whereas the electric field only has
the transverse component. For radiation by magnetic current density K, the
electric and magnetic fields can be obtained via the duality relationship.
Now we consider a nonmagnetic dielectric 2D scatterer. For the TM case,
the induced electric current is along the z direction and there is no magnetic
current. Consequently, the EFIE (2.47) has only the z component and the corresponding scalar equation is
Ez (r) − Ezi (r) = k02
∫∫
g(r, r′ )[𝜖r (r′ ) − 1]Ez (r′ )dr′
(2.58)
The MFIE is seldom used to solve the TM scattering problem. For the TE case,
the induced electric current is in the transverse plane and there is no magnetic current. The vectorial property of the induced electric current makes the
equation more complex than (2.58) no matter whether the EFIE or the MFIE
is used. For the EFIE, we extract the x and y components of (2.47) where the
right-hand side is given by (2.55). For the MFIE, we extract the z component of
(2.48), where the right-hand side is given by (2.57).
The above discussions about the nonmagnetic dielectric scatterer are also
applicable to the PEC cylinder scattering problem. It is worth noting that, in
the TE case, the induced transverse electric current follows on the surface of
the PEC cylinder, which consequently means that the current flow direction
is perpendicular to both the z and the n̂ directions; that is, tangential to the
surface of the PEC scatterer.
23
24
Electromagnetic Inverse Scattering
Finally, we mention in passing that, in the two-dimensional case, the Sommerfeld radiation conditions for scattered fields are given by
( s
)
√
𝜕Ez
lim 𝜌
(2.59)
− ik0 Ezs = 0
𝜌→∞
𝜕𝜌
( s
)
√
𝜕Hz
lim 𝜌
(2.60)
− ik0 Hzs = 0
𝜌→∞
𝜕𝜌
for the TM and TE cases, respectively, where 𝜌 is the radial variable in cylindrical coordinates.
2.8 Scattering by Small Scatterers
It is important to analyze scattering by small particles, which is often referred
to as Rayleigh scattering. When the size of a scatterer is much smaller than
the wavelength, then the scattering problem is in the quasistatic regime and
the scattered field can be well approximated by the lowest orders of multipole
expansion.
2.8.1
Three-Dimensional Case
For dielectric small particles, Rayleigh scattering results from the electric polarizability of the particles. The oscillating electric field of incident wave acts on
the charges within a particle, causing them to move at the same frequency. The
particle therefore becomes a small radiating electric current dipole that radiates
what we see as a scattered field. The induced electric current dipole Il depends
on the permittivity, size, and shape of the small particle, and is related to the
incident electric field E by
Il = 𝜉 ⋅ E
(2.61)
where 𝜉 is in general a 3 × 3 matrix, which is called the electric polarization
tenor, the polarization strength, or the depolarization factor. Note that the incident electric field E (as well as the incident magnetic field H) is nearly constant
over the particle’s volume. For example, for a small spherical particle of radius
a, with isotropic permittivity 𝜖, the electric polarization tenor can be solved by
electrostatic principles (section 6.1 of [1]),
1 𝜖 − 𝜖0
𝜉 = −i4𝜋k0 a3
I ,
(2.62)
𝜂0 𝜖 + 2𝜖0 3
where I 3 is a three-dimensional identity matrix. That is, the electric polarization
tenor is a scalar for an isotropic sphere. The scattered fields Es (r) and Hs (r) at
Fundamentals of Electromagnetic Wave Theory
position r, which are the reradiated fields by the induced electric current dipole
at position r0 , have the analytical expressions (section 4.3 of [1]),
)
)]
[(
(
RR
i
3i
1
3
s
Il − 2 ⋅ Il 1 +
E (r) = i𝜔𝜇0 g(r, r0 ) 1 +
−
−
k0 R k02 R2
R
k0 R k02 R2
(2.63)
(
)
ik0
1
(2.64)
− 2
Hs (r) = R × Il g(r, r0 )
R
R
where R = r − r0 and R = |R|. In fact, these reradiated fields can be also derived
from the general radiation formula given in Section 2.4. Since the size of scatterer is much smaller than the wavelength, we can treat it centered at r0 and
consequently the integral (2.35) and (2.41) are carried out over a small volume
V0 centered at r0 . When the observation point r is sufficiently far, compared
with the size of the scatterer, from the scatterer, (2.35) and (2.41) are simplified
to be
Es (r) = i𝜔𝜇0 G(r, r0 ) ⋅
Hs (r) = ∇g(r, r0 ) ×
∫ ∫ ∫V0
∫ ∫ ∫V0
J(r′ )dr′
J(r′ )dr′
(2.65)
(2.66)
respectively. By comparing (2.63–2.66) with the closed-form expressions of
G(r, r0 ) (shown in (2.40)) and ∇g(r, r0 )× (shown in (2.42)), we find that the
electric current dipole moment is in fact defined as
Il =
∫ ∫ ∫V 0
J(r′ )dr′
(2.67)
which has the unit ampere ⋅ meter. Consequently, the unit of the electric polarization tensor 𝜉, defined in (2.61), is meter square per ohm.
For magnetic small particles, magnetic current dipoles will be induced by a
time-harmonic magnetic field due to the magnetization process in the material
associated with the atomic scale current loops. The induced magnetic current
dipole Kl is related to the incident magnetic field H by the magnetic polarization tensor 𝜁,
Kl = 𝜁 ⋅ H
(2.68)
The unit of Kl is volt ⋅ meter and the unit of 𝜁 is meter square ⋅ ohm. For
example, for a small spherical particle of radius a, with isotropic permeability
𝜇, the magnetic polarization tenor can be solved by magnetostatic principles
25
26
Electromagnetic Inverse Scattering
(sections 5.10 and 5.11 of [6]),
𝜇 − 𝜇0
𝜁 = −i4𝜋k0 a3 𝜂0
I ,
𝜇 + 2𝜇0 3
(2.69)
which is in fact the dual of (2.62). The scattered electric and magnetic fields,
which are the reradiated fields by the induced magnetic current dipole, can be
directly written out from (2.63) and (2.64) by duality,
)
)]
[(
(
RR
i
3i
1
3
s
Kl − 2 ⋅ Kl 1 +
H (r) = i𝜔𝜖0 g(r, r0 ) 1 +
−
−
k0 R k02 R2
R
k0 R k02 R2
(2.70)
(
)
ik0
1
(2.71)
− 2
Es (r) = −R × Kl g(r, r0 )
R
R
For small PEC scatterers, both electric and magnetic current dipoles are
induced, due to the polarization of surface charge density and the circulation of
surface current, which is different from the volumetric behavior for the cases
of dielectric and magnetic scatterers. The electric and magnetic polarization
tensors of a small spherical PEC scatterer are
1
I
𝜂0 3
(2.72)
𝜁 = +i2𝜋k0 a3 𝜂0 I 3
(2.73)
𝜉 = −i4𝜋k0 a3
respectively.
We briefly mention the electric and magnetic polarization tensors of some
frequently seen small scatterers. For anisotropic spheres, the electric and magnetic polarization tensors are (section 5.6 of [5])
[ (1)
]
𝜖 − 𝜖0 𝜖 (2) − 𝜖0 𝜖 (3) − 𝜖0
1
𝜉 = −i4𝜋k0 a3 ⋅ diag (1)
, (2)
, (3)
(2.74)
𝜂0
𝜖 + 2𝜖0 𝜖 + 2𝜖0 𝜖 + 2𝜖0
[ (1)
]
𝜇 − 𝜇0 𝜇(2) − 𝜇0 𝜇(3) − 𝜇0
3
𝜁 = −i4𝜋k0 a 𝜂0 ⋅ diag (1)
,
,
(2.75)
𝜇 + 2𝜇0 𝜇(2) + 2𝜇0 𝜇(3) + 2𝜇0
respectively, where 𝜖 (1) , 𝜖 (2) , and 𝜖 (3) (𝜇(1) , 𝜇(2) , and 𝜇(3) ) are the diagonal values
of permittivity (permeability) tensor. The electric and magnetic polarization
tensors for a needle made of PEC is given by ([8])
[
]
3
a
k0
8𝜋
4𝜋
8𝜋
1
𝜉 = −i ⋅ diag
a b2 ,
a b2 ,
(2.76)
𝜂0
3 1 1 3 1 1 3 ln(a1 ∕b1 )
]
[
8𝜋
8𝜋
4𝜋
(2.77)
𝜁 = −ik0 𝜂0 ⋅ diag − a1 b21 , − a1 b21 , − a1 b21 .
3
3
3
Since a1 ∕b1 ≫ 1, the induced electric dipole along the needle direction is
much more dominant than other electric dipoles and all magnetic dipoles.
Fundamentals of Electromagnetic Wave Theory
The electric and magnetic polarization tensors for a disk made of PEC is given
by ([8])
]
[
k
16 3 16 3 4𝜋 2
(2.78)
𝜉 = −i 0 ⋅ diag
a2 , a2 ,
a2 b 2 ,
𝜂0
3
3
3
]
[
4𝜋
4𝜋
8
(2.79)
𝜁 = −ik0 𝜂0 ⋅ diag − a22 b2 , − a22 b2 , − a32 ,
3
3
3
Since a2 ∕b2 ≫ 1, two components of induced electric dipoles, both of which
are in the plane of the disk, and one component of induced magnetic dipoles
that is aligned with the normal direction of the disk, are far more dominant
than other components of electric and magnetic dipoles.
2.8.2
Two-Dimensional Case
For a two-dimensional small scatterer, its cross section in the xy plane is a small
area A0 . The induced sources are defined in a way similar to (2.67),
Il =
∫ ∫A0
J(r′ )dr′
(2.80)
However, the physical meaning is different from the three-dimensional case.
The z component of Il is a two-dimensional monopole and the x, y components are two-dimensional dipoles. The z component and the x, y components
of the current generate TM and TE waves, respectively, where the formula of
the electric and magnetic fields are given in (2.54)–(2.57). The electric and magnetic polarization tensors, 𝜉 and 𝜁, can be defined in the same way as in the
three-dimensional case. Here, we list the electric and magnetic polarization
tensors of some commonly seen two-dimensional scatterers.
For a small circular anisotropic dielectric cylinder, with radius a and permittivity tensor diag[𝜖 (1) , 𝜖 (2) , 𝜖 (3) ], the electric polarization tensor is given by
[ (1)
]
𝜖 − 𝜖0 𝜖 (2) − 𝜖0 𝜖 (3) − 𝜖0
−i2𝜋k0 a2
𝜉=
diag (1)
, (2)
,
(2.81)
𝜂0
2𝜖0
𝜖 + 𝜖0 𝜖 + 𝜖0
The expression of 𝜁 can be obtained by duality. Obviously, (2.81) is applicable
to circular isotropic dielectric cylinder as well.
For a small elliptic isotropic dielectric cylinder, the electric polarization tensor is given by
𝜉=
−i𝜋k0 a2
cosh u0 sinh u0 diag
𝜂0
[
]
eu0 (𝜖 − 𝜖0 )
eu0 (𝜖 − 𝜖0 )
𝜖 − 𝜖0
,
,
𝜖 sinh u0 + 𝜖0 cosh u0 𝜖 cosh u0 + 𝜖0 sinh u0
𝜖0
(2.82)
where a is the distance between its local origin and either foci and u0 defines
the contour of the ellipse. The expression of 𝜁 can be obtained by duality.
27
28
Electromagnetic Inverse Scattering
For a small circular PEC cylinder, the radius of which is a, the polarization
tensors are given by
[
]
−i2𝜋
1
𝜉=
diag k0 a2 , k0 a2 ,
,
(2.83)
𝜂0
k0 ln(k0 a)
𝜁 = i𝜋k0 𝜂0 a2 diag[2, 2, 1],
(2.84)
In these three examples for the 2D case, the z component of 𝜉 and the x and y
components of 𝜁 are for the TM case, and the z component of 𝜁 and the x and
y components of 𝜉 are for the TE case.
2.8.3
Scattering by a Collection of Small Scatterers
Suppose there are M three-dimensional small scatterers illuminated by a
time-harmonic electromagnetic wave. The centers of the scatterers are located
at r1 , r2 , … , rM , and their polarization tenors are 𝜉 m and 𝜁 m , m = 1, 2, … , M.
When multiple scattering between scatterers are taken into account, the total
incident field Et (rm ) (Ht (rm )) upon the mth scatterer includes both the incident
field directly from antennas Ei (rm ) (Hi (rm )) and the scattered fields from other
scatterers. The total incident fields are governed by the Foldy–Lax equation,
∑
{i𝜔𝜇0 G(rm , rm′ ) ⋅ 𝜉 m′ ⋅ Et (rm′ )
Et (rm ) = Ei (rm ) +
m′ ≠m
− ∇g(rm , rm′ ) × [𝜁 m′ ⋅ Ht (rm′ )]}
Ht (rm ) = Hi (rm ) +
∑
(2.85)
{i𝜔𝜖0 G(rm , rm′ ) ⋅ 𝜁 m′ ⋅ Ht (rm′ )
m′ ≠m
+ ∇g(rm , rm′ ) × [𝜉 m′ ⋅ Et (rm′ )]}
(2.86)
Thus, the unknowns Et (rm ) and Ht (rm ), m = 1, 2, … , M, can be solved from
the linear equation systems (2.85) and (2.86), where the number of scalar
unknowns and the number of scalar equations are both 6M.
2.8.4
Degrees of Freedom
Since scattered field is the reradiated field by induced sources, the degrees of
freedom of the scattered field are equal to the degrees of freedom of induced
sources. For example, a small nonmagnetic dielectric sphere is able to provide
three electric dipoles, whereas a small magnetic dielectric sphere provides three
electric and three magnetic dipoles. A small PEC sphere is also able to provide
three electric and three magnetic dipoles. A small PEC circular cylinder provides a single dominant source, that is, the monopole of the z-direction electric
current, when it is under TM illumination, whereas it provides three dominant
sources, that is, two electric dipoles in the xy plane and one magnetic monopole
Fundamentals of Electromagnetic Wave Theory
in the z direction, when it is under TE illumination. This information is of great
importance in solving inverse scattering problems involving small scatterers,
since the degrees of freedom of induced sources can be directly inferred from
the measured scattering data. Consequently, the number of small scatterers
might be accurately estimated if certain a priori information is available.
2.9 Scattering by Extended Scatterers
For scattering by extended scatterers, there is no close-form solution except
for few special geometries and, consequently, we have to resort to numerical
approaches. This section does not intend to present an in-depth treatment of
computational methods, but instead to provide the formula of discrete linear
equation systems that can be directly used to solve scattering problems. Readers
are referred to [4, 9, 10] for in-depth treatments of computational electromagnetics. We first deal with nonmagnetic dielectric scatterers and then deal with
PEC scatterers.
2.9.1
Nonmagnetic Dielectric Scatterers
Consider an extended nonmagnetic dielectric scatterer. It is isotropic and the
distribution of its relative permittivity is 𝜖r (r). It is discretized into M small cells
and the permittivity within each cell can be considered to be homogenous. By
“small,” the rule of thumb is that the maximum linear extent of each cell is no
larger
√ than one-tenth of the wavelength of the medium inside the cell; that is,
𝜆0 ∕ 𝜖r , where 𝜆0 is the wavelength in free space.
First, we introduce the method of moments (MoM), which is also referred
to as the weighted-residual method. The solutions of differential or integral
equations are assumed to be well approximated by a finite sum of basis functions. Since the solution is approximate, the original equation will not be satisfied exactly and we will be left with a residual. The MoM tries to minimize the
residual in a weighted average sense, with respect to certain chosen weighting
functions (also known as testing functions). For a 2D TM scattering problem,
both the electric field and the induced current are along the z direction and
consequently the EFIE becomes a scalar equation as shown in (2.58). It is convenient to rewrite the equation in terms of the current J = −i𝜔(𝜖 − 𝜖0 )E. The cross
section of of the scatterer is divided into square cells, whose area is denoted as
S. Applying the pulse basis function and the delta test function, we obtain the
following M × M linear equation system,
i
E =Z⋅J
i
(2.87)
where E = [Ezi (r1 ), Ezi (r2 ), … , Ezi (rM )]T is an M-dimensional column vector,
consisted of the incident electric field evaluated at the center of each cell,
29
30
Electromagnetic Inverse Scattering
where the superscript T denotes the transpose operator. The M-dimensional
column vector J is defined in a similar way. The M × M matrix Z has the
following entries,
Zm,m′
𝜂 𝜋a
⎧ 0 J1 (k0 a)H (1) (k0 |rm − rm′ |),
if m ≠ m′
0
⎪ 2
=⎨
i𝜂0 𝜖r (rm )
𝜂 𝜋a
⎪ 0 H1(1) (k0 a) +
, if m = m′
⎩ 2
k0 (𝜖r (rm ) − 1)
(2.88)
√
where a = S∕𝜋 is the radius of the circle of the same area of the square cell.
Once the J is obtained from (2.87), the scattered field is then directly evaluated
from (2.54). The application of the pulse basis function and delta testing function to 2D TE case and 3D case can be found in sections 2.6 and 10.11 of [4],
respectively. The limitations of the pulse basis function and delta testing function are also discussed in these two sections. Other choices of basis and testing
functions can be found in chapters 5 and 9 of [4]. For a small scale problem,
(2.87) can be solved by a direct matrix inversion. However, for large scale problems, it is computational costly to apply a direct matrix inversion since its computational complexity is O(M3 ). Instead, the linear equation can be solved by
iterative methods, such as the conjugate gradient (CG) method (see Appendix
C), where the computational complexity in each iteration is O(M log M) if the
fast Fourier transform (FFT) is used to evaluate the matrix-vector product (see
Appendix D).
Next, we introduce the coupled dipole method (CDM), which is also
known as the discrete dipole approximation (DDA) [11]. In this method,
the volume of the scatterer is divided into small cells and scattered field is
considered to be the superposition of reradiation of induced dipoles. The
dipole moment is derived from the original EFIE (2.47). It is important to
note that the CDM is based on the concept of an exciting field rather than
the actual field. The exciting field for a cell refers to the total field incident
upon the cell, consisting of the original incident field coming directly from
transmitters and the scattered field coming from all other cells. For a cell,
the exciting field coming from other cells is derived from the original EFIE
(2.47) by moving the cell’s self-contribution term on the right-hand side of
(2.47) to the left-hand side. The dipole moment induced within a cell is the
product of polarization tensor (also known as depolarization tensor) and the
exciting field, which is exactly the topic of Section 2.8. Thus, from this point
of view, compared with the MoM, the CDM is easier to learn and implement
since the singular integral present in the cell’s self-contribution term has been
automatically considered in the polarization tensor that was introduced in
Fundamentals of Electromagnetic Wave Theory
Section 2.8. For a 2D TM scattering problem, the linear equation system is
given by
∑
i𝜔𝜇0 g(rj , rm )𝜉m Ezt (rm )
(2.89)
Ezt (rj ) = Ezi (rj ) +
m≠j
for j = 1, 2, … , M. The 𝜉m for the mth cell is the third component of the polarization tensor (2.81). Thus, the M unknowns Ezt (rj ) can be solved by M linear equations. For a 2D TE scattering problem, the linear equation system is
given by
∑
i𝜔𝜇0 G(rj , rm ) ⋅ 𝜉 m ⋅ Et (rm )
(2.90)
Et (rj ) = Ei (rj ) +
m≠j
for j = 1, 2, … , M. The 𝜉 m for the mth cell is the first two components of
−i2𝜋k a2 𝜖 −𝜖
the polarization tensor (2.81), that is, 𝜉 m = 𝜂 0 𝜖m +𝜖0 I 2 for the isotropic
0
m
0
medium. The 2M unknowns Ext (rj ) and Eyt (rj ) can be solved from the 2M linear
equations. For a 3D scattering problem, the linear equation system has the
same format as (2.90), except that a 3D Green’s function is used and 𝜉 m is given
by (2.62). In fact, the equation system is just (2.85), with 𝜁 = 0 for nonmagnetic
materials. The 3M unknowns Et (rj ), j = 1, 2, … , M, can be solved from the 3M
linear equations.
It has been proven in [12] that the CDM is equivalent to the MoM that
employs a pulse basis function and delta testing function. For an extensive
review of the CDM (or equivalently, the DDA), including both theoretical and
computational aspects, the reader is referred to [12, 13] and references therein.
2.9.2
Perfectly Electrically Conducting Scatterers
We consider a 2D TM PEC scattering problem. Since both the electric field
and the induced current are in the z direction, the EFIE (2.52) becomes a scalar
equation. The integral over the boundary is now along a closed loop 𝜕D, so that
the reduced EFIE reads
Ei (r) + i𝜔𝜇0
∫𝜕D
g(r, r′ ) ⋅ J(r′ )dr′ = 0
(2.91)
The boundary 𝜕D is discretized into a total number of M small line
cells, with the center of them being at rm , m = 1, 2, … , M. Define the
following M-dimensional vectors: the vector of induced current deni
sity J = [J(r1 ), J(r2 ), … , J(rM )]T and the vector of incident field E =
[Ei (r1 ), Ei (r2 ), … , Ei (rM )]T . Applying the pulse basis function and the delta test
function, we obtain the following M-dimensional linear equation system,
i
E =Z⋅J
(2.92)
31
32
Electromagnetic Inverse Scattering
where the M × M matrix Z has the entries
Zm,m′
⎧ k𝜂𝑤
⎪
if m ≠ m′
H0(1) (k|rm − rm′ |),
⎪ 4
{
[ (
)
]}
=⎨
⎪ k𝜂𝑤 1 + i 2 ln 𝛾k𝑤 − 1
, if m = m′
⎪ 4
𝜋
4
⎩
(2.93)
where 𝑤 is the length of the line element, and 𝛾 ≈ 1.781. The M-dimensional
unknown J can be solved from (2.92), which can be then used in (2.54) to calculate the scattered field.
2.10 Far-Field Approximation
When the observation point is very far away from the source, approximations
can be made in the evaluation of radiated fields in the far zone. The source,
either active or induced source, is assumed to be located in a locally finite
domain. By “far away,” we mean the distance of the observation point r = r̂r to
the source is much larger than the wavelength 𝜆0 , or equivalently k0 r ≫ 1. It is
convenient to first introduce the concept of angular spectrum representation
of electromagnetic fields in a homogenous medium. The electric field E in free
space, as an example of a homogenous medium, satisfies the vector Helmholtz
equation,
(∇2 + k02 )E = 0.
(2.94)
In any plane where the z coordinate is a constant, the two-dimensional Fourier
transform of the field E is represented as
∞
∞
̂ x , ky ; z) = 1
E(k
E(x, y, z)e−i(kx x+ky y) dxdy
4𝜋 2 ∫−∞ ∫−∞
(2.95)
The inverse Fourier transform reads as
∞
E(x, y, z) =
∞
∫−∞ ∫−∞
̂ x , ky ; z)ei(kx x+ky y) dkx dky
E(k
(2.96)
Plugging (2.96) into (2.94), we find that the Fourier spectrum Ê evolves along
the z-axis as
̂ x , ky ; z) = E(k
̂ x , ky ; 0)e±ikz z
E(k
where the wavenumber along the z direction is defined as
⎧ √ 2
k0 − kx2 − ky2 k02 − kx2 − ky2 ≥ 0
⎪
kz = ⎨ √
⎪ i kx2 + ky2 − k02 k02 − kx2 − ky2 < 0
⎩
(2.97)
(2.98)
Fundamentals of Electromagnetic Wave Theory
This definition assures outgoing waves and a finite energy density at infinity. In
the exponent in (2.96), the + sign refers to a wave propagating in the positive
z direction and the − sign denotes a wave propagating in the negative z direction. Without loss of generality, we consider only the +z-going wave from here
onward. Consequently, inserting the result of (2.97) into (2.96), we obtain
∞
E(x, y, z) =
∞
∫−∞ ∫−∞
̂ x , ky ; 0)ei(kx x+ky y+kz z) dkx dky
E(k
(2.99)
which is known as the angular spectrum representation; that is, any wave in a
source free region can be represented by a superposition of plane waves.
When the observation is in far zone, kz ≫ 1, the second type of the plane
wave in (2.98), referred to as the evanescent wave, exponentially decays and
thus does not contribute to the far field. The first type of the plane wave in
(2.98), referred to as the travelling wave or propagation wave, is able to reach
to far field. Thus, the domain of integral in (2.99) is reduced kx2 + ky2 ≤ k02 . The
asymptotic behavior of the double integral as kr ≫ 1 can be evaluated by the
method of stationary phase (section 3.3 of [14]), where the fast oscillation of
the phase cancels out the integrand everywhere except at the spectral point
where the phase is stationary. It turns out that the stationary phase occurs at
the spectral point satisfying kx ∕x = ky ∕y = k0 ∕r. If the wave vector k is written
̂ then the stationary phase occurs at
as k0 k,
k̂ = r̂
(2.100)
which applies that at the far zone r = r̂r, the field consists of only a single plane
wave with the propagation direction identical to the observation direction.
This result is of great significance, since several important conclusions can be
derived from it.
• Since the observation is in far zone, the receiver senses a locally plane wave
within its finite neighborhood, just like we feel flat ground at a local level due
to the large radius of the Earth. In fact, the statement that the far field is an
outward going locally plane wave is just the radiation boundary conditions
formulated in (2.26) and (2.27).
• Since both the electric and magnetic fields of a plane wave are perpendicular
̂ we conclude that the electric and magnetic
to the propagation direction k,
fields in far zone do not have the radial component, considering the fact that
k̂ = r̂ . In other words, in spherical coordinate systems, the electric and magnetic fields in the far zone only have the 𝜃̂ and 𝜙̂ components.
̂ no matter whether it rep• For a plane wave, the ∇ can be replaced by k0 k,
resents the curl, divergence, or gradient. This replacement greatly simplifies
the far-field expression of the electric field in (2.35).
Finally, we provide the far-zone field radiated by an electric current
dipole that is located at r0 . Considering the far field condition k0 r ≫ 1 and
33
34
Electromagnetic Inverse Scattering
|r − r0 | ≈ r − r̂ ⋅ r0 , we simplify the electric current dipole radiation equation
(2.63) to
eik0 r −ik0 r̂ ⋅r0
(Il − r̂ r̂ ⋅ Il)
e
4𝜋r
which can be also written as
Es (r) = i𝜔𝜇0
Es (r) = −i𝜔𝜇0
eik0 r −ik0 r̂ ⋅r0
r̂ × r̂ × Il
e
4𝜋r
(2.101)
2.11 Reciprocity
In the radiation or scattering of electromagnetic waves, the background
medium in which an electromagnetic wave propagates plays an important role.
One of the properties of the background medium is the reciprocity condition,
which is an important concept that will be frequently used in this book, such
as in Section 2.12, and Chapters 3 and 9.
Consider a source a, denoted as Ja and Ka , illuminated by a field Eb and Hb
that is generated by the source b, denoted as Jb and Kb . The reaction of the
source a with the field b is defined as
⟨a, b⟩ =
∫ ∫ ∫V
Ja ⋅ Eb − Ka ⋅ Hb dV
(2.102)
The reaction can be thought of as generalized measurements. A medium is said
to be reciprocal, if it satisfies
⟨a, b⟩ = ⟨b, a⟩
(2.103)
It can be proven that isotropic medium, with both 𝜖(r) and 𝜇(r) being scalar, is
reciprocal.
The purpose of this section is mainly to present the Green’s function representation of reciprocity condition. In an isotropic medium, with the distribution of permittivity 𝜖(r) and permeability 𝜇(r). The electric field is related to the
electric source by the vector wave equation,
[
]
1
∇×
(2.104)
∇ × E(r) − k02 𝜖r (r)E(r) = i𝜔𝜇0 J(r)
𝜇r (r)
We define the electric Green’s function GE (r, r′ ) that satisfies
]
[
1
∇×
∇ × GE (r, r′ ) − k02 𝜖r (r)GE (r, r′ ) = 𝛿(r − r′ )I
𝜇r (r)
(2.105)
Note that the physical meaning of GE is the same as the G that has been used
in previous sections. The subscript E is added in order to differentiate the magnetic Green’s function to be introduced later. We easily see that the electric and
Fundamentals of Electromagnetic Wave Theory
magnetic fields generated by an electric current source are given by
+∞
E(r) = i𝜔𝜇0
∫ ∫ ∫−∞
GE (r, r′ ) ⋅ J(r′ )d3 r′
(2.106)
+∞
H(r) =
1
∇ × GE (r, r′ ) ⋅ J(r′ )d3 r′
∫ ∫ ∫−∞ 𝜇r (r)
(2.107)
By duality, we define the magnetic Green’s function GM (r, r′ ) that satisfies
[
]
1
′
∇×
(2.108)
∇ × GM (r, r ) − k02 𝜇r (r)GM (r, r′ ) = 𝛿(r − r′ )I
𝜖r (r)
The magnetic and electric fields generated by a magnetic current source are
given by
+∞
H(r) = i𝜔𝜖0
∫ ∫ ∫−∞
GM (r, r′ ) ⋅ K(r′ )d3 r′
(2.109)
+∞
E(r) =
−1
∇ × GM (r, r′ ) ⋅ K(r′ )d3 r′
∫ ∫ ∫−∞ 𝜖r (r)
(2.110)
Without derivation, we provide the following relationships between Green’s
functions, which are implied by the reciprocity condition.
T
GE (r, r′ ) = GE (r′ , r)
T
GM (r, r′ ) = GM (r′ , r)
1
1
∇ × GE (r, r′ ) =
[∇′ × GM (r′ , r)]T
𝜇r (r)
𝜖r (r′ )
(2.111)
(2.112)
(2.113)
where the superscript T denotes the transpose operator.
2.12 Huygens’ Principle and Extinction Theorem
Huygens’ principle states that the electromagnetic fields in a source-free region
V are completely determined by the tangential fields specified over the surface
S enclosing V . The extinction theorem is a closely related theorem. Both
of them are important in solving electromagnetic radiation and scattering
problems.
The derivation of the Huygens’ principle starts from the following two
equations,
]
[
1
(2.114)
∇ × E(r) − k02 𝜖r (r)E(r) = i𝜔𝜇0 J(r),
∇×
𝜇r (r)
[
]
1
∇×
(2.115)
∇ × GE (r, r′ ) − k02 𝜖r (r)GE (r, r′ ) = 𝛿(r − r′ )I
𝜇r (r)
35
36
Electromagnetic Inverse Scattering
After lengthy but simple vector manipulations, we arrive at
∫ ∫ ∫V
−
∯S
E(r′ ) ⋅ 𝛿(r − r′ )d3 r′ = i𝜔𝜇0
∫ ∫ ∫V
GE (r, r′ ) ⋅ J(r′ )d3 r′
̂ ′ ) × H(r′ )] + GEM (r, r′ ) ⋅ [E(r′ ) × o(r
̂ ′ )]dS(r′ ),
i𝜔𝜇0 GE (r, r′ ) ⋅ [o(r
(2.116)
where V is an arbitrarily chosen volume and its boundary is a closed surface
̂ GEM (r, r′ ) is the dyadic Green’s function,
S with outward normal direction o.
expressing the relationship of electric-field and magnetic-source,
+∞
E(r) =
∫ ∫ ∫−∞
GEM (r, r′ ) ⋅ K(r′ )d3 r′ ,
(2.117)
and it is related to the electric dyadic Green’s function by GEM (r, r′ ) =
[
]T
−1
′
′
∇
×
G
(r
,
r)
, which can be seen from (2.110) and (2.113).
E
𝜇r (r′ )
If we choose V to be an arbitrary space Vs that encloses all electric current
sources, that is, its complementary space Vs f = ℝ3 \V is a source-free region,
then we obtain
∯S
̂ ′ ) × H(r′ )] + GEM (r, r′ ) ⋅ [E(r′ ) × n(r
̂ ′ )]dS(r′ )
i𝜔𝜇0 GE (r, r′ ) ⋅ [n(r
{
E(r), if r ∈ source-free region
=
(2.118)
0,
if r ∈ source region
Note that the direction n̂ is from source region Vs to source-free region Vs f ,
as shown in Fig. 2.2. It is important to stress that if there are multiple sources,
including both active and induced sources, we proceed with the linear superposition principle. When dealing with one of the sources, we let other sources
be zero and then apply (2.118). We mention in passing that (2.118) will remain
the same if the integration domain V in (2.116) is chosen as Vs f .
z
Source
y
Vs
S0
n
Vsf
x
Figure 2.2 Configuration for the
derivation of Huygens’ principle:
The space Vs encloses all electric
current sources and its
complementary space Vs f = ℝ3 \V
is a source-free region. The
direction n̂ is from source region
Vs to source-free region Vs f .
Fundamentals of Electromagnetic Wave Theory
By duality, the magnetic counterpart of the surface integral (2.118) yields the
exact magnetic field in the source free region and zero in the source region.
The following discussions on (2.118) are worth emphasizing:
1) In the source free region, the electric field is uniquely determined by the
tangential electric and magnetic fields on the surface S enclosing the source.
In comparison, in the source region, the surface integral in (2.118) does not
represent physical fields and it is numerically equal to zero.
2) A model that makes (2.118) physically meaningful in both the source and
source-free region does this by inserting at S the surface electric current
̂ The
density Js = n̂ × H and surface magnetic current density Ks = E × n.
surface current densities reproduce the electric and magnetic fields in the
source-free region, which is known as the Huygens’ surface equivalent principle. The surface current densities yield null fields in the source region. In
addition, the surface current densities accounts for the discontinuity of the
fields across the surface S; that is, the boundary condition is satisfied.
3) Since (2.118) is derived for an arbitrary isotropic medium, the surface
integral still yields null fields in the source region if the medium therein is
replaced by a different one. The replacement of the medium in the source
region changes neither the electric/magnetic fields in the source region nor
the electric/magnetic surface current densities, which consequently means
that the tangential electric/magnetic fields at the boundary S on the side of
the source-free region are not altered. To summarize, changing the isotropic
medium in the source region does not change the right-hand side of (2.118)
in either the source or source-free region. Care should be taken to ensure
that the dyadic Green’s function in the left-hand side should be changed to
one accounting for the new medium in the source region and the original
medium in the source-free region. The freedom of choosing an arbitrary
medium in the source region sometimes allows us to arrive at an analytical
dyadic Green’s function. Such freedom brings significant convenience in
solving inverse scattering problems, as demonstrated in Chapter 9 and [15].
4) Equation (2.118) is derived for a radiation problem in presence of an arbitrary isotropic background medium, which can be either homogeneous or
inhomogeneous. For a scattering problem, scatterers can be considered to be
a perturbation of permittivity with respect to the background medium, and
the scattered field can be considered to be the reradiation of induced sources
in the scatterer. Thus, when (2.118) is applied to the scattered field, the surface integral reproduces the scattered field in the region of scatterer-free and
yields null fields in the region of scatterer.
For example, we can apply (2.118) to solve the following scattering problem.
As shown in Fig. 2.2, the electric current source is located in the region Vs ,
which is enclosed by two surfaces, S0 that is next to the source free region Vs f ,
and S∞ that is located at infinity. The scatterer is located in the region Vs f and
37
38
Electromagnetic Inverse Scattering
the medium in region Vs is homogenous with constant relative permittivity 𝜖r .
The total electric field consists of the incident field and the scattered field,
E(r) = Ei (r) + Es (r).
(2.119)
The incident field is due to the active source located in region Vs , radiating in a
homogenous medium. The application of (2.118) yields
∯S0
̂ ′ ) × Hi (r′ )] + GEM (r, r′ ) ⋅ [Ei (r′ ) × n(r
̂ ′ )] dS(r′ )
i𝜔𝜇0 GE (r, r′ ) ⋅ [n(r
{
Ei (r),
=
0,
if r ∈ Vsf
if r ∈ Vs
(2.120)
where GE and GEM are the dyadic Green’s function for a homogeneous medium
with relative permittivity 𝜖r , and they can be expressed analytically. It is important to note that the integral over S∞ that is located at infinity is equal to zero
due to the property of the radiation condition. That is why the integral in (2.120)
is carried out only on S0 .
The scattered field is due to the induced source in region Vs f . Consequently,
the normal direction at the surface S0 is now −n̂ since it is defined as going
from the source region to the source free region. When applying (2.118) to the
scattered field yield, the corresponding source region Vs f can be changed to
an arbitrary isotropic medium. It is easy to see that the analytical formula of a
dyadic Green’s function will be available if the medium Vs f is replaced by the
medium in region Vs . Consequently, the application of (2.118) yields
∯S0
̂ ′ ) × Hs (r′ )] + GEM (r, r′ )
i𝜔𝜇0 GE (r, r′ ) ⋅ [(−n)(r
̂ ′ )]dS(r′ )
⋅ [Es (r′ ) × (−n)(r
{
0,
if r ∈ Vs f
=
s
E (r), if r ∈ Vs
(2.121)
We have seen that (2.121) − (2.120) leads to
Ei (r) −
∯ S0
̂ ′ ) × H(r′ )] + GEM (r, r′ )
i𝜔𝜇0 GE (r, r′ ) ⋅ [n(r
̂ ′ )]dS(r′ )
⋅ [E(r′ ) × n(r
{
0,
if r ∈ Vs f
=
E(r), if r ∈ Vs
(2.122)
This equation is evocative of Huygens’ principle. It says that when the observation point is in the source region (i.e., in the exterior of scatterer region), then
the total field consists of the incident field and the contribution of field due to
equivalent surface sources on S0 . But if the observation point is in a source-free
Fundamentals of Electromagnetic Wave Theory
region (i.e., in the region of scatterer), then the equivalent surface sources on S0
generate a field that exactly cancels out the incident field making the total field
in this region zero. This is in fact the vectorial analogy of the Ewald–Oseen
extinction theorem in optics [16] (section 2.4.2).
In this example, if the region Vs is not homogeneous, then the analytical
dyadic Green’s functions GE and GEM are generally not available. In this case,
the conclusion in (2.122) still applies, but the dyadic Green’s functions have to
be numerically obtained.
Finally, the Huygens’ principle and the extinction theorem are applicable to
scalar waves as well, for example, acoustic waves. Consider a scalar wave field
u(r), the scalar counterpart of (2.118) reads
[
] 𝜕g(r, r′ )
𝜕u
g(r, r′ ) − (r′ ) +
u(r′ )dS(r′ )
∯S
𝜕n
𝜕n(r′ )
{
u(r), if r ∈ source-free region
=
0,
if r ∈ source region
(2.123)
Note that the direction n̂ is from source region Vs to source-free region Vs f .
(r′ ) is called the amplitude of the source
In the mathematical community, − 𝜕u
𝜕n
′
for single-layer potential and u(r ) is called the amplitude of the source for
= ∇u ⋅ n is the directional derivative of u
double-layer potential. Note that 𝜕u
𝜕n
in the direction of n.
For electromagnetic waves, there is no 3D scalar wave. However, in the
extreme case of an electrostatic regime; that is, k0 = 0, the u in (2.123)
represents the electric potential, the source for single-layer potential is the
surface charge, and the source for double-layer potential is the dipole layer
(section 1.8 of [6]). Instead, the scalar wave occurs in the 2D TM case, E = Ez ẑ .
′
̂ ′ ) × H(r′ )] in (2.118)
It can be easily seen
[ that the
] first term i𝜔𝜇0 GE (r, r ) ⋅ [n(r
𝜕E
̂ ′ )]
reduces to g(r, r′ ) − 𝜕nz (r′ ) ẑ , and the second term GEM (r, r′ ) ⋅ [E(r′ ) × n(r
𝜕g(r,r′ )
in (2.118) reduces to 𝜕n(r′ ) Ez (r′ )̂z. These observations imply that, for the 2D
TM case, the first term in (2.123) represents the radiation by the z-direction
surface electric current and the second term represents the radiation by the
transverse direction surface magnetic current.
References
1 Kong, J.A. (2000) Electromagnetic wave theory, EMW, Cambridge, MA.
2 Harrington, R.F. (1961) Time harmonic electromagnetic fields, McGraw-Hill,
New York.
3 Chew, W.C. (1995) Waves and fields in inhomogeneous media, IEEE Press.
39
40
Electromagnetic Inverse Scattering
4 Peterson, A.F., Ray, S.L., and Mittra, R. (1998) Computational methods for
electromagnetics, IEEE Press, New York.
5 Bohren, C.F. and Huffman, D.R. (1983) Absorption and scatttering of light by
small particles, John Wiley & Sons, Inc., New York.
6 Jackson, J.D. (1998) Classical electrodynamics, 3rd Edn., John Wiley & Sons,
Inc., New York.
7 Jones, D.S. (1986) Acoustic and electromagnetic waves, Clarendon Press,
Oxford: Oxford University Press, New York.
8 Chambers, D.H. and Berryman, J.G. (2006) Target characterization using
9
10
11
12
13
14
15
16
decomposition of the time-reversal operator: electromagnetic scattering
from small ellipsoids. Inverse Probl., 22, 2145–2163.
Chew, W., Michielssen, E., Song, J.M., and Jin, J.M. (Eds) (2001) Fast and
efficient algorithms in computational electromagnetics, Artech House, Inc.,
Norwood, MA, USA.
Jin, J.M. (2015) The finite element method in electromagnetics, John Wiley &
Sons, Inc.
Purcell, E.M. and Pennypacker, C.R. (1973) Scattering and absorption of
light by nonspherical dielectric grains. Astrophys. J., 186, 705–714.
Lakhtakia, A. (1992) Strong and weak forms of the method of moments and
the coupled dipole method for scattering of time-harmonic electromagnetics fields. Int. J. Mod. Phys. C, 3, 583–603.
Yurkin, M.A. and Hoekstra, A.G. (2007) The discrete dipole approximation:
an overview and recent developments. J. Quant. Spectros Radiative Trans.,
106 (1), 558–589.
Mandel, L. and Wolf, E. (1995) Optical coherence and quantum optics,
Cambridge University Press.
Takenaka, T. and Moriyama, T. (2012) Inverse scattering approach based on
the field equivalence principle: inversion without a priori information on
incident fields. Opt. Lett., 37 (16), 3432–3434.
Born, M. and Wolf, E. (1999) Principles of optics (7th Edn), Cambridge University Press.
41
3
Time-Reversal Imaging
A time-reversal mirror (TRM) is a device that records an incoming wave
and then re-transmits its time-reversed version. The re-transmitted wave
propagates back through the same medium and refocuses on the original
source. TRM properties can be used not only to locate active sources; that is,
to solve an inverse source problem, but also to detect passive targets in the
frame of inverse scattering problems. Time-reversal principles apply to both
acoustic wave and electromagnetic waves, as long as the medium is reciprocal
and lossless. Time-reversal techniques have been an actively researched
field with wide applications, such as source localization, medical imaging,
telecommunications, and medical therapy.
Since the time-domain signal can be expressed as a linear superposition of
its Fourier components, this chapter focuses on frequency-domain representation. This choice is convenient when discussing spatial resolution of imaging
in terms of wavelength.
The organization of the chapter is as follows. Section 3.1 presents TRM for
active sources that radiate by themselves, where the implementation steps and
fundamental theory are discussed. The scalar wave is presented first, followed
by the vectorial wave. Section 3.2 presents the application of TRM for target
detections, where targets are illuminated by other active sources and consequently secondary sources are induced on targets. Section 3.3 briefly discusses
several topics, such as the application of TRM in a random medium and the
properties of time-domain TRM.
3.1 Time-Reversal Imaging for Active Sources
3.1.1
Explanation Based on Geometrical Optics
A straightforward explanation of TRM, which is easy to understand but not
rigorous, is based on geometrical optics. In Fig. 3.1(a), the ray coming from a
source object (labelled “O”) in front of a flat mirror is reflected, and all reflected
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
42
Electromagnetic Inverse Scattering
O
O
(a)
(b)
O
(c)
Figure 3.1 Basic principle of time-reversal imaging. (a) Imaging by a mirror; (b) the
functionality of mirror can be considered as a device in the black box; and (c) the received
signal is sent back along the original path to yield an image at the position of the original
object.
rays seem to come from a point (labelled “I”) that is behind the mirror, which
is referred to as the virtual image. In addition, at the virtual image position, the
rays radiating in all direction turn out to be in phase, and what’s more, the common phase turns out to just be the phase of the original source. This fact can be
easily proven. For each ray that is radiated from the source object and intersects
the mirror at a point, it takes the same travelling time from the object “O” or the
virtual image “I” to the intersection point. Since all rays at the source position
“O” are in phase, all conjugate virtual rays are also in phase at “I.” To conclude,
the virtual image seems to radiate with the same phase as the original source
object. In fact, we can treat the mirror as a type of device, placed in the black
box in Fig. 3.1(b), which is able to generate symmetric rays and consequently
generate a focal spot at which all rays are in phase. Any device placed inside
the black box that is able to realize such functionalities can be said an imaging
device. Examples of such devices are the time-reversal mirror (TRM) and phase
conjugation mirror (PCM). The PCM has been known and used for decades in
the optical domain, and readers are referred to [1] for the physical realization
and working principle of the PCM. This chapter discusses only the TRM.
The TRM is an array of transceivers that first record the received signal and
then send the time-reversed signal back to the space of the original source
point. Figure 3.1(c) sketches the intuitive idea of time-reversal imaging and
Time-Reversal Imaging
shows that the TRM has realized the functionality of the mirror that is shown in
Fig. 3.1(a), except that the generated image is in front of the mirror. In addition,
the TRM image is focused by physical rays and thus the image is a physically
true image, instead of a virtual one.
3.1.2
Implementation Steps
Time-reversal is equivalent to phase conjugation of every frequency component, which will be proven in Section 3.1.3. For ease of presentation, we
start with frequency domain TRM and postpone the time domain TRM
to Section 3.1.3. In the frequency domain, TRM imaging is realized by the
following two implementation steps: (1) during the recording step, the TRM is
passive and records the outgoing fields that are radiated from the source; and
(2) during the time-reversal step, the TRM is active with the driving source
being the complex conjugate of the recorded fields in step (1), and the radiated
fields focus at the original source position. At this step, the original source
is absent, since the position of the source is exactly the parameter that the
imaging problem looks for. The two steps are illustrated in Fig. 3.2.
It can be proven that the fields radiated by all TRM transmitters are in
phase at the original source position. Since the original source simultaneously
radiates rays in all redirections, rays propagating to all directions are in phase
at the source position, which is denoted as 𝜙. Consider the ith transceiver,
which is separated by a distance of di from the source. At the first step of the
implementation, the phase of this ray becomes 𝜙 + kdi when it reaches the ith
Received
signal
Time
reverse
Source
Receiving antennas
Transmitting
signal
Image
Transmitting antennas
Figure 3.2 Illustration of the time-reversal implementation steps. The TRM is a
transmitter-receiver array. The first step consists of recording fields radiated by the source
and the second step consists of transmitting time-reversed fields back through the same
medium, which tend to focus at the original source position. Adapted from: Liu 2005, IEEE
Trans. Antennas Propag., 53, 3058–3066. [39] Reproduced with permission of IEEE.
43
44
Electromagnetic Inverse Scattering
transceiver, where k is the wavenumber of the medium. At the second step of
the implementation, the driving source at the ith transmitter is the complex
conjugate of the recorded signal in step (1); that is, the driving source has a
phase of −𝜙 − kdi . When the field radiated by the ith transmitter reaches the
original source position, the phase becomes (−𝜙 − kdi ) + kdi , which is simply
−𝜙, independent of di . That is, all rays that propagate back in the second step
are in phase (−𝜙) at the original source position.
At the first sight, the above analysis seems to reach the conclusion that, while
the physical flat mirror shown in Fig. 3.1(a) generates a virtual image that is
an exact replica of the original source, the TRM generates a true image that is
a phase-conjugate of the original source. However, this conclusion is obtained
for an observation domain that is much larger than the wavelength; that is, in
the sense of geometrical optics. If the observation domain is comparable to the
wavelength, the conclusion is different. A question that is naturally asked is that,
if the source is an ideal point source, that is a delta function in space, will the
image generated by the TRM be also a delta function; that is, a perfect image?
This question can be answered in the spectrum domain. A spherical wave due to
a point source can be expressed in terms of plane waves (section 2.2.2 of [2]) as
eikr
i
eikx x+iky y+ikz |z|
dkx dky
(3.1)
=
4𝜋r
8𝜋 2 ∫−∞ ∫−∞
kz
where the wavenumber along the longitudinal direction (i.e., the z direction)
is defined as
⎧ √ 2
k − kx2 − ky2 k 2 − kx2 − ky2 ≥ 0
⎪
kz = ⎨ √
(3.2)
⎪ i kx2 + ky2 − k 2 k 2 − kx2 − ky2 < 0
⎩
This definition assures outgoing waves and a finite energy density at infinity.
Equation (3.1) is referred to as the Weyl identity. The integrand that corresponds to the first type of kz in Eq. (3.2) represents a travelling wave (a.k.a.
propagating wave), and the integrand that corresponds to the second type of
kz represents an evanescent wave. The travelling wave changes only the phase
during propagation, whereas an evanescent wave changes both phase and
magnitude during propagation. It is obvious that if the information of point
source is fully recovered by the TR process, the phase of travelling wave has
to be compensated for and at the same time both the phase and amplitude
of evanescent wave have to be compensated for. First, the evanescent wave
exponentially decays and cannot be received by receivers that are usually
placed in the far field. This is the first reason why the resolution of TR image is
limited. Second, for the transceivers shown in Fig. 3.1(c), the array covers only
a finite aperture so that only a portion of travelling waves are collected. Thus,
the phase compensation by the TR process is for only a portion of travelling
waves. Due to these two reasons, the TR image is not a perfect image anymore.
The TR image of the original delta-function source will be an expanded spot.
∞
∞
Time-Reversal Imaging
In the spatial-frequency space, all components satisfying kx2 + ky2 > k 2 and part
of components satisfying kx2 + ky2 ≤ k 2 are lost during the TR imaging process.
3.1.3
Fundamental Theory
Time-reversal invariance (or T-symmetry) is the theoretical symmetry of physical laws under a time-reversal transformation: T ∶ t → −t; that is, the related
physical quantities transform in a consistent fashion so that the form of the
equation is unchanged (section 6.10 of [3]). For a scalar wave equation in a
lossless medium,
1 𝜕 2 u(r, t)
∇2 u(r, t) − 2
=0
(3.3)
𝑣 (r) 𝜕t 2
it is obvious that the u(r, t) is even under time-reversion transformation:
Tu(r, t) = u(r, −t) since the wave equation only contains an even order
time-derivative operator. Considering the fact that u(r, t) is a real-valued
signal, we find that its Fourier transform
∞
1
u(r, t)e−i𝜔t dt
(3.4)
2𝜋 ∫−∞
implies u(r, −𝜔) = u∗ (r, 𝜔), where the superscript * denotes complex conjugate.
Thus, time-reversal is equivalent to phase conjugation of every frequency component Tu(r, 𝜔) = u∗ (r, 𝜔). Hereafter the discussion centers around monochromatic waves.
The time-reversal principle is based on Green’s theorem. Consider a
monochromatic scalar wave u(r) propagating in a medium characterized by
the spatially varying wavenumber k(r), which is defined as 𝜔∕𝑣(r), where 𝑣(r)
is the local velocity of the wave. Such a u(r) satisfies the Helmholtz equation
u(r, 𝜔) =
∇2 u(r) + k 2 (r)u(r) = −f (r)
(3.5)
where the source f (r) is distributed in a finite spatial domain. Green’s function
G(r, r′ ) of the wave equation Eq. (3.5) satisfies
∇2 G(r, r′ ) + k 2 (r)G(r, r′ ) = −𝛿(r − r′ )
(3.6)
The multiplication of G(r, r′ ) with the complex conjugate of Eq. (3.5) minus the
multiplication of u∗ (r) with Eq. (3.6) yields
G(r, r′ )∇2 u∗ (r) − u∗ (r)∇2 G(r, r′ ) = u∗ (r)𝛿(r − r′ ) − G(r, r′ ) f ∗ (r)
(3.7)
Integrate Eq. (3.7) over a domain V , the boundary S of which encloses all
sources f (r), and we obtain
𝜕u∗ (r)
𝜕G(r, r′ )
G(r, r′ )
u∗ (r)𝛿(r − r′ )
− u∗ (r)
dS(r) =
∯S
∫ ∫ ∫V
𝜕n(r)
𝜕n(r)
− G(r, r′ ) f ∗ (r)d3 r,
(3.8)
where the integration by part and the divergence theorem have been used.
Note that 𝜕u
= ∇u ⋅ n is the directional derivative of u along n that is the
𝜕n
45
46
Electromagnetic Inverse Scattering
normal direction at the surface, pointing outside of the domain V . Since r
and r′ coordinates are conventionally reserved for representing the field and
source, respectively, we swap the r and r′ in Eq. (3.8). Use the reciprocity
property G(r, r′ ) = G(r′ , r), and we obtain
∯S
G(r, r′ )
𝜕u∗ (r′ ) 𝜕G(r, r′ ) ∗ ′
u∗ (r′ )𝛿(r′ − r)
−
u (r )dS(r′ ) =
∫ ∫ ∫V
𝜕n(r′ )
𝜕n(r′ )
− G(r, r′ ) f ∗ (r′ )d3 r′ ,
(3.9)
For an observation point r that is inside the domain V , that is, r ∈ V , the
left-hand side of Eq. (3.9) is defined as the time-reversal field, denoted as uTR ,
uTR (r) =
∯S
G(r, r′ )
𝜕u∗ (r′ ) 𝜕G(r, r′ ) ∗ ′
−
u (r )dS(r′ ).
𝜕n(r′ )
𝜕n(r′ )
(3.10)
Its value can be calculated from the right-hand side of Eq. (3.9),
uTR (r) = u∗ (r) −
G(r, r′ ) f ∗ (r′ )d3 r′
∫ ∫ ∫V
[
]∗
=
G(r, r′ ) f (r′ )d3 r′ −
G(r, r′ ) f ∗ (r′ )d3 r′
∫ ∫ ∫V
∫ ∫ ∫V
=
∫ ∫ ∫V
=
∫ ∫ ∫V
[G∗ (r, r′ ) − G(r, r′ )] f ∗ (r′ )d3 r′
[−2i[G(r, r′ )]] f ∗ (r′ )d3 r′
(3.11)
where [⋅] denotes the imaginary part operator.
This theoretical model of the closed surface TR imaging needs further
discussions.
First, the formula of time-reversal field Eq. (3.10) shows that a theoretical
TRM contains both monopole and dipole sources that are located at a closed
surface S enclosing all sources. The axes of the dipoles are perpendicular to the
surface S. The monopole sources emit the phase conjugated normal derivative
signals and the dipole sources emit the phase conjugated signals. In the special case of spherical surface S with a large radius, which is often encountered
in practical applications, the TR re-transmitting sources are greatly simplified.
The field in far-field zone (r ≫ 𝜆) satisfies the Sommerfeld radiation condition,
𝜕G(r, r′ )
(3.12)
≈ ikG(r, r′ )
𝜕r
which represents an outgoing spherical wave in the far field. The same condition
applies to the u(r). Since the normal direction of a spherical surface is just the
radial direction, the TR field Eq. (3.10), where r′ is in the far zone, is reduced to
either
uTR (r)
G(r, r′ )u∗ (r′ )dS(r′ )
(3.13)
=
∯S
−2ik
Time-Reversal Imaging
or
uTR (r)
𝜕G(r, r′ ) 𝜕u∗ (r′ )
=
dS(r′ ).
−2i∕k ∯S 𝜕n(r′ ) 𝜕n(r′ )
(3.14)
In (3.13) and (3.14), the same device is used as an emitter and a receiver, with
the former being monopole only and the latter dipole only. From a practical
point of view, this property brings significant conveniences compared with
the ideal time-reversal device. In addition, the time-reversal fields for the
monopole-transceiver-only case (3.13) and the dipole-transceiver-only case
(3.14) are different from the ideal time-reversal field just by a factor (−2ik
and −2i∕k, respectively). Consequently, the patterns of the time-reversal
field are the same in all the three cases. It is worth mentioning that this
conclusion is valid only for the case when the surface S is spherical with a large
radius. For other cases, the properties of the TR field obtained by monopoletransceiver-only TRM and dipole-transceiver-only TRM are discussed in [4, 5].
Second, the derivation of the TR field (3.10) indicates that the formula is valid
regardless of whether the boundary S is near or far field. In other words, even
if the boundary S is near zone of the original source, the measurement of near
field signal does not change the result of the TR field (3.11) and consequently
the resolution. In fact, this should not be a surprise. In order to fully recover the
information of a point source by the TR process, the phase of travelling wave
has to be compensated for and at the same time both the phase and amplitude of evanescent wave have to be compensated
for. For an evanescent wave,
√
k 2 +k 2 −k 2 |z|
−
x
y
the amplitude decays by a factor of e
when captured by the receiver.
When the evanescent wave is sent back√
to the original source position, the wave
−
k 2 +k 2 −k 2 |z|
x
y
. Thus, even in an ideal case
further decays by the same factor, e
where the sensors are placed in the near zone and are unlimitedly accurate in
sensing weak evanescent
waves, the time-reversed signal has to be first ampli√
2
k 2 +k 2 −k 2 |z|
x
y
before it is sent back. Since the TR process does
fied by a factor of e
not involve this amplification, the measurement of near field does not enhance
the resolution of TR imaging.
Third, the medium in which a wave propagates has to be reciprocal and lossless, which are two necessary conditions to achieve time-reversal imaging. If
the medium is lossy, there will be a term in (3.3) containing the first-order time
derivative of u(r, t). Since the equation contains both odd and even order time
derivatives, the invariance of the wave equation under time-reversal is lost.
Fourth, we discuss the time-domain implementation of TR imaging. Since
time-reversal operator Tu(r, t) = u(r, −t) is equivalent to phase conjugation
of every frequency component Tu(r, 𝜔) = u∗ (r, 𝜔), when implementing the
second step of the TR process that is described in Section 3.1.2, the driving
source is chosen as u(ri , −t) at the ith antenna. In practice, the time t is often
nonnegative, and thus it is more practical to formulate the driving source as
47
48
Electromagnetic Inverse Scattering
u(ri , T0 − t), where T0 is a large number, representing a sufficiently long break
in time between the two events: the source emits signal and the TRM sends
back signal. That is, the received signal at receivers during the recording
step should be negligibly small for t > T0 . Intuitively speaking, the fact that
driving current is chosen as u(ri , T0 − t) at the ith antenna means that a
receiver that receives the signal later compared to another receiver in the array
needs to send the signal back earlier, which ensures a synchronization of all
re-transmitted waves when they reach the original source position.
Fifth, the time-reversal principle for closed surface TRM presented here is
based on Green’s theorem. For a finite aperture case, Green’s theorem cannot
be used. In this case, the TR imaging technique can be understood as a matched
filter when the TRM consists of only monopole transceivers. In the language
of signal processing, if a delta function is the original signal, then the received
signal at the TRM is the impulse response (here, referred to as Green’s function)
of the channel (here, referred to as the background medium). It is well known
that the TRM sends the reversed version of the impulse response back through
the same channel, which means autocorrelation. This autocorrelation function
has a peak at the position where the original source is. In addition, we notice
that the time-reversal processing carries out a matched filtering in both time
and spatial domains [6–8].
Lastly, for acoustic waves, the wave equation presents a slightly different format compared with (3.3) [6, 9],
)
(
𝜕 2 u(r, t)
∇u(r, t)
1
= 0,
(3.15)
−
∇⋅
2
𝜌(r)
𝜌(r)𝑣 (r) 𝜕t 2
where the density 𝜌(r) varies with the space. Following the steps of deriving
(3.10), we are able to reach the time-reversal field for acoustic wave [6],
]
[
∗ ′
𝜕G(r, r′ ) ∗ ′
1
′ 𝜕u (r )
(3.16)
uTR (r) =
G(r, r )
−
u (r )
dS(r′ ).
∯S
𝜕n(r′ )
𝜕n(r′ )
𝜌(r′ )
The only difference from (3.10) is the factor 1∕𝜌(r′ ) appearing in the integrand.
3.1.4
Analysis of Resolution
Note that the result (3.11), that is,
uTR (r) =
∫ ∫ ∫V
[−2i[G(r, r′ )]] f ∗ (r′ )d3 r′
for full aperture TRM is applicable to any background medium, provided that
it is lossless and reciprocal. For the special case of homogeneous background
medium, the Green’s function is eikr ∕(4𝜋r) when the source is placed at the
origin and thus
uTR (r) =
−i sin(kr)
∝ sinc(kr).
2𝜋r
(3.17)
Time-Reversal Imaging
1
–10
0.8
0.8
–5
0.6
0.6
0
0.4
0.4
5
0.2
0.2
y
–15
10
0
15
–15 –10
–5
0
x
(a)
5
10
15
–0.2
1
Sinc(x)
x
–15
–10
5
–5
10
15
–0.2
–0.4
(b)
Figure 3.3 The spatial distribution of the TR field for 3D scalar wave case: (a) in the xy plane;
(b) along the x-axis. The values are normalized so that the peak value is 1.
The spatial distributions of the TR field in the xy plane and along the x-axis
are plotted in Fig. 3.3 (a) and (b), respectively. The wavenumber k is 1. The first
zero of the sinc function appears at r = 𝜋; that is, when the distance is at half
wavelength from the point source. If the resolution of image is defined as the
distance between the main lobe and the first zero, then the resolution of TR
imaging is half a wavelength. Note that there are other definitions on resolution,
but their results do not significantly differ from each other. Thus, the obtained
resolution of half wavelength in an homogeneous background medium can be
used as a rule of thumb, and consequently any imaging result that achieves a
better resolution than half a wavelength can be referred to as super-resolution
imaging.
In an inhomogeneous background medium, it is possible to achieve superresolution TR imaging. A focal spot as small as one-thirtieth of the wavelength
has been reported when the background medium is chosen as a microstructured medium that is strongly heterogeneous [10]. In such a case, the focus is
still given by the imaginary part of the Green’s function, but the near-field scattering of the wave off the medium’s heterogeneities (i.e., the microstructure)
allows the Green’s function to fluctuate faster than the wavelength.
For finite aperture TRM case, there is in general no analytical solution for the
distribution of TR field except for some special cases. For example, the resolution of TR image in both longitudinal and transverse directions for the case of
a paraxial wave TRM is discussed in [5].
3.1.5
Vectorial Wave
The time-reversal technique can be easily generalized to 3D electromagnetic
waves [11, 12]. The 𝜖(r) is the permittivity that accounts for inhomogeneity.
49
50
Electromagnetic Inverse Scattering
We assume that all materials√
under study are nonmagnetic; that is, 𝜇(r) = 𝜇0 .
The wavenumber is k(r) = 𝜔 𝜖(r)𝜇0 . The electric field E(r) that is generated
by a source distribution J(r) satisfies the following equation,
∇ × ∇ × E(r) − k 2 (r)E(r) = i𝜔𝜇0 J(r)
(3.18)
The dyadic Green’s function G(r, r′ ) for the background medium 𝜖(r) is
defined as
∇ × ∇ × G(r, r′ ) − k 2 (r)G(r, r′ ) = 𝛿(r − r′ )I
(3.19)
The transceivers are distributed within a closed surface S, the outward normal
̂ The time-reversal electric field interior to S
direction of which is denoted as n.
is defined as
ETR (r) =
∯S
̂ ′ )) × (−H∗ (r′ ))]
i𝜔𝜇0 G(r, r′ ) ⋅ [(−n(r
̂ ′ ))]dS(r′ )
+ GEM (r, r′ ) ⋅ [E∗ (r′ ) × (−n(r
(3.20)
where GEM (r, r′ ) = −[∇′ × G(r′ , r)]T , which means, by the reciprocity principle,
the dyadic Green’s function expressing the electric field generated by a unit
magnetic dipole source. Following the steps of deriving (3.11), the time-reversal
electric field is found to be
ETR (r) =
∫ ∫ ∫V
− 2i[G(r, r′ )] ⋅ (−i𝜔𝜇0 )J∗ (r′ )d3 r′
(3.21)
In electromagnetic wave TR process, the receivers in the TRM first measure the tangential components of electric and magnetic fields, and then
̂ and (−n)
̂ × (−H∗ ) are chosen as the driving magnetic and electric
E∗ × (−n)
current dipole source, respectively. Note that −n̂ is used since the observation
point is inside S whereas n̂ points outside of S. In practice, it is not convenient
to set up both electric and magnetic current sources. In the special case
of spherical surface S with a radius much larger than wavelength, which is
often encountered in practical applications, the TR retransmitting sources are
greatly simplified. The field in far-field zone satisfies radiation condition,
E∗ (r′ )
𝜂
̂ ′ ) × E∗ (r′ ) ≈ H∗ (r′ )𝜂
n(r
̂ ′ ) × H∗ (r′ ) ≈ −
n(r
(3.22)
(3.23)
̂ ) × G(r , r)] ⋅ = G(r, r ) ⋅ ik n(r
̂ )×
GEM (r, r )⋅ ≈ −[ik n(r
′
′
′
T
′
′
(3.24)
Thus, the first item in the integrand is equal to the second one. If a short-wire
receiver measures the electric field and then the same device is used as a
retransmitting antenna with the electric current dipole moment being the
Time-Reversal Imaging
complex-conjugated electric field (scaled by −1∕𝜂), then the time-reversed
field in this case is equal to
∯S
i𝜔𝜇0 G(r, r′ ) ⋅
E (r)
E∗ (r′ )
dS(r′ ) = TR
−𝜂
2
(3.25)
It is worth noting that this conclusion is valid only for the case when the surface
S is spherical with a large radius. Similarly, if small-loop antennas are used
to measure magnetic fields and then they act as a magnetic dipole radiating
antenna, the same result will be obtained.
It is important to note that Maxwell equations are form-invariant, under
the time-reversal transformation t → −t, if and only if D → D, E → E, 𝜌 → 𝜌,
B → −B, H → −H, and J → −J. It is easy to show that D and E can be directly
derived from electric charge density 𝜌, which doesn’t involve a time derivative. They are said to be even under time reversal. In comparison, J (involving
time derivative), B and H are said to be odd under time reversal. In the frequency domain, it is easy to prove by the definition of Fourier transform that
̂
time-reversal operator means E∗ and −H∗ . That is why (3.20) presents E∗ × (−n)
̂ × (−H∗ ) as the driving magnetic and electric current dipole source,
and (−n)
respectively.
For an electric dipole source located at the origin, oriented along the ẑ direction, the x, y, and z components of the TR electric field are proportional to
[Gxz (r, 0)], [Gyz (r, 0)], and [Gzz (r, 0)], respectively.
[(
)
]
3
3
1 xz
−1 + 2 2 sin kr −
[Gxz (r, 0)] =
cos kr
(3.26)
2
4𝜋r r [(
k r )
kr
]
3
3
1 yz
−1 + 2 2 sin kr −
(3.27)
cos kr
[Gyz (r, 0)] =
2
4𝜋r r[(
k r
kr
)
1
z2
3z2
1
[Gzz (r, 0)] =
1 − 2 2 − 2 + 2 4 sin kr
4𝜋r
k r
r
k r
(
)
]
2
1
3z
+
cos kr
(3.28)
−
kr kr3
The x, y, and z components of the (normalized) ETR in the xy, yz, and zx planes
are plotted in Fig. 3.4, where we see that the z-component of ETR is much larger
than the x and y components. The z components of ETR along the x- and z-axes
are plotted in Fig. 3.5. We see that the z component of ETR peaks at the original
source position along both the x- and z-axes. The first zero of the z component
of ETR along the x- and z-axes is x = 0.44 𝜆 and z = 0.71 𝜆, respectively.
These results show that the position of the source is identified as where |ETR |,
the magnitude of the TR field, is at a maximum. The quantitative information of
the x, y, z components of the vector source, and consequently its polarization,
can be obtained by solving an optimization problem that is formulated in [13].
51
Electromagnetic Inverse Scattering
–4λ
–4λ
1
0.5
2λ
2λ
4λ
4λ
–4λ
0
–2λ
0
x
(b)
2λ
4λ
z
0.5
2λ
–4λ
0
y
(d)
2λ
4λ
0
0
4λ
–4λ
0
0.1
–2λ
0
x
(c)
2λ
4λ
–4λ
2λ
–2λ
4λ
–4λ
0
0.05
–2λ
0
0
2λ
–4λ
1
–2λ
z
0.5
–0.05
–2λ
–4λ
0
y
(e)
2λ
1
0.2
0
0.1
2λ
4λ
–4λ
4λ
0
–2λ
–4λ
0
y
(f)
2λ
4λ
–0.03
2λ
–2λ
0.5
0
x
0
0
–2λ
x
0.03
2λ
–0.1
0.3
0.06
–2λ
–0.1
0.3
–2λ
z
0
x
(a)
–4λ
4λ
–4λ
0
2λ
–2λ
0.3
–2λ
y
0
4λ
–4λ
–4λ
1
–2λ
y
y
–2λ
x
52
0.2
0
0.1
2λ
0
–0.06
4λ
–4λ
–2λ
0
z
(g)
2λ
4λ
4λ
–4λ
–2λ
0
z
(h)
2λ
4λ
0
4λ
–4λ
–2λ
0
z
(i)
2λ
4λ
–0.1
Figure 3.4 The plot of the (normalized) ETR : Top, middle, and bottom rows show the field
distribution in the xy, yz, and zx planes, respectively; Left, middle, and right columns show
the x, y, and z components of the field, respectively.
0.4
0.4
0.2
0.2
0
0
–0.2
–4λ
–2λ
0
x
(a)
2λ
4λ
–0.2
–4λ
–2λ
Figure 3.5 The z component of ETR along the x- and z-axes.
0
z
(b)
2λ
4λ
Time-Reversal Imaging
3.2 Time-Reversal Imaging for Passive Sources
Regarding the applications of TRM imaging, one of the most promising areas is
the detection and imaging of passive targets. In fact, the problem of imaging of
passive targets is closely related to the imaging of active sources, which has been
discussed in Section 3.1, since scattered field generated by passive targets can be
considered as the radiated field by an induced source, which is also known as
the secondary source.
The detection and imaging of passive targets is in fact an inverse scattering problem. A schematic of the problem is shown Fig. 1.1, where a target is
illuminated by an array of transmitters and the scattered fields are measured
at an array of receivers. In the application of TRM imaging, we use the same
antenna array to function as both transmitters and receivers, which are referred
to as transceivers. The scattering system can be described a multistatic response
(MSR) matrix K. If N is the number of transceivers, the MSR matrix is of size
N × N, with the element Ki,j representing the received field at the ith receiver
when the jth transmitter is driven by a unit source.
To study the problem of detecting passive targets, it is desirable to first look
into a simple case, where an analytical expression of the MSR matrix K can
be easily identified. In the special case of a total number of M point-like targets (size being much smaller than wavelength) that are sufficiently separated
from each other so that multiple scattering effects can be ignored, the MSR
matrix can be written as the product of three matrices: (1) a forward propagation matrix GI that describes the illumination process, that is, the propagation
from transmitters to scatterers, (2) a scattering matrix C, which is diagonal, with
the element being the scattering strength, which is also known as the reflectivity coefficients, of point targets, and (3) the backward propagation matrix GS
that describes the process of radiation of scattered field; that is, the propagation
from scatterers to receivers.
K = GS ⋅ C ⋅ GI
(3.29)
The incidence matrix GI is of size M × N, with the element GIij being Green’s
function evaluated at the position of the ith point-like targe and the jth transmitter. The scattering matrix C is an M-dimensional diagonal matrix, with the
element Ci denoting the scattering strength of the ith target that is defined as
the amplitude of induced source due to a unit excitation field. The scattering
matrix GS is of size N × M, which is just the transpose of GI due to reciprocity.
As a consequence of the well-known reciprocity theorem, the MSR matrix K is
transpose symmetric.
53
54
Electromagnetic Inverse Scattering
3.2.1
Imaging by an Iterative Time-Reversal Process
Imaging of passive targets can be realized by an iterative time-reversal process.
The steps of implementation are as follows. Firstly, the transceiver array generates waves to illuminate the region of interest. Then, the transceiver array
records the scattered field that is due to the presence of passive targets. Next,
the recorded field is complex-conjugated, which is then chosen as the amplitude
of the driving source at the transceiver. These steps of illuminating, recording,
and time reversal are iteratively implemented. After some iterations, the process converges and produces a wavefront focused on the most reflective target
under some conditions.
If S0 is the initial driving source at the transceiver array, then the driving
source after the time-reversal operation is the phase conjugate of the received
signals
S1 = (K ⋅ S0 )∗
(3.30)
After two time-reversal operations, the driving source is given by
∗
S2 = K ⋅ K ⋅ S0
(3.31)
It is easy to see that the driving source after 2n iterations of TR process is
given by
∗
S2n = (K ⋅ K)n ⋅ S0
(3.32)
∗
The matrix K ⋅ K is called the time-reversal operator (TRO). Since K is symmetric, the time-reversal operator is Hermitian and its eigenvalues are all real.
In addition, it is easy to prove that all eigenvalues are nonnegative.
∗
K ⋅ K ⋅ V i = 𝜆i V i ,
i = 1, 2, .., N
(3.33)
Since the set of N eigenvectors form an orthogonal basis in  N , the vector of
initial driving source S0 can be decomposed to
S0 =
N
∑
𝛼i V i
(3.34)
i=1
Consequently, we obtain [14]
S2n =
N
∑
𝜆ni 𝛼i V i
(3.35)
i=1
Consider the special case of well-resolved targets. For a given transceiver
array, two targets are well resolved if time-reversal field that is due to the
passive source induced at one target generates a zero illumination at the other
target [14, 15]. Mathematically, a collection of well-resolved targets means
T
∗
that GS,i ⋅ GS,j = 0 for i ≠ j, where GS,i , i = 1, 2, ..., M are the columns of the
Time-Reversal Imaging
scattering matrix GS . In other words, the columns of GS are orthogonal in the
sense of vector inner product. In this case, the time-reversal operator has a
simplified formula,
∗
T
∗
2
K ⋅ K = GS ⋅ Diag[|C12 ||GS,1 |2 , |C22 ||GS,2 |2 , ..., |CM
||GS,M |2 ] ⋅ GS
(3.36)
It is straightforward to verify that the eigenvectors of the time-reversal operator
are,
∗
V i = GS,i ,
i = 1, 2, .., M
(3.37)
associated with the eigenvalues
𝜆i = |Ci2 ||GS,i |4 ,
i = 1, 2, .., M
(3.38)
√
The term |Ci ||GS,i |2 , which is equal to 𝜆i , is referred to as the apparent
reflectivity. If the first scatterer has the greatest scattering ability so that
𝜆1 > 𝜆2 ≥ ..., ≥ 𝜆M , then for a large integer n, the driving source after 2n times
TR process (3.35) reduces to
∗
S2n ≈ 𝜆n1 𝛼1 GS,1
(3.39)
The value of n that is large enough to reach the above approximation depends on
the ratio 𝜆1 ∕𝜆2 . The driving source after 2n times TR process is proportional to
the signal after one TR process in the fictitious case when the first target alone
were present. Thus, the TR field after 2n times TR process focuses on the first
target.
3.2.2
Imaging by the DORT Method
The iterative TR process in fact can be considered to experimentally obtain the
eigenvector associated with the largest eigenvalue of the TRO matrix. Alternatively, a numerical approach by implementing an eigenvalue decomposition
of the TRO matrix will be more powerful, since all eigenvalues and associated
eigenvectors can be obtained simultaneously. This imaging approach is referred
to as the DORT method, which is the French acronym for “Décomposition de
l’Opérateur de Retournement Temporel” (decomposition of the time-reversal
operator) [16].
The DORT method is performed in three steps: The first step is to obtain the
MSR matrix K by conducting a total number of N 2 transmit-receive operations.
The second step is to perform an eigenvalue decomposition of the time-reversal
∗
operator K ⋅ K. The third step is to apply each eigenvector as the driving source
at transceivers and send waves back to the region of interest. This can be done
either experimentally or numerically.
For a set of well-resolved scatterers, the iterative time-reversal process allows
a selective focusing on the most reflective scatterer, whereas the DORT method
is able to focus on each scatterer provided that their apparent reflectivities are
55
56
Electromagnetic Inverse Scattering
different. If the eigenvalues 𝜆i formulated in (3.38) are different from each, written in decreasing order 𝜆1 > 𝜆2 > … , > 𝜆M , then the eigenvector V i that is
formulated in (3.37) is the signal that would act as the driving source after a
time-reversal process if only the ith target were present. Thus, the TR field due
to V i focuses on the ith target. In addition, we see that the number of non-zero
eigenvalues is equal to the number of scatterers. As a side note, for the continuous counterpart of K, the focusing property of its eigenfunctions is extensively
discussed in [17].
3.2.3
Numerical Simulations
Since the iterative TR process can be considered to be a special case of the
DORT method in the sense that the former obtains only the first eigenvector
of the TRO matrix, all imaging results presented in this section are computed
by the DORT method. Before conducting numerical simulations, it is worth
discussing well-resolved scatterers and the physical interpretation of the DORT
method.
For a collection of well-resolved scatterers, the requirement is for any pair of
scatterers that the time-reversal field that is due to the passive source induced
at one scatterer generates a zero illumination at the other scatterer. However,
it is rare that scatterers are well-resolved. For a given array of transceivers, the
positions of scatterers have to be carefully chosen so as to meet the condition
of being well-resolved. For example, in the case of full aperture transceivers,
the condition of well-resolved scatterers is that the imaginary part of Green’s
function [G(ri , rj )] is equal to zero for any pair of scatterers. In practice, a
good approximation to the condition of being well-resolved is that scatterers
are sufficiently separated from each other. Nevertheless, to apply the DORT in
such an approximation condition is of little interest, since the imaging problem
is not challenging for sufficiently separated scatterers. Indeed, there are many
existing algorithms to image sufficiently separated scatterers. Considering the
above, our numerical simulations in this section do not require that scatterers
have to be well resolved.
For scatterers that are not well resolved, the physical meaning of the
DORT method can be interpreted from the singular value decomposition
H
(SVD) of the MSR matrix K, which satisfies K ⋅ 𝑣p = 𝜎p up and K ⋅ up = 𝜎p 𝑣p ,
p = 1, 2, … , N, where 𝜎p is singular value, left singular vectors up are mutually orthogonal unit vectors, and so are the right singular vectors 𝑣p . The
superscript H denotes the Hermitian operation. Since the K is symmetric, the
∗
H
time-reversal operator K ⋅ K can be rewritten as K ⋅ K, which is well known
to have eigenvalues 𝜎p2 and the corresponding eigenvectors 𝑣p . When the initial
driving source at transceivers is chosen to be 𝑣p , the driving source after two
time-reversal operations (3.31) is given by
∗
ind
K ⋅ K ⋅ 𝑣p = (GS ⋅ Sp )∗
(3.40)
Time-Reversal Imaging
ind
where the M-dimensional vector Sp = 𝜎p C ⋅ GI ⋅ u∗p is referred to as the
eigen-source, which is a vector of the induced sources at all scatterers.
Equation (3.40) is the signal that would act as the driving source after a
ind
time-reversal process if the eigen-source Sp were present. It is important to
note that the interpretation (3.40) using the concept of eigen-source applies
to the case of multiply scattering scatterers as well, with the only difference
ind
being that the vector of induced source Sp is more complex than the single
scattering case. The single scattering model (3.29) is for the purpose of ease
in presentation, but the DORT imaging method is not limited to the single
scattering case.
From (3.40), we tell that the number of non-zero singular values is equal
to the degrees of freedom of eigen-sources [18]. For scalar wave equations,
the induced source at point-like scatterers is monopole source and thus the
number of independent of sources is equal to the number of scatterers. For
vectorial waves, dipole sources are induced at point-like scatterers and consequently one scatterer corresponds to two or three eigenvalues in the 2D or 3D
case, respectively. If electric and magnetic dipoles are simultaneously induced
in small scatterers, then one scatterer corresponds up to six nonzero singular
values [19]. As a side note, the distribution of singular values is quite different
for an extended scatterer [20].
In numerical simulations, both scalar and vector wave DORT imaging results
are presented. The system is a two-dimensional problem, which is invariant
in the z-axis. The background is air, and the wavelength 𝜆 is 1 m. The array
of transceivers consists of 100 line elements that are uniformly distributed in
a circle of radius 200 𝜆. If the line transceiver is electric current line source
when operating in the transmitting mode and it measures the z-component of
electric field when operating in receiving mode, the setup is the TM mode. By
duality, if the line transceiver is magnetic current line source and it measures
the z-component of magnetic field, it is in the TE mode. The scatterers considered in numerical simulations have a relative permittivity 𝜖r = 2.25 and a small
radius R = 𝜆∕200. The numerical method to calculate the MSR matrix can be
found in [21] and the noise-free data are treated as measured data.
First, we consider a single scatterer located at (0.3 𝜆, −0.3 𝜆). For the TM
mode, the dominant induced source is the electric current along the z–axis and,
consequently, there should be only a single leading singular value, with others
being close to zero. The first 15 singular values of the MSR matrix are shown in
Fig. 3.6 (a), where we see that there is indeed a single dominant singular value.
The DORT imaging result is shown in Fig. 3.7, where we see concentric patterns centred at the position of the scatterer. Since the application of DORT to
detect a single scatterer is equivalent to locating a single induced source, it is
not a surprise to see that Fig. 3.7 bears the resemblance to Fig. 3.3(a). For TE
mode, the dominant induced sources are electric dipoles that have two degrees
57
5
–5
0
–10
log10(σj)
log10(σj)
Electromagnetic Inverse Scattering
–5
–15
–10
–15
–20
0
5
10
Singular value number, j
(a)
15
–25
5
5
0
0
log10(σj)
log10(σj)
58
–5
–10
–15
0
5
10
Singular value number, j
(b)
15
0
5
10
Singular value number, j
(d)
15
–5
–10
0
5
10
Singular value number, j
(c)
15
–15
Figure 3.6 The base-10 logarithm of the first 15 singular values of the MSR matrix. (a) A
single scatterer under TM illumination; (b) a single scatterer under TE illumination; (c) two
scatterers separated by d = 0.6 𝜆 under TM illumination; and (d) two scatterers separated by
d = 0.2 𝜆 under TM illumination.
2
1
0.8
1
0.6
0
0.4
–1
–2
–2
0.2
–1
0
1
2
Figure 3.7 DORT imaging result
for a single scatterer under TM
illumination.
Time-Reversal Imaging
2
1
0.8
1
2
1
0.8
1
0.6
0.6
0
0
0.4
0.4
–1
–2
–2
0.2
–1
0
(a)
1
2
–1
–2
–2
0.2
–1
0
1
2
(b)
Figure 3.8 DORT imaging results for a single scatterer under TE illumination. Images (a) and
(b) are obtained by the singular vectors corresponding to the first and the second singular
values, respectively.
of freedom in the xy plane. Consequently, there should be two leading singular
values. The first 15 singular values of the MSR matrix are shown in Fig. 3.6(b),
where we see that there are indeed two dominant singular values. Their corresponding singular vectors are used in the DORT to generate the imaging results
that are shown in Fig. 3.8. We see that the DORT imaging results obtained by
both singular vectors present a two-sidelobe pattern, which is centered at the
position of the scatterer. However, the TR field is zero at the position of the
scatterer, rather than being strongest.
Next, we consider two scatterers located at (−d∕2, 0) and (+d∕2, 0), respectively, where d is the distance between the two scatterers. Under the TM mode,
the DORT imaging results for two different values of d, that is, d = 0.6 𝜆 and
d = 0.2 𝜆, are plotted in Fig. 3.9 and Fig. 3.10, respectively. The first 15 singular
values of the MSR matrix are shown in Fig. 3.6(c) and (d) for the two separations, where we see two dominant singular values since two electric line sources
are induced in each case. For a larger separation, d = 0.6 𝜆, Fig. 3.9(a) shows
that the DORT imaging result generated by the first singular vector is able to
successfully locate the positions of the two scatterers. Figure 3.9(b) shows that
the second singular vector fails to do so. For a smaller separation, d = 0.2 𝜆,
Fig. 3.10 shows that neither singular vector is able to successfully locate the
positions of the two scatterers.
Lastly, we consider a partial aperture of transceivers, which covers only the
angle 0 ≤ 𝜙 ≤ 𝜋∕2. All other conditions are the same as those for Fig. 3.7 and
Fig. 3.8. The DORT imaging result under TM illumination for partial aperture
of transceivers is shown in Fig. 3.11. Compared with the result Fig. 3.7 for full
aperture, Fig. 3.11 presents a focal spot that is elongated along the axis of symmetry of the partial aperture transceivers. The focal spot peaks at the location
of the scatterer.
59
60
Electromagnetic Inverse Scattering
2
1
0.8
1
2
1
0.8
1
0.6
0.6
0
0
0.4
0.4
–1
–2
–2
0.2
–1
0
1
–1
–2
–2
2
0.2
–1
0
1
2
(b)
(a)
Figure 3.9 DORT imaging results for two scatterers under TM illumination. The separation
of the two scatterers is d = 0.6 𝜆 and their exact positions are marked by black dots. Images
(a) and (b) are obtained by the singular vectors corresponding to the first and the second
singular values, respectively.
2
1
0.8
1
2
1
0.8
1
0.6
0.6
0
0
0.4
0.4
–1
–2
–2
0.2
–1
0
(a)
1
2
–1
–2
–2
0.2
–1
0
1
2
(b)
Figure 3.10 DORT imaging results for two scatterers under TM illumination. The separation
of the two scatterers is d = 0.2 𝜆 and their exact positions are marked by black dots. Images
(a) and (b) are obtained by the singular vectors corresponding to the first and the second
singular values, respectively.
The phenomenon of axial elongation of the focal spot is also typical in lens
imaging in optics [22]. Since the finite-aperture transceivers used in TR imaging
are somewhat equivalent to a finite-size optical lens, it is worth further discussing the axial elongation of the focal spot. An intuitive understanding of the
axial elongation of focal spot is as follows. Since the transceivers are in the far
field, the radiation by a transceiver driven by a time-reversed signal is a plane
wave in a finite region around the scatterer, with the propagation direction
Time-Reversal Imaging
Figure 3.11 DORT imaging result for a
single scatterer under TM illumination,
where transceivers cover only the angle
0 ≤ 𝜙 ≤ 𝜋∕2.
4
1
0.8
2
0.6
0
0.4
–2
–4
–4
0.2
–2
0
2
4
from the transceiver to the scatterer. If the origin of a local coordinate system
is chosen at the position of scatterer, the axial direction is labelled as l, and
the transverse plane is labelled as t, then the plane wave can be formulated as
exp [i(k l ⋅ rl + k t ⋅ rt )]. For a pair of transceivers that are distributed symmetrically about the axis, the signs of their k t s are opposite, so that a standing wave
pattern is formed along the transverse direction. On the other hand, their k l s
have the same sign so that no standing wave is formed along the axial direction.
It is well known that the magnitude of standing wave exhibits a spatial variation,
whereas the magnitude of a plane wave does not. Thus, the magnitude of the
TR field around the position of scatterer changes more slowly along the axial
direction than along the transverse direction; that is, the focal spot is elongated
along the axial direction.
The DORT imaging result under TE illumination for partial aperture of
transceivers is shown in Fig. 3.12. While the first singular vector generates an
axial-elongated spot that peaks at the location of scatterer, the second one
generates an axial-elongated two-sidelobe pattern, centered at the location of
scatterer with a minimum magnitude.
These numerical simulations use noise-free data. The performance of the
DORT algorithm under additive noise and translational perturbations of
transceivers is presented in [23]. For 3D simulation of the DORT algorithm,
[24] presents several numerical illustrations. In particular, an interesting result
is presented for a spherical scatterer, the permeability and permittivity of
which are both different from those of the background medium. Although
induced dipole sources have up to six freedoms, that is, three electric dipoles
and three magnetic dipoles, [24] demonstrates that the MSR has only five
nonzero singular values. The reason for this discrepancy is that the transceivers
used in [24] are all oriented in the z direction, which generates a magnetic field
that has only x and y components (see (2.64)). Thus, the one freedom that is
missed is the z-oriented magnetic dipole.
61
62
Electromagnetic Inverse Scattering
4
1
0.8
2
4
1
0.8
2
0.6
0.6
0
0
0.4
0.4
–2
–4
–4
0.2
–2
0
(a)
2
4
–2
–4
–4
0.2
–2
0
2
4
(b)
Figure 3.12 DORT imaging results for a single scatterer under TE illumination, where
transceivers cover only the angle 0 ≤ 𝜙 ≤ 𝜋∕2. Images (a) and (b) are obtained by the
singular vectors corresponding to the first and the second singular values, respectively.
3.3 Discussions
The advantage of time-reversal imaging is that it is fast and easy to implement,
and it is able to provide an approximate image of the source or scatterer. The
weakness of the TR imaging is its limited resolution. The resolution limit is
inherent, even in absence of noise. Considering the aforementioned properties,
TR imaging is a convenient and useful tool in the following scenarios. Some real
world applications only require detection of the presence/absence of source or
scatterer and do not need super-resolution. For example, the aim is to detect
submarines in underwater acoustics. If some applications require high resolution, the rich information that TR imaging quickly provides, such as rough
estimation of scatterers’ positions, sizes, and number, can be used as initial
guess or a priori knowledge for other super-resolution imaging algorithms.
Time-reversal imaging has found wide applications in the area of imaging and
detection, such as ultrasound imaging in medical diagnostics [25], structural
health monitoring [26], microwave breast cancer detection [27], and detection
of buried objects [28]. The performance of TR imaging has been validated both
numerically and experimentally. Since this chapter discusses the application of
time-reversal in imaging and detection, we only mention in passing that time
reversal has many other applications, such as communication, filter design, and
energy harvesting.
This chapter mainly presents the TR in frequency domain. In addition to the
reason that time-domain signal can be expressed as a linear superposition of
its Fourier components, many real-world applications of TRM indeed deal with
quasi-monochromatic waves. For temporal dispersion materials, it is more convenient to present in the frequency domain. On the other hand, the TR in the
Time-Reversal Imaging
time domain is also a rich research topic. Since time is also a variable, there is an
additional freedom in time-domain TR. Time reversal provides both temporal
and spatial matched filtering. The theory and implementation of time-domain
time reversal can be found in [9, 29–33]. A closely related topic is ultrawideband
(UWB) time-reversal imaging, which is discussed in [28, 34].
It is important to distinguish physical and computational time reversal. In
physical time reversal, transceivers retransmit the time reversed signals to the
medium. The beauty of physical time reversal is that we need not know any
details of the medium, as long as it is lossless and reciprocal. The retransmitted
signal will automatically focus on the vicinity of the original source or scatterer.
For example, physical time-reversal has been applied to kidney stone destruction [35]. In comparison, computational time reversal simulates the process of
physical time-reversal in computers, and in this situation we need to know the
details of the medium so that the wave propagation can be numerically calculated or analytically derived. All numerical results presented in Section 3.2.3
are obtained by computational time-reversal.
For electromagnetic time reversal, practically speaking, electric field is never
directly measured, but instead voltages or currents on antennas are measured.
In this direction, the time-reversal theory in terms of impedance is developed
in [36], which is closer to microwave experiments. Impedance matrix formalism is very useful because it naturally takes into account the coupling between
radiating antennas.
Various numerical and experimental results have demonstrated that
super-resolution time-reversal imaging can be achieved in rich scattering
environments, such as in random media, in turbid media, and in cluttered
environments [37–40]. The mathematical foundation of random-media
time-reversal imaging can be found in [41, 42]. Generally speaking, the
more inhomogeneous the media is, the higher the focusing resolution that is
achieved. Intuitively, time reversal in rich scattering environments is equivalent to generating a virtual aperture that is larger than its actual physical size,
yielding a much higher resolution.
For point-like scatterers, when they are illuminated by transmitters, the scatterers can be identified as secondary sources. Since each incidence generates
one set of secondary sources on point-like scatterers, the total number of sets
of secondary sources is equal to the number of transmitters Nt . Mathematically, the sensing of secondary source has no difference from the sensing of
primary source. Thus, we can effectively treat the MSR matrix as the measured
signals that are due to the radiation of primary sources at Nt different time
slots. Consequently, [43] shows that the time-reversal operator can be written
as a covariance matrix, like the one introduced in classical primary-source
detection. Signal processing methods that work on the covariance matrix that
have been widely used in the field of source detection can be applied to solve
inverse scattering problem involving point-like scatterers, which will be topic
of the next chapter.
63
64
Electromagnetic Inverse Scattering
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67
4
Inverse Scattering Problems of Small Scatterers
The chapter considers scatterers whose dimensions are much smaller than
the wavelength, which are often referred to as small scatterers, point-like
scatterers, or Rayleigh scatterers. The inverse scattering problem (ISP) of small
scatterers consists of determining the locations and scattering strengths of
small scatterers. For inverse scattering problems, it is very important to discuss
small scatters before solving the problems involving extended scatterers. The
reasons are threefold: First, ISPs involving small scatterers can be tackled by
semi-analytical and fast algorithms, and their mathematical foundation is
solid; Second, physical insights into scattering mechanisms of small scatterers
can be obtained, which are significantly important in understanding the
scattering problem involving extended scatterers; Third, imaging resolution is
usually defined as the ability to resolve two small scatterers.
This chapter first addresses scalar wave ISPs of small scatterers in
Sections 4.1–4.3, and then discusses the vector wave counterpart in
Section 4.4. Section 4.1 introduces the Foldy–Lax equation, which is the
governing equation for the forward scattering problem. Section 4.2 presents
the uniqueness theorem that states that the locations and scattering strengthes
of small scatterers can be uniquely determined. Numerical methods for solving
ISPs of small scatterers are presented in Section 4.3, which consists of two
parts; the first one being the multiple signal classification (MUSIC) method
that determines the locations of scatterers and the second one being the
two-step least-squares method that retrieves scattering strengths of scatterers
in a noniterative way. Section 4.4 discusses the vector wave ISP, which is
not a simple generalization of its scalar counterpart. Physical insights and
mathematical manipulations are presented so that the inversion algorithm
not only is able to reconstruct small scatterers with special shapes but also
improves imaging resolution. Section 4.5 briefly discusses several topics, such
as the relation of the small-scatterer ISP to the problem of determining the
direction of arrival (DOA).
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
68
Electromagnetic Inverse Scattering
4.1 Forward Problem: Foldy–Lax Equation
The Foldy–Lax equation has been widely used as an efficient numerical approximation of multiple scattering of time-harmonic wave through a medium with
many separated scatterers, the radii of which are much smaller than the
wavelength. For a scatterer that is much smaller than wavelength, its scattering
mechanism can be well approximated by a point with scattering strength 𝜏,
which is defined as the ratio of the amplitude of induced source to the field
illuminating at this scatterer. Assume that there are M point-like scatterers
located at rm , m = 1, 2, … , M, with scattering strength 𝜏m , m = 1, 2, … , M,
respectively. For a given time-harmonic incident field 𝜓 i , the total field 𝜓 t as a
consequence of the interaction of the incident field with the M point scatterer
consists of two parts, with the first being the original incident field and the
second being the scattered field; that is
𝜓 t (r) = 𝜓 i (r) +
M
∑
G(r, rm )[𝜏m 𝜓 t (rm )],
(4.1)
m=1
where G denotes the background Green’s function. Equation (4.1) applies to
all spatial points except at the point scatterers r = rm , where the background
Green’s function is singular. It is important to note that a point-like scatterer
physically occupies an infinitely small but still finite volume. The scattering
strength 𝜏m is defined as the ratio of the amplitude of induced source to the
field illuminating at this scatterer, which automatically absorbs the self-field
contribution at the point scatterer rm . Consequently, the total field 𝜓 t (rm )
denotes the total field illuminating at rm , instead of the field generated at the
center rm of the small but finite scatterer. When evaluated at the M point-like
scatterers, Eq. (4.1) is replaced by the so-called Foldy–Lax equations ([1] and
section 7.2 of [2])
∑
𝜏m′ G(Xm , Xm′ )𝜓 t (rm′ ),
(4.2)
𝜓 t (rm ) = 𝜓 i (rm ) +
m′ ≠m
Written in a compact format, the total field 𝜓 t is related to the incident field
𝜓 i by
𝜓 t = (I M − Φ ⋅ Λ)−1 ⋅ 𝜓 i ,
(4.3)
where 𝜓 t is an M-dimensional column vector, 𝜓 t = [𝜓 t (r1 ), 𝜓 t (r2 ), … , 𝜓 t (rM )]T
and the superscript T denotes the transpose. The column vector 𝜓 i is defined
similarly. I M is an M-dimensional identity matrix. Λ is a diagonal matrix, with
Λ(m, m) = 𝜏m for m = 1, 2, … , M. Φ is an M × M matrix, with the mth row
and m′ th column (m, m′ = 1, 2, … , M) being Φ(m, m′ ) = G(rm , rm′ ) for m ≠ m′
and zero otherwise. Once the total field 𝜓 t is computed at every point-like
Inverse Scattering Problems of Small Scatterers
scatterer, the scattered field can be evaluated at an arbitrary spatial point from
the definition of Eq. (4.1),
𝜓 s (r) =
M
∑
G(r, rm )[𝜏m 𝜓 t (rm )].
(4.4)
m=1
In practice, consider a total number of Ni incidences that are due to transmitters located at rip , p = 1, 2, ..., Ni . For each incidence, the scattered electric field
is measured by an array of Ns receivers, which are located at rsq , q = 1, 2, ..., Ns .
The scattering property of the collection of point-like scatterers is represented
by a multistatic response (MSR) matrix K. The MSR matrix is of size Ns × Ni ,
with the element Kij representing the received field at the ith receivers when
the jth transmitter is driven by a unitary source. From (4.3) and (4.4), we obtain
the expression of K as
K = GS ⋅ Λ ⋅ (I M − Φ ⋅ Λ)−1 ⋅ GI ,
(4.5)
where GS is the scattering matrix of size Ns × M, with the element being
GS (q, m) = G(rsq , rm ), and GI is the incident matrix of size M × Ni , with the
element being GI (m, p) = G(rm , rip ).
4.2 Uniqueness Theorem for the Inverse Problem
The inverse scattering problem of determining the locations and scattering
strengths of small scatterers can be uniquely solved, under mild conditions.
The key of the theoretical foundation lies in the injectivity of the so-called
source-to-field mapping operator, which is presented in the Theorem 2.1 of [3].
In a nutshell, if we consider a radiation problem where sources are placed at
the locations of point-like scatterers, then it is impossible for two different
sources to generate the same radiation fields at a set of discrete measurement
positions, as long as the number of measurement points is large enough.
This section is somewhat oriented to theoretical and mathematical aspects.
Readers should feel free to skip this section if their primary interests are not
in theories and mathematical proofs. Since this section does not involve any
implementation details, the skipping of this section will not affect the understanding of the rest sections. However, a good understanding of this section
will deepen the understanding of inverse scattering problems involving small
scatterers.
Mathematically, the radiation problem can be defined as an operator Gs ,
which maps the complex amplitude of source 𝜆 ∈ ℂM at the positions of M
point-like scatterers to the space C(ℝ3 ⧵{r1 , r2 , … , rM }) of continuous function
69
70
Electromagnetic Inverse Scattering
on the three-dimensional spatial space excluding the M point-like scatterers
r ∈ ℝ3 ⧵{r1 , r2 , … , rM },
(Gs 𝜆)(r) ∶=
M
∑
G(r, rm )𝜆m
(4.6)
m=1
The ℂM denotes the M-dimensional complex vector space. In practice, radiated field is measured at a finite, say N, discrete positions r′1 , r′2 , … , r′N . The key
ingredient of the Theorem 2.1 of [3] makes the following statement:
Define the operator Γ: ℂM to ℂN , which is a discrete version of the Gs operator, by
𝜆 → {(Gs 𝜆)(r′1 ), (Gs 𝜆)(r′2 ), … , (Gs 𝜆)(r′N )}T .
(4.7)
Then there exists an integer N0 such that the operator Γ is one-to-one (i.e., injective) for N ≥ N0 .
The proof of the injectivity of Γ will not be outlined here and interested
readers are referred to [3]. Nevertheless, it is worth mentioning that an
important step in proving this statement is to first prove the injectivity of Gs .
This step is simple but important and we briefly discuss it here. Let 𝜆 ∈ ℂM ,
such that (Gs 𝜆)(r) = 0 for all r ∈ ℝ3 ⧵{r1 , r2 , … , rM }. When r tends to one of
the points rm , the fact that G(r, rm ) approaches infinity yields that 𝜆m = 0 for
every m = 1, 2, … , M. Thus, if two vectors of source 𝜆1 and 𝜆2 generate the
same radiation field (Gs 𝜆1 )(r) = (Gs 𝜆2 )(r) for all r ∈ ℝ3 ⧵{r1 , r2 , … , rM }, then
we conclude 𝜆1 = 𝜆2 . Intuitively, the injectivity of the operator Γ is equivalent
to the injectivity of the operator Gs for all r ∈ ℝ3 ⧵{r1 , r2 , … , rM } when the
number of measurements is sufficiently large.
Roughly speaking, from the perspective of linear algebra, the injectivity of
Γ can be understood as follows: when the number of equations is sufficiently
large so that the rank of the resultant linear equational system is equal to the
number of unknowns, there is one and only one solution.
As corollaries, the injectivity of the source-to-field operator Gs theoretically
justifies the uniqueness of position-locating and the retrieval of scattering
strengths, which will be discussed in detail hereafter.
4.2.1
Inverse Source Problem
The inverse source problem consists of determining the positions and amplitudes of a collection of point sources. When the number of measurements is
sufficiently large, the solution to the inverse source problem is unique, which
can be proven by contradiction. Assume, to the contrary, that there are positions {r01 , r02 , … , r0M′ } of another set of point sources that are different from
the positions {r1 , r2 , … , rM } of exact point sources, such that the former set
Inverse Scattering Problems of Small Scatterers
of sources 𝜆′m′ generates the same radiation field as the second set of sources
𝜆m does; that is,
M
∑
′
G(r, r0m′ )𝜆′m′ =
m′ =1
M
∑
G(r, rm )𝜆m ,
(4.8)
m=1
Consider the case when the number of measurement point r is sufficiently large.
Then the new set of amplitudes of sources {𝜆′1 , … , 𝜆′M′ , −𝜆1 , … , −𝜆M }, corresponding to the group of M′ + M points {r01 , … , r0M′ , r1 , … , rM }, generates
null fields at sufficiently many measurement points. Since the injectivity of the
operator Γ applies to any finite number of point sources, (4.8) contradicts the
injectivity of the operator Γ. After the uniqueness of the position-locating is
proven, the amplitudes of sources at those positions are unique, which is a
direct conclusion of the injectivity of the operator Γ.
Although this proof first proves the uniqueness of position-locating and then
proves the uniqueness of the amplitudes of sources, in practice of solving an
inverse source problem, it is difficult to first determine the positions of sources
and then determine the amplitudes. A common approach is as follows. If the
domain of interest (DOI) is discretized into Q points that are sufficiently dense
so that all M point-like scatterers are located at those points. We use 𝜆 ∈ ℂQ ,
the vector of complex Q-tuples, to denote the amplitude of source at Q points.
Since the amplitude of the source at points unoccupied by scatterers is zero,
the vector 𝜆 is sparse due to M ≪ Q. The radiated field E(r′n ) is measured at
sufficiently many discrete positions r′1 , r′2 , … , r′N . The inverse source problem
of determining the locations and amplitudes of sources amounts to solving the
following linear problem,
Q
∑
G(r′n , rq )𝜆q = E(r′n ),
n = 1, 2, … , N
(4.9)
q=1
subject to the sparsity of the vector 𝜆. The positions of point-like scatterers are
automatically identified as entries with nonzero amplitude. This approach naturally treats the uniqueness and the reconstruction in a unified way. Though
the solution to the problem is unique, it is unstable in presence of noise since
this inverse problem is severely ill-posed. Nevertheless, [4] presents the mathematical theory that can recover point sources precisely, that is, determining the
exact locations and amplitudes, by solving a convex optimization problem, provided that the distance between sources is above a certain threshold. Interested
readers are referred to [4] and [5].
4.2.2
Inverse Scattering Problem
Different to the inverse source problem, the inverse scattering problem
considers passive scatterers that are illuminated by several incident waves
71
72
Electromagnetic Inverse Scattering
generated by certain known primary sources. For each incidence, scattered
field can be considered as the re-radiation of secondary sources induced on
scatterers. Thus, for a single incidence, the inverse scattering problem reduces
to an inverse source problem. Since there are multiple incidences, the procedure of solving the inverse scattering problem is significantly different to that
for inverse source problems. For an inverse scattering problem, it is feasible to
first determine the positions of sources and then determine the amplitudes.
The theorems presented in this section not only prove the uniqueness of the
inverse scattering problem but also directly lead to numerical methods that
will be presented in Section 4.3.
Locating Positions
When the numbers of incidences Ni and measurements Ns are large enough so
that the rank of the MSR matrix K is equal to the number M of scatterers, the
positions of point-like scatterers can be uniquely determined.
For any position r, the Ns -dimensional vector Gs (r) = [G(rs1 , r), … , G(rsN , r)]T ,
s
which is referred to as the background Green’s function vector evaluated at r, is
in the range (K) of the MSR matrix K if and only if r ∈ {r1 , r2 , … , rM }.
The range (K) of the Ns × Ni dimensional matrix K denotes the space of
K ⋅ V for all Ni -dimensional vector V . In proving the sufficient and necessary
conditions of the above theorem, it is important to realize from (4.5) the fact
that the range of K is the span of the columns of the GS matrix; that is, Gs (rm ),
m = 1, 2, … , M.
Proof of sufficient condition: Consider, for example, r = r1 . Since Ni > M
and the rank of the MSR matrix K is equal to the number M of scatterers,
there exists an Ni -dimensional vector V that satisfies the linear equation system Λ ⋅ (I M − Φ ⋅ Λ)−1 ⋅ GI ⋅ V = [1, 0, … , 0]T . In other words, since the number of transmitters is larger than the number of scatterers, there are freedoms in
choosing the amplitude of driving current at transmitters such that the incident
fields illuminating at all scatterers, except the first one, are equal to zero.
Proof of necessary condition: If there is an r0 , which is different from
r1 , r2 , … , rM , so that its Ns -dimensional radiation vector is in the range of K,
then the new set of radiation vectors Gs (rm ), m = 0, 1, 2, … , M., corresponding
to the group of M + 1 points {r0 , r1 , r2 , … , rM }, is linearly dependent that,
however, contradicts the injectivity of Γ.
The theorem lays the foundation for a numerical method for positionlocating, that is, the multiple signal classification (MUSIC) imaging method,
which will be introduced in Section 4.3.1.
Retrieving Scattering Strength
After the positions of scatterers are obtained, we will determine the scattering strengths of point-like scatterers. Since the inverse scattering problem for
a single incidence is equivalent to the inverse source problem, the injectivity of
the Γ operator that is defined in (4.7) can be used in this section.
Inverse Scattering Problems of Small Scatterers
Once the positions of scatterers are obtained, for any single incidence, the
amplitude of source 𝜏m 𝜓 t (rm ) induced at those scatterers can be uniquely
determined from the measured scattered field by solving (4.4). Consequently,
the total field 𝜓 t (rm ) incident onto the mth scatterer is obtained from the
Foldy–Lax equation (4.2). Finally, the scattering strength 𝜏m of the mth
scatterer is obtained as the numerical ratio of 𝜏m 𝜓 t (rm ) to 𝜓 t (rm ). These steps
will be implemented in detail in Section 4.3.2.
4.3 Numerical Methods
The theorems presented in Section 4.2.2 that prove the uniqueness of the solution to inverse scattering problems directly lead to numerical methods of locating the positions and determining the scattering strengths of scatterers. Note
that we assume that the numbers of incidences Ni and measurements Ns are
so large that the rank of the MSR matrix K is equal to the number M of scatterers. In practice, the condition that the rank of the MSR matrix K is equal to
M can be easily realized as long as M < min(Ni , Ns ), unless transmitters and
receivers are placed in extremely special positions relative to scatterers, which
rarely happens and thus is out of the scope of discussion [6].
4.3.1
Multiple Signal Classification Imaging
As stated in the theorem in Section 4.2.2, whether a spatial point r is the
position of scatterers can be determined by testing whether the background
Green’s function vector Gs (r) = [G(rs1 , r), … , G(rsN , r)]T is in the range of the
s
MSR matrix K. The range of a matrix can be obtained by the singular value
decomposition (SVD) of the matrix. From section 2.6 of [7], the SVD of the K
of size Ns × Ni could be represented as
H
K =U ⋅Σ⋅V ,
(4.10)
where U is an Ns × Ns unitary matrix, that is, its columns are unitary vectors
that are mutually orthogonal, V is an Ni × Ni unitary matrix, and Σ is a diagonal
Ns × Ni matrix with nonnegative real numbers on the diagonal. The superscript
H denotes Hermitian. The columns up of U are called the left singular vectors
and the columns 𝑣p of V the right singular vectors. The diagonal elements 𝜎p
of Σ are called the singular values and they are sorted in nonincreasing order,
𝜎1 ≥ 𝜎2 … ≥ 𝜎M > 0 and 𝜎p = 0 for p > M. The SVD of K has the following
properties,
K ⋅ 𝑣p = 𝜎p up ,
H
K ⋅ up = 𝜎p 𝑣p
(4.11)
Due to the orthogonality of unitary vectors up , p = 1, 2, … , Ns , they form a set
of orthonormal bases of ℂNs , the vector space of complex Ns -tuples. It is obvious
73
74
Electromagnetic Inverse Scattering
that the space ℂNS can be decomposed into a direct sum of two orthogonal and
complementary subspaces,  = span{up , 𝜎p > 0} and  = span{up , 𝜎p = 0},
that is,
ℂNs =  ⊕  ,
 ⟂ 
(4.12)
The range (K) of K denotes the space of K ⋅ V for all vector V ∈ ℂNi . It is
obvious that unitary vectors 𝑣p , p = 1, 2, … , Ni form a set of orthonormal bases
of ℂNi . From the property of (4.11), we conclude that the range (K) coincides
with the subspace . Consequently, the orthogonality between  and  yields
|uH
p ⋅ Gs (rm )| = 0 for all 𝜎p = 0, m = 1, 2, … , M. Define a pseudospectrum as
Φ(r) = ∑
1
H
𝜎p =0 |up
⋅ Gs (r)|2
,
(4.13)
To determine whether the background Green’s function vector Gs (r) is in the
range of the MSR matrix K is equivalent to determine whether the pseudospectrum Φ(r) goes to infinity. In other words, the positions of scatterers are identified as where the pseudospectrum Φ(r) blows up.
The aforementioned subspace approach to determining the locations of scatterers is called the multiple signal classification (MUSIC). MUSIC was originated by Schmidt in 1979 and was used to determine the parameters of multiple
wavefronts arriving at an antenna array from measurements made on the signals received at the array elements [8]. At its early stage, MUSIC was used to
estimate of individual frequencies of multiple time-harmonic signals and to
determine the direction of arrival (DOA) of signals, among other applications.
Later, Devaney applied the MUSIC method to locate point-like scatterers in
2000, where the name TR-MUSIC method was coined since the method blends
ideas of standard MUSIC with the decomposition of the time-reversal operator
technique (DORT) [9–12]. Standard MUSIC works on determining parameters of active sources by decomposing the covariance matrix, whereas DORT
detects the positions of scatterers where passive sources are induced by decomposing the time-reversal operator. The analogy between the covariance matrix
and time-reversal operator is established in [13]. Actually, there is no need to
differentiate the subtle differences of the previously mentioned algorithms, and
they can all be referred to as MUSIC-type algorithms. In the signal processing
community, the subspace  is referred to as the signal subspace and the  as
the noise subspace.
Here we consider a numerical example of the MUSIC method. The system
under consideration is invariant in the z direction and the 2D problem in the x-y
plane is considered. The incident electric field is parallel to the z-axis, and this
scattering problem is called the transverse magnetic (TM) mode, where scalar
wave equation is involved. Let N = 20 antennas that function as transceivers be
Inverse Scattering Problems of Small Scatterers
uniformly located in a line (y = 10𝜆, −10𝜆 ≤ x ≤ 10𝜆) outside the DOI. Let two
cylinders be present in the DOI, which is otherwise free space. The cylinders
are isotropic with relative permittivity 𝜖r = 10. The radius of the cylinders is
𝜆∕30, which is so small in comparison with the wavelength that cylinders can
be treated as point-like scatterers. The cylinders are located at r1 = (−d∕2, 0)
and r2 = (d∕2, 0), where d is the distance between the two cylinders. The size
of the MSR matrix K is 20 × 20.
First we consider the case when the separation d = 𝜆∕3. When there is no
noise in the measured data, the base 10 logarithm of the singular values of
the MSR matrix for noise-free is plotted in Fig. 4.1(a), where two singular values are significantly larger than the rest. This observation indicates that there
are two point-like scatterers. The base 10 logarithm of the pseudospectrum is
shown in Fig. 4.1(c), where two peaks occur at the positions of the two scatterers. Next, we add white Gaussian noise 𝜅 to the exact K. The noisy matrix
5
0
−5
−10
Noise Subspace
0
−1
−15
−20
Signal Subspace
1
Noise Subspace
log10(σj)
log10(σj)
2
Signal Subspace
0
−2
5
10
15
20
Singular Value Number (j)
0
(b)
(a)
2
2
1
10
0
5
−1
y/ λ
y/ λ
5
10
15
20
Singular Value Number (j)
−0.5
1
−1
0
−1.5
−1
−2
0
−2
−2
0
x/ λ
(c)
2
−2
−2
0
x/ λ
2
(d)
Figure 4.1 MUSIC method is applied to locate two point-like scatterers separated by 𝜆∕3.
(a) and (b): The base 10 logarithm of the singular values of the MSR matrix for noise-free and
SNR = 30 dB cases, respectively. (c) and (d): The base 10 logarithm of the pseudospectrum
for noise-free and SNR = 30 dB cases, respectively.
75
76
Electromagnetic Inverse Scattering
K + 𝜅 is treated as the measured MSR matrix. The noise level is quantified
||K ||F
by the signal-to-noise ratio (SNR) in dB defined as 20 log10
, where || ⋅ ||F
||𝜅 ||F
denotes the Frobenius norm of a matrix, also called the Euclidean norm, which
is defined as the square root of the sum of the absolute squares of all its elements. When SNR = 30 dB, Fig. 4.1(b) shows that the gap between the singular
values of signal subspace and noise subspace becomes smaller compared with
the noise-free case. Figure 4.1(d) shows two peaks at the positions of two scatterers, although the two peaks spread out and their contrast with respect to the
background is reduced.
Next, we investigate the effect of noise on the imaging resolution. Four separation distances d = 𝜆∕8, 𝜆∕4, 𝜆∕2, and 𝜆 are considered under the influence
of four different levels of SNR: noise free, 20, 10, and 5 dB. The base 10 logarithm of the pseudospectrum for the line passing the two scatterers is plotted
in Fig. 4.2. We observe that the noise has a significant impact on the resolution;
that is, the larger the noise the poorer the resolution. It is important to note
1 d = λ/ 8
1 d = λ/ 4
0.5
0
−0.2
0.5
−0.1
0
0.1
0.2
0
−0.5
−0.25
x/ λ
0.25
0.5
0.5
1
x/ λ
1 d = λ/ 2
1 d=λ
0.5
0
−1
0
0.5
−0.5
0
x/ λ
0.5
1
0
−1
−0.5
0
x/ λ
Figure 4.2 MUSIC pseudospectrum for two identical cylinders with four different separation
distances under four different noise levels. The MUSIC pseudospectrum is plotted for the
line passing the two scatterers and it is normalized so that the maximum value in each
subfigure is 1. The SNR levels are: No noise (solid line), 20 dB (dash line), 10 dB (dotted line),
and 5 dB (dash-and-dot line).
Inverse Scattering Problems of Small Scatterers
that the two scatterers separated by d = 𝜆∕8 can be resolved well when there is
no noise.
4.3.2
Noniterative Retrieval of Scattering Strength
Once the positions of scatterers are determined, the problem of retrieving
scattering strengths from the MSR matrix Equation (4.5) is nonlinear. Such
a nonlinear inverse problem is usually dealt with by an iterative numerical
approach, where the forward problem is repeatedly solved during iterations
of the optimization procedure, but iterative algorithms do not always yield a
convergent result [9]. This section presents a two-step least-squares method,
which is noniterative and can be easily implemented.
Section 4.2.2 shows that once the positions of scatterers are determined, the
scattering strengths of scatterers can be uniquely obtained for a single incidence. For example, consider the first incidence where the first transmitter is
driven by a unitary source. The scattered field is given by the first column of the
MSR matrix,
K 1 = GS ⋅ Λ ⋅ (I M − Φ ⋅ Λ)−1 ⋅ GI 1 ,
(4.14)
Define the total field 𝜓 t1 = (I M − Φ ⋅ Λ)−1 ⋅ GI 1 and induced source J 1 = Λ ⋅
t
𝜓 1 . The least-squares solution (section 7.4 of [7]) for the overdetermined prob-
lem K 1 = GS ⋅ J 1 is
†
J 1 = GS ⋅ K 1
(4.15)
where the superscript † denotes a pseudoinverse of a matrix. If the singular
†
H
value decomposition of GS is U ⋅ Σ ⋅ V , then its the pseudoinverse is GS = V ⋅
′
H
Σ ⋅ U where the diagonal element Σ′ii = Σ−1
for Σii ≠ 0 and Σ′ii = 0 for Σii = 0.
ii
†
H
H
In fact, this definition of pseudoinverse is equivalent to GS = (GS ⋅ GS )−1 ⋅ GS ,
which is the formula presented in Appendix B. Then from (4.2), the total field
is given by
𝜓 t1 = GI 1 + Φ ⋅ J 1
(4.16)
From the definition of scattering strength, we obtain
𝜏m = (J 1 )m ∕(𝜓 t1 )m ,
m = 1, 2, … , M
(4.17)
The above solution obtained under a single incidence is exact when there is
no noise in a scattered field. In presence of noise, multiple incidences help to
improve the accuracy of the solution. For all Ni incidences, induced sources
can be written in a matrix by stacking column-wise single-incidence solution
(4.15),
†
J = GS ⋅ K
(4.18)
77
78
Electromagnetic Inverse Scattering
Similarly, the total field for all incidences is given by
t
𝜓 = GI + Φ ⋅ J
(4.19)
t
For the mth scatterer, the mth rows of J and 𝜓 satisfy
t
Cm (J) = Cm (𝜓 )𝜏m ,
m = 1, 2, … , M
(4.20)
where Cm (⋅) denotes extracting the mth row of a matrix and then writing it
as a column vector. By solving an overdetermined problem, the least-squares
solution of 𝜏m is given
t
𝜏m = [Cm (𝜓 )]† ⋅ Cm (J),
m = 1, 2, … , M
(4.21)
We observe that the proposed two-step least-squares method does not
require to iteratively evaluate the forward scattering problem. It is worth
mentioning that the first noniterative analytical algorithm was proposed in
[14], where the first step is to design a driving source vector that yields an
exactly zero total field at positions of all scatterers except one. Numerical
results have shown that this noniterative retrieval method is superior to
the iterative numerical approach. Another noniterative retrieval algorithm
proposed in [15] is essentially the same as the one presented in this section,
but the physical meaning of the former is not as clear as that of the latter.
Here we present a numerical example of retrieving the scattering strength.
The experiments are in two-dimensional free space, which considers M = 4
point scatterers under illumination by a set of Ns = Ni = 7 transceivers; that
is, transmitters coincidence with receivers. The transceiver array is a uniform
linear array distributed in the x-axis and with six wavelength inter-element
separation, where the wavelength 𝜆 = 1. The four targets are located in a
Cartesian grid at positions (−2.25, −14.75), (−0.25, −14.75), (0.75, −15.75),
and (2.75, −15.75), all in units of a wavelength. All targets have unit-amplitude
scattering strengths; that is, 𝜏m = 1, m = 1, 2, 3, 4. We assume that the correct
positions of the scatterers are already known. We add additive white Gaussian
noises to the exact scattering data and treat the noise-contaminated data as
measured data. The accuracy of the estimates is quantified by a normalized
percentage error, which is also the same as that used in [14], defined by
̂
E = 100 ⋅ ‖𝜏−𝜏‖
, where ‖⋅‖ denotes the L2 vector norm (its Euclidean length),
‖𝜏‖
𝜏 = [𝜏1 , 𝜏2 , … , 𝜏M ]T is the actual value of the scattering strengths, and 𝜏̂ is the
estimation of the scattering strengths. The errors in our numerical experiments
are averages over 1000 repetitions. Both the presented two-step least-squares
method and the noniterative method described in [14] are employed to
calculate the values of 𝜏m . The retrieval results are shown in Fig. 4.3. We clearly
see that, the least-squares retrieval method gives a better estimation than the
method in [14] does. The reason why the least-squares method outperforms
the one proposed in [14] is that one of the key intermediate formulas in [14]
Inverse Scattering Problems of Small Scatterers
10
Least Squares Retrieval Method
CRB
Result by Marengo
8
% Errors
Figure 4.3 Comparison of
the result obtained by the
least-squares retrieval
method and that given by
Marengo in [14]. The errors
are averages over 1000
repetitions. The
Cramer–Rao bound (CRB)
of the estimation is also
shown. Reproduced from
Chen 2007, J. Acoust. Soc.
Am., 122, 1325–1327, [15],
with the permission of the
Acoustical Society of
America.
6
4
2
0
20
22
24
26
SNR (dB)
28
30
32
cannot be fully satisfied in the presence of noise and consequently it propagates
errors to the next step, whereas the least-squares retrieval method minimizes
the error in L2 norm in every step. The retrieval result is also compared with
the Cramer–Rao bound (CRB), which expresses a lower bound on the variance
of estimators [16] (chapter 3). Using the formulas in the appendix of Ref. [6],
we calculate the Fisher matrix I, and take the trace of the inverse of I as the
CRB of the total variance of the four scattering strengths. In Fig. 4.3, we
observe that the result obtained √by the proposed algorithm is close to that
specified by the CRB (ECRB = 100
tr{I −1 }
‖𝜏‖
).
4.4 Inversion of a Vector Wave Equation
For a three-dimensional electromagnetic wave inverse scattering problem, the
locations of small scatterers can be obtained by the MUSIC method and their
scattering strength tensors can be determined by the two-step least-squares
retrieval method as well. However, the methods for solving a vector wave ISP is
not a simple generalization of its scalar counterpart. A major difference is that
electromagnetic waves are able to induce up to six independent sources inside
a smaller scatterer, whereas the scalar wave induces only one inside a small
scatterer. Physical insights and mathematical manipulations are presented that
not only are able to reconstruct small scatterers with special shapes but also
improve the resolution of imaging.
4.4.1
Forward Problem
Suppose there are M three-dimensional objects illuminated by time-harmonic
electromagnetic waves radiated by an array of N transceiver antenna units. We
79
80
Electromagnetic Inverse Scattering
mention in passing that, after minor modifications, the formulas presented in
this section can be easily applied to the case where transmitters are separated
from receivers. The transceiver antenna units are located at rs1 , rs2 , … , rsN , each
of which consists of three small electric dipole antennas oriented in the x, y
and z direction with electric current dipole moment Ilxn , Ilyn , Ilzn , respectively,
n = 1, 2, … , N. The size of each of the M scatterers is much smaller than the
wavelength so that Rayleigh scattering is observed. These scatterers can be of
any shape, but here we consider only spherical and ellipsoidal objects for ease
of presenting. The centers of the scatterers are located at r1 , r2 , … , rM . The scatterers may be made of dielectric and magnetic (permeable) materials, or perfect
electric conductor (PEC), and may be isotropic or anisotropic. As introduced
in Section 2.8, the shape and composing material of each small scatterer determine its polarization tenor 𝜉 m (𝜁 m ), m = 1, 2, … , M, which relates the induced
electric (magnetic) current dipole Il(rm ) (Kl(rm )) inside the object to the total
incident electric field Et (rm ) (magnetic field Ht (rm )) by
Il(rm ) = 𝜉 m ⋅ Et (rm )
(4.22)
Kl(rm ) = 𝜁 m ⋅ Ht (rm )
(4.23)
The analytical expression of 𝜉 m (𝜁 m ) for isotropic and anisotropic spheres and
ellipsoids can be found in Section 2.8 and [17]. The units of Il, 𝜉 and Et are A⋅m,
m2 ∕Ω, and V/m, respectively, and their magnetic counter parts Kl, 𝜁and Ht
have the units of V⋅m, m2 ⋅ Ω, and A/m, respectively.
When multiple scattering between scatterers is taken into account, the total
incident field Et (rm ) (Ht (rm )) upon the mth scatterer includes both the incident
field directly from antennas Ei (rm ) (Hi (rm )) and the scattered fields from other
scatterers. The total incident fields are governed by the Foldy–Lax equation,
∑
{i𝜔𝜇0 G(rm , rm′ ) ⋅ 𝜉 m′ ⋅ Et (rm′ )
Et (rm ) = Ei (rm ) +
m′ ≠m
− ∇g(rm , rm′ ) × [𝜁 m′ ⋅ Ht (rm′ )]}
∑
{i𝜔𝜖0 G(rm , rm′ ) ⋅ 𝜁 m′ ⋅ Ht (rm′ )
Ht (rm ) = Hi (rm ) +
(4.24)
m′ ≠m
+ ∇g(rm , rm′ ) × [𝜉 m′ ⋅ Et (rm′ )]}
(4.25)
where 𝜖0 and 𝜇0 are the permittivity and permeability of the homogeneous
)g(r, r′ ) is the dyadic
background medium, respectively, G(r, r′ ) = (I 3 + ∇∇
k2
0
Green’s function of the background medium. Since electric parameters E and 𝜉
have different SI units from their magnetic counterpart H and 𝜁, respectively,
it is convenient to normalize them so that each pair has the same units. With
the help of normalization, we write Eqs. (4.24) and (4.25) in the following
Inverse Scattering Problems of Small Scatterers
compact matrix form,
𝜓 t = 𝜓 i + Φ ⋅ Λ ⋅ 𝜓 t,
(4.26)
where both 𝜓 t and 𝜓 i are 6M-dimensional vectors,
𝜓 t = [Et (r1 )T , Et (r2 )T , … , Et (rM )T , 𝜂0 Ht (r1 )T , … , 𝜂0 Ht (rM )T ]T
𝜓 i = [Ei (r1 )T , Ei (r2 )T , … , Ei (rM )T , 𝜂0 Hi (r1 )T , … , 𝜂0 Hi (rM )T ]T
The elements in both 𝜓 t and 𝜓 i have the same SI unit, V/m. Λ is a block diagonal
matrix,
Λ = diag[P1 , P2 , … , PM , PM+1 , PM+2 , … , P2M ],
(4.27)
where Pm = 𝜂0 𝜉 m for m ≤ M, Pm = (1∕𝜂0 )𝜁 m−M for m > M, and 𝜂0 is the
impedance of the background medium. The elements in Λ have the same unit,
m2 . Φ is a 6M-by-6M matrix,
]
[
𝛼, −𝛽
(4.28)
Φ=
𝛽, 𝛼,
where both 𝛼 and 𝛽 consist of M × M sub-matrices whose formulas
in the mth row and m′ th column (m, m′ = 1, 2, … , M) are given by
𝛼(m, m′ ) = ik0 G(rm , rm′ ) and 𝛽(m, m′ ) = 𝜒(rm , rm′ ) in case of m ≠ m′ and
are both equal to zero otherwise. The matrix operator 𝜒(r, r′ ) is defined in
a way such that 𝜒(r, r′ ) ⋅ A = ∇g(r, r′ ) × A for an arbitrary vector A, and the
exact expression is given by (2.42), which is copied here,
0
−(z − z′ ) (y − y′ )
⎡
0
−(x − x′ )
𝜒(r, r′ ) = ⎢ (z − z′ )
⎢
′
′
0
⎣ −(y − y ) (x − x )
(
)
⎤
⎥ g(r, r′ ) ik0 − 1
(4.29)
⎥
R
R2
⎦
The elements in Φ have the same unit, m−2 . It can be seen that every parameter
in (4.26) is a normalized parameter.
By using the Foldy–Lax equation, we obtain a 3N × 3N multi-static response
(MSR) matrix that relates the scattered electric fields to the driving electric current dipoles (Il)
K = GS ⋅ Λ ⋅ (I 6M − Φ ⋅ Λ)−1 ⋅ GI .
(4.30)
where I 6M is a 6M-dimensional identity matrix. GI is a 6M × 3N matrix, GI =
𝜂0 [G, X]T , and both G and X consist of N × M sub-matrices whose formulas
in the nth row and mth column (n = 1, 2, … , N, m = 1, 2, … , M) are given by
G(n, m) = ik0 G(rsn , rm ) and X(n, m) = 𝜒(rsn , rm ). Finally, GS = [G, −X]. The elements in both G and X have the same SI unit, m−2 .
81
82
Electromagnetic Inverse Scattering
In electromagnetic wave inverse scattering problems, we need to consider
nondegenerate and degenerate cases. In a nondegenerate case, three independent electric dipole components and three independent magnetic dipole components are induced in each of the scatterers, ending up with 6M independent
dipole sources. Thus, the rank of the MSR matrix is equal to 6M in case of 6M <
3N. Note that the condition 6M < 3N is assumed throughout the chapter. In a
degenerate case, which is due to special shapes or composing materials of the
scatterers, the rank of the MSR matrix is less than 6M.
4.4.2
Multiple Signal Classification Imaging
Nondegenerate Case
In a nondegenerate case, six independent dipole components are induced in
each scatterer, and the range of MSR matrix K is spanned by the background
Green’s function vectors corresponding to the x, y, and z components of the
electric and magnetic dipoles evaluated at the position of each scatterer; that is,
{
}
Gx (rm ), Gy (rm ), Gz (rm ), X x (rm ), X y (rm ), X z (rm );
(K) = span
m = 1, … , M
where Gx (rm ), Gy (rm ) and Gz (rm ) are the [3(m − 1) + 1]th, [3(m − 1) + 2]th,
and [3(m − 1) + 3]th column of the matrix G, respectively, and X x (rm ), X y (rm )
and X z (rm ) the [3(m − 1) + 1]th, [3(m − 1) + 2]th, and [3(m − 1) + 3]th column of matrix X, respectively. The Green’s function vectors Gl (r) and X l (r)
(l = x, y, z) evaluated at an arbitrary position r can be similarly defined. On
the other hand, the singular value decomposition of the MSR matrix could be
H
represented as K ⋅ 𝑣p = 𝜎p up and K ⋅ up = 𝜎p 𝑣p , p = 1, 2, … , 3N. The vector
space of complex 3N-tuples ℂ3N can be decomposed into the direct sum of
the range Sr = span{up , 𝜎p > 0} and the orthogonal complement subspace
Sn = span{up , 𝜎p = 0} that is referred to as the noise subspace. The orthogonality between the subspace Sr and the noise subspace Sn yields |uH
p ⋅ Gl (rm )| = 0
and |uH
p ⋅ X l (rm )| = 0, for 𝜎p = 0, m = 1, 2, … , M and l = x, y, z. Define the
MUSIC pseudospectrum as
1
Φ(r) = ∑
,
(4.31)
H
2
𝜎p =0 |up ⋅ f (r)|
where the test function f (r) can be any linear combination of Gx (r), Gy (r),
Gz (r), X x (r), X y (r), and X z (r). The pseudospectrum becomes infinite at the
position of each scatterer.
When scatterers are all nonmagnetic and there are 3M independent induced
electric dipoles, the test function f (r) can be chosen as any linear combination
of Gx (r), Gy (r), Gz (r). Similarly, if there is zero electric dipole but 3M independent magnetic dipoles are induced, the test function f (r) can be chosen as
any linear combination of X x (r), X y (r), and X z (r). These two special cases are
referred to as electric and magnetic nondegenerate cases, respectively.
Inverse Scattering Problems of Small Scatterers
Degenerate Case
In the degenerate case, only one or two independent electric (or magnetic)
dipole components are induced inside some of the scatterers. The following
analysis deals with degenerate electric dipoles, and the case of degenerate magnetic dipoles can be analyzed similarly. If only one independent electric dipole
component is induced inside one of the scatterers, that is, all dipoles induced in
this scatterer are parallel to each other, the test function f (r) should represent
the background Green’s function corresponding to an electric dipole source
aligned with the induced dipole. Otherwise, the pseudospectrum cannot produce an infinite peak at the position of this scatterer. If two independent electric
dipole components are induced inside one of the scatterers, that is, all dipoles
induced in this scatterer are located in a plane, f (r) should represent the background Green’s function corresponding to an electric dipole source oriented
in the aforementioned plane. Therefore, when the test function f (r) is a proper
linear combination of the background Green’s functions associated with the x,
y, and z components of the electric dipole source,
f (r) = [Gx (r), Gy (r), Gz (r)] ⋅ 𝛼,
̂
(4.32)
where 𝛼̂ = [𝛼x , 𝛼y , 𝛼z ]T is a unit vector representing the direction of the test electric dipole, so that the test dipole source is within the subspace spanned by the
physically induced independent electric dipole components, the pseudospectrum will produce an infinite peak at the location of the scatterer. However, it
is nontrivial to choose the correct value of 𝛼̂ since for an arbitrary test point r
in the DOI, there may be zero, one, two, or three independent electric dipole
components induced. If there is no electric dipole induced at a position, there
is no point in finding the “correct” 𝛼̂ since there is no 𝛼̂ at all that produces
an infinite peak at the test position. Moreover, since the discrete small scatterers occupy only a small fraction of the DOI, it is desirable for an algorithm to
make a quick decision on whether there is an electric dipole induced at the test
position.
If the test function f (r) is properly chosen, the pseudospectrum will produce
an infinite peak at a position where an electric dipole is induced; that is,
uH
̂ = 0,
p ⋅ [Gx (r), Gy (r), Gz (r)] ⋅ 𝛼
for 𝜎p = 0.
(4.33)
Writing Eq. (4.33) in the matrix form, we have
W (r) ⋅ 𝛼̂ = 0,
(4.34)
where matrix W (r) is of size p0 × 3, where p0 is the number of vanishing singular
values and it is larger than 3 in practice. For a test position r, if the rank of W (r)
is equal to 3, the solution to Eq. (4.34) is a null vector, which contradicts with
the fact that 𝛼̂ is a unit vector. In this case, there is no electric dipole induced
at the test position. If the rank of W (r) is equal to 0, any unit vector 𝛼̂ is a
solution to Eq. (4.34), which indicates that three independent electric dipole
components are induced at the position. If the rank of W (r) is equal to 1, the
83
84
Electromagnetic Inverse Scattering
solution to Eq. (4.34) forms a subspace spanned by two linearly independent
unit vectors 𝛼̂ 1 and 𝛼̂ 2 , which is identical to the plane of the physically induced
electric dipoles. If the rank of W (r) is equal to 2, there is one and only one
independent unit vector 𝛼̂ satisfying Eq. (4.34), which is identical to the single
direction of the physically induced electric dipoles.
The above analysis is valid only when there is no noise in the MSR matrix. The
extension of it to noisy data can be done by solving a minimization problem:
H
𝛼̂ H ⋅ W (r)W (r) ⋅ 𝛼.
̂
𝛼̂ = arg min
H
𝛼̂ ⋅𝛼=1
̂
(4.35)
Note that “arg min” means the argument of the minimum. The well-known
Rayleigh theorem [7](section 4.2) gives the solution of 𝛼,
̂ which is the eigenvector corresponding to the minimum eigenvalue of the 3 × 3 Hermitian matrix
H
W (r)W (r). The MUSIC pseudospectrum generated by the test electric dipole
whose direction is given by (4.35) exhibits peaks at positions where at least
one independent electric dipole is induced. We mention in passing that the
test dipole direction given by Eq. (4.35) is in fact a generalization of the 2D
solution [18] to a 3D scenario.
The proposed best-testing-direction MUSIC method is tested through the
following numerical example. The background medium is free space, the wavelength at the operating frequency is 𝜆 = 3 m, and the transceiver antenna units
are distributed in a 9 × 9 grid pattern with increment 2.5 𝜆 in a plane z = 1.5 𝜆.
The center of the grid is directly above the origin. The length of each antenna
is short and can be treated as an electric dipole antenna. Two small scatterers
are present. One is a needle and the other is a disk. Both the needle and the
disk are made of PEC, and their geometries are described as spheroids. The
major and minor semi-axes of the needle are a1 = 1∕30 𝜆 and b1 = 1∕1200 𝜆,
respectively, and the major and minor semi-axes of the disk are a2 = 1∕60 𝜆 and
b2 = 1∕2400 𝜆, respectively. Note that the ratios of their major axes to minor
axes are both equal to 40. The electric and magnetic polarization tensors for a
needle made of PEC is given by (Section 2.8.1)
[
]
a31
k
8𝜋
2 8𝜋
2 4𝜋
𝜉 = −i ⋅ diag
,
(4.36)
ab,
ab,
𝜂0
3 1 1 3 1 1 3 ln(a1 ∕b1 )
]
[
8𝜋
8𝜋
4𝜋
(4.37)
𝜁 = −ik𝜂0 ⋅ diag − a1 b21 , − a1 b21 , − a1 b21 .
3
3
3
Since a1 ∕b1 ≫ 1, the induced electric dipole along the needle direction (the
z-axis) is much more dominant than other electric dipoles and magnetic
dipoles [19]. The electric and magnetic polarization tensors for a disk made of
PEC is given by (Section 2.8.1)
]
[
k
16 3 16 3 4𝜋 2
(4.38)
𝜉 = −i ⋅ diag
a2 , a2 ,
a2 b 2 ,
𝜂0
3
3
3
]
[
4𝜋
4𝜋
8
(4.39)
𝜁 = −ik𝜂0 ⋅ diag − a22 b2 , − a22 b2 , − a32 .
3
3
3
Inverse Scattering Problems of Small Scatterers
Since a2 ∕b2 ≫ 1, two components of induced electric dipoles, both of which
are in the plane of the disk (the xy plane), and one component of induced
magnetic dipoles that is aligned with the normal direction of the disk (the
z-axis) are much more dominant than other components of electric and
magnetic dipoles [19].
The needle is located at the origin, with orientation Euler angles (𝜋∕3, 𝜋∕4,
0) [20] (section 3.3), and the disk is located at (0.05𝜆, 0.05𝜆, 0.05𝜆), with orientation Euler angles (0, 𝜋∕6, 0). Thus the dominant electric dipole induced in the
needle is in the direction (sin(𝜋∕4) cos(𝜋∕3), sin(𝜋∕4) sin(𝜋∕3), cos(𝜋∕4)). The
dominant magnetic dipole induced in the disk is in the normal direction of the
disk, (sin(𝜋∕6), 0, cos(𝜋∕6)), and the dominant components of electric dipoles
are in the plane of the disk.
In numerical simulations, we plot a logarithmic normalized MUSIC pseudospectrum
1
,
(4.40)
Φ(r) = log10 √
√∑
̂ (r)|2
√ 𝜎 =0 |uH
⋅
f
p
√ p
p0
instead of the original one defined in Eq. (4.31), where p0 is the number of singular values that are vanishing (in practice, smaller than a certain threshold)
and f̂ (r) = f (r)∕||f (r)|| is a normalized test function. Since both up and f̂ (r) are
unit vectors, the denominator of Eq. (4.40) represents an averaged orthogonality between the test vector and the singular vectors corresponding to vanishing
singular values.
First, we use noise-free MSR matrix to produce MUSIC pseudospectrum,
where the test function is associated with the x̂ -oriented electric or magnetic
dipole. We can also choose other arbitrary testing directions, such as ŷ and ẑ ,
since we do not know a priori the geometry of scatterers. Nevertheless, the
arbitrarily chosen direction is unlikely to be the same as the directions of the
aforementioned four dominant dipoles. The pseudospectra for test positions
in the z = 0 and z = 0.05 𝜆 planes are shown in Fig. 4.4. We see that both scatterers are detected. Similar results are produced by the test function associated
with the ŷ or ẑ oriented electric and magnetic dipoles. The reason for this is
that although the aforementioned four dipole components are dominant, the
other eight dipole components also contribute to the scattered field so that
the range of the MSR matrix is spanned by a total of 12 Green’s functions
associated with the three electric and three magnetic dipoles at each scatterer.
We plot the 30 largest singular values of the MSR matrix in Fig. 4.5 (a), where
we see four dominant singular values and the next eight singular values are
much larger than the rest so that the range of the MSR matrix is spanned by
the singular vectors corresponding to the 12 leading singular values.
However, the results are quite different when the MSR matrix is contaminated with noise. When 30 dB Gaussian noise is added to the MSR matrix, its
30 largest singular values are plotted in Fig. 4.5(b). Compared with Fig. 4.5(a),
85
Electromagnetic Inverse Scattering
0.1
12
10
0
8
6
–0.05
14
12
0.05
y (λ)
y (λ)
0.1
14
0.05
10
0
8
6
–0.05
4
4
0.1
0
x (λ)
(a)
0.05
–0.1
–0.1 –0.05
0.1
0.1
14
12
0.05
10
0
–0.05
0
x (λ)
(b)
0.05
0.1
14
12
0.05
8
y (λ)
–0.1
–0.1 –0.05
y (λ)
86
6
–0.05
10
0
8
6
4
–0.1
–0.1 –0.05
0
x (λ)
(c)
0.05
0.1
4
–0.1
–0.1 –0.05
0
x (λ)
(d)
0.05
0.1
Figure 4.4 Logarithmic pseudospectrum for test positions in the z = 0 plane for (a) and (b),
and in the z = 0.05 𝜆 plane for (c) and (d). The MSR matrix is noise-free. The testing sources
are: (a) an electric dipole oriented in the x-axis (the position of the needle is correctly
detected); (b) a magnetic dipole oriented in the x-axis (the position of the needle is correctly
detected); (c) an electric dipole oriented in the x-axis (the position of the disk is correctly
detected); and (d) a magnetic dipole oriented in the x-axis (the position of the disk is
correctly detected). Source: Chen 2008, Journal of Physics: Conference Series, 124, 012016. [39]
Reproduced with permission of IOP Publishing.
Fig. 4.5(b) shows that although the first four singular values are still dominant the next eight are contaminated with noise and are in the same level in
magnitude as the rest singular values. The pseudospectra with the test function corresponding to the x̂ oriented electric or magnetic dipoles are plotted
in Fig. 4.6. Compared with Fig. 4.4, the values of peaks in the pseudospectra
in Fig. 4.6 are much smaller and the peaks spread out. Moreover, the positions
of the peaks do not agree with those of the scatterers. Thus the pseudospectra generated from the test function associated with the x̂ oriented electric or
magnetic dipole that is different from the directions of the aforementioned four
dominant dipoles fail to locate the needle and the disk.
Next, for the noisy MSR matrix, we produce the MUSIC pseudospectrum in
Fig. 4.7 with the orientation of the test dipole given by Eq. (4.35). The needle is
−8
−10
−12
−14
−16
−18
−20
−22
−24
log10(σj)
log10(σj)
Inverse Scattering Problems of Small Scatterers
0
5
10 15 20 25
Singular value number, j
−8
−10
−12
−14
−16
−18
−20
−22
−24
30
0
5
10 15 20 25
Singular value number, j
(a)
30
(b)
Figure 4.5 Thirty largest singular values of the MSR matrix. (a) Noise-free and (b) 30 dB
Gaussian noise. Source: Chen 2008, Journal of Physics: Conference Series, 124, 012016. [39]
Reproduced with permission of IOP Publishing.
0.1
2
1.9
0
1.8
1.75
0.05
y (λ)
2.1
0.05
y (λ)
0.1
2.2
1.7
0
1.65
1.8
–0.05
0.1
1.6
0
x (λ)
(a)
0.05
0.1
y (λ)
–0.1
–0.1 –0.05
0.1
2.2
2.1
0.05
2
1.9
0
1.6
–0.05
1.55
0
x (λ)
(b)
0.05
0.1
1.8
1.75
1.7
0
1.65
1.8
–0.05
–0.1
–0.1 –0.05
1.7
1.6
0
x (λ)
(c)
0.05
0.1
1.5
1.85
0.05
y (λ)
–0.1
–0.1 –0.05
1.7
1.6
–0.05
1.55
–0.1
–0.1 –0.05
0
x (λ)
(d)
0.05
0.1
1.5
Figure 4.6 Logarithmic pseudospectrum for test positions in the z = 0 plane for (a) and (b),
and in the z = 0.05 𝜆 plane for (c) and (d). The MSR matrix is contaminated with Gaussian
noise, where SNR = 30 dB. The testing sources are: (a) an electric dipole oriented in the x-axis
(the position of the needle is not detected); (b) a magnetic dipole oriented in the x-axis
(the position of the needle is not detected); (c) an electric dipole oriented in the x-axis (the
position of the disk is not detected); and (d) a magnetic dipole oriented in the x-axis
(the position of the disk is not detected).
87
Electromagnetic Inverse Scattering
0.1
3
0
2.5
–0.05
2.4
2.3
2.2
2.1
2
1.9
1.8
1.7
0.05
y (λ)
y (λ)
0.1
3.5
0.05
0
–0.05
2
0.1
0
x (λ)
(a)
0.05
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1.8
0.05
0
–0.05
–0.1
–0.1 –0.05
–0.1
–0.1 –0.05
0.1
0
x (λ)
(c)
0.05
0.1
0.1
0
x (λ)
(b)
0.05
0.1
3.2
3
2.8
2.6
2.4
2.2
2
1.8
0.05
y (λ)
–0.1
–0.1 –0.05
y (λ)
88
0
–0.05
–0.1
–0.1 –0.05
0
x (λ)
(d)
0.05
0.1
Figure 4.7 Logarithmic pseudospectrum for test positions in the z = 0 plane for (a) and (b),
and in the z = 0.05 𝜆 plane for (c) and (d). The MSR matrix is contaminated with Gaussian
noise, where SNR = 30 dB. The orientation of the test dipole is given by Eq. (4.35). (a) An
electric test dipole (the position of the needle is correctly detected); (b) a magnetic test
dipole (the position of the needle is not correctly detected); (c) an electric test dipole (the
position of the disk is correctly detected); and (d) a magnetic test dipole (the position of the
disk is correctly detected).
correctly located by the test function associated with an electric dipole and its
position, that is, the peak of the pseudospectrum as shown in Fig. 4.7 (a), reads
(0.002 𝜆, 0.001 𝜆, 0.000 𝜆), which is very close to the actual position (the origin).
In comparison, Fig. 4.7 (b) shows that the needle is not correctly located by
the test function associated with a magnetic dipole since there is no dominant
magnetic dipole in the needle. Figure 4.7 (c) and (d) show that the position of
the disk is correctly predicted by the electric and the magnetic test dipole. We
read, from the pseudospectrum, the position of the disk to be (0.050 𝜆, 0.051 𝜆,
0.050 𝜆), very close to actual location of the disk.
4.4.3
Noniterative Retrieval of Scattering Strength Tensors
The polarization tensors of scatterers can be retrieved after the positions of
scatterers are located by the MUSIC method. This section presents a two-step
Inverse Scattering Problems of Small Scatterers
least-squares noniterative retrieval algorithm that applies to both nondegenerate and degenerate objects.
Consider the case where either electric or magnetic dipole is not induced
inside some scatterers; this is, Pm in Eq. (4.27) is a zero matrix for those scatterers. Define the matrix Lm (m = 1, 2, … , M), which is a zero matrix of size
3 × 3 if no electric dipole is induced in the mth scatterer, and is an identity
matrix I 3 otherwise. The matrix Lm for m = M + 1, M + 2, … , 2M is defined
similarly. Note that Lm can be easily determined from MUSIC pseudospectrum
corresponding to an electric or magnetic test dipole. A matrix L is formed by
diagonally connecting Lm (m = 1, 2, … , 2M) in sequence, followed by removing columns whose elements are all zero. The size of L is 6M × 3Itot , where Itot
is the sum of the number of scatterers within which at least one electric dipole
is induced and the number of scatterers within which at least one magnetic
dipole is induced. Then the original rank deficient matrix in Eq. (4.27), of size
6M × 6M, can be effectively expressed as
′
T
Λ=L⋅Λ ⋅L ,
(4.41)
′
where Λ , of size 3Itot × 3Itot , is constructed by diagonally concatenating the
nonzero Pm , m = 1, 2, … , 2M.
The substitution of Eq. (4.41) into Eq. (4.30) yields the MSR matrix K in terms
′
of Λ
′
T
′
T
K = GS ⋅ L ⋅ Λ ⋅ L ⋅ (I 6M − Φ ⋅ L ⋅ Λ ⋅ L )−1 ⋅ GI
(4.42)
Define a 6M × 3N matrix 𝜓 tot , which is the total fields incident onto the scatterers, as
′
T
𝜓 tot = (I 6M − Φ ⋅ L ⋅ Λ ⋅ L )−1 ⋅ GI
(4.43)
and define a 3Itot × 3N matrix J, which is the induced current dipole moment,
as
′
T
J = Λ ⋅ L ⋅ 𝜓 tot
(4.44)
The noniterative retrieval algorithm consists of two steps. In the first step, we
treat J as an unknown in Eq. (4.42), the least-squares solution of which is given
by
J = (GS ⋅ L)† ⋅ K,
(4.45)
where † denotes pseudoinverse of a matrix. In the second step, since 𝜓 tot comprises of the incident wave coming directly from transmitting antennas and
incident wave due to multiple scattering, we can write (4.44) as
′
T
J = Λ ⋅ L ⋅ (GI + Φ ⋅ L ⋅ J).
(4.46)
89
90
Electromagnetic Inverse Scattering
′
Since the matrix Λ consists of diagonal blocks, (4.46) can be written as
three-row block-wise equations,
T
′
C3,i (J) = Λi ⋅ C3,i [L ⋅ (GI + Φ ⋅ L ⋅ J)],
i = 1, 2, … , Itot ,
′
Λi
(4.47)
′
where 3 × 3 matrix is the ith block matrix in the diagonal of Λ and the 3 ×
3N matrix C3,i (J) denotes the corresponding ith block row of J. Finally, the least
′
squares solution of Λ is obtained as
′
T
Λi = C3,i (J) ⋅ {C3,i [L ⋅ (GI + Φ ⋅ L ⋅ J)]}† ,
i = 1, 2, … , Itot .
(4.48)
Note that (4.48) yields an exact solution when there is no noise in the measurement of the MSR matrix K and the positions of scatterers are exactly estimated by the MUSIC pseudospectrum. The retrieval error noticeably depends
on the accuracy of the estimation of the locations of the scatterers. Various
numerical simulation results show that the retrieval error is drastically reduced
when the error in the estimation of the positions of the scatterers is decreased.
This observation agrees with the retrieval method presented in [9, 21].
As an example, we apply the presented two-step least-squares retrieval
method to retrieve the polarization tensors of the scatterers discussed in
Section 4.4.2. In the presence of 30 dB of white Gaussian noise, after the positions of the scatterers are obtained, the retrieval method yields a normalized
percentage error E = 3.23%.
We briefly mention that the first noniterative analytical algorithm to retrieve
scattering strengths in the electromagnetic inverse scattering problem is
presented in [21]. However, it applies only to the nondegenerate case, where
six independent electric and magnetic dipole components are induced in
each scatterer; that is, both 𝜉 m and 𝜁 m , m = 1, 2, … , M, are invertible. This
restriction refrains the method from applying to two types of media. The first
one is those in which only one or two electric or magnetic independent dipoles
can be induced. The second one is those in which either electric or magnetic
dipole is absent. In comparison, the noniterative retrieval algorithm presented
in this section applies to both nondegenerate and degenerate objects.
4.4.4
Subspace Imaging Algorithm with Enhanced Resolution
Electromagnetic MUSIC imaging is different from its counterpart in acoustic MUSIC imaging in two aspects. The first is regarding degenerate scatterers
inside which only one or two independent components of an electric (or magnetic) dipole are induced due to special shapes or composing materials of the
scatterers, and this topic has been discussed in Section 4.4.2. The second is
regarding the effect of test source on the resolution of MUSIC pseudospectrum. Different from the monopole test source in acoustic imaging, the test
source in electromagnetic imaging is an electric or magnetic dipole. Although
Inverse Scattering Problems of Small Scatterers
the test dipole can be oriented in any direction in the noise-free case for nondegenerate scatterers, the MUSIC pseudospectrum depends noticeably on the
orientation of the test dipole in noisy scenarios.
This section presents an algorithm to obtain the direction of the test dipole
that yields enhanced resolution, and what is equally important is that the proposed method can deal with degenerate cases as well. Different from standard
MUSIC algorithms that search for the test dipole direction so that the corresponding Green’s function vector is orthogonal to the noise subspace, the proposed algorithm determines the test dipole direction so that the corresponding
Green’s function vector is located in a subspace of the signal subspace.
In a nutshell, the motivation of the algorithm is that it is not mandatory for all
singular vectors that are in signal subspace to involve in evaluating the pseudospectrum. For example, for a single small nonmagnetic isotropic spherical
scatterer, the MSR matrix has three nonzero singular values and the signal
subspace is spanned by the first three singular vectors. If all three singular vectors are used to evaluate the pseudospectrum, that is, determining whether the
test Green’s vector is within the signal subspace, then the test dipole can be
in any direction. However, since we have the freedom to choose the direction
of the test dipole, we may choose it as the direction of dipole that is induced
in the spherical scatterer when the largest singular value occurs. Consequently,
the second and third singular vectors are not involved in calculating the pseudospectrum. It can be seen that the pseudospectrum generated in this way is
better than standard MUSIC pseudospectrum because the singular vector associated with the largest singular value is more stable than the other two singular
vectors when noise is present.
In this section, we know a priori that scatterers are all nonmagnetic and consequently we only consider the electric dipole. To find the optimal test dipole
direction is equivalent to determine 𝛼̂ ∈ ℂ3 subject to ||𝛼||
̂ = 1, so that the solution x ∈ ℂ3N to the equation
K ⋅ x = D3 (r) ⋅ 𝛼̂
(4.49)
is most robust in the presence of noise, where the 3N × 3 matrix D3 (r) is defined
as [Gx (r), Gy (r), Gz (r)]. In fact, the right-hand side of (4.49) is a shorthand of
(4.32). The SVD of K is given by
K=
3N
∑
ui 𝜎i 𝑣H
i .
(4.50)
i=1
where 𝜎1 ≥ 𝜎2 ≥ … , ≥ 𝜎3N ≥ 0. The least-squares solution of x is given by
x=
3N
∑
uH
⋅ D3 (r) ⋅ 𝛼̂
i
i=1
𝜎i
𝑣i
for 𝜎i ≠ 0
(4.51)
91
92
Electromagnetic Inverse Scattering
Note that the value of
1
𝜎i
is large for a small 𝜎i . To obtain a stable solution x,
⋅ D3 (r) ⋅ 𝛼̂ is nonzero for only the first few items.
we should find 𝛼̂ so that uH
i
It worth mentioning that even if in case of truncated singular value decomposition, that is, the regularization method is used, it is desirable to have a
fast decaying series of uH
⋅ D3 (r) ⋅ 𝛼.
̂ Since ui form a set of orthogonal bases in
i
ℂ3N , we need to find 𝛼̂ so that D3 (r) ⋅ 𝛼̂ is a linear combination of the first few
(say L) ui .
L
∑
𝜆i ui = D3 (r) ⋅ 𝛼.
̂
(4.52)
i=1
The proposed MUSIC algorithm is based on the analysis of the induced electric
current dipoles in the eigenstate, which is referred to as the eigen-dipole here(i)
after. We use J m ∈ ℂ3 to denote the dipole induced in the mth scatterer in the
ith eigen-state. We have
ui =
M
∑
(i)
D3 (rm ) ⋅ J m ,
i = 1, 2, … , L.
(4.53)
m=1
The substitution of (4.53) into (4.52) yields
M
∑
D3 (rm ) ⋅
L
∑
m=1
(i)
𝜆i J m = D3 (r) ⋅ 𝛼.
̂
(4.54)
i=1
There are two cases to be considered: (1) when the test position r is not at any
of the scatterers, rm , and (2) r is at one of the scatterers.
It is important to stress that the map from induced dipoles to scattered electric fields is one-to-one, which has been proven in Section 4.2. When the test
position r is not at any of the scatterers, (4.54) holds only if 𝛼̂ = 0 and 𝜆i = 0.
H
Therefore, for any dipole direction 𝛼,
̂ which satisfies ||𝛼||
̂ = 1, ui ⋅ D3 (r) ⋅ 𝛼̂ is
not equal to zero for all 3N left singular vectors ui .
When r is at one of the scatterers, for example, r = r1 , (4.54) requires
L
∑
(i)
J 1 𝜆i = 𝛼̂
(4.55)
i=1
L
∑
(i)
J m 𝜆i = 0,
m = 2, 3, … , M
(4.56)
i=1
Equation (4.56) amounts to determining the minimum value of L so that J̃ (1) ,
J̃ (2) , …, J̃ (L) are linearly dependent, where J̃ (i) is a column vector of length
(i)
3(M − 1) consisting of J m , m = 2, 3, … , M. Therefore, the value of minimum
L is equal to one plus the total number of independent dipoles induced in all
other scatterers (numbering 2, 3, … , M). For example, for M isotropic spheres,
Inverse Scattering Problems of Small Scatterers
there are 3M non-zero singular values in total and the value of L equals to
3M − 2. It is stressed that the algorithm also applies to degenerate cases. For
example, if only the first scatterer is disk-like, then there are 3M − 1 nonzero
singular values in total and the value of L is equal equal to 3M − 2. If only the
first scatterer is needle-like, then there are 3M − 2 nonzero singular values in
total and the value of L is still equal to 3M − 2. When the sampling point r is
at other scatterers, the results can be similarly obtained.
To summarize, when the test point r is at one of the scatterers, the value
of minimum L is equal to one plus the total number of independent dipoles
induced in other scatterers. In practice, the value of L can be easily obtained
after the total number of dominant singular values is found from the SVD spectrum. From the previous analysis, it is easy to conclude that the value of integer
L has three possibilities. It might be equal to the total number D of dominant
singular values, D − 1, or D − 2, depending on the case of degeneracy of the
scatterers. Thus, for a given SVD spectrum, we try three times at most in order
to obtain the best value of L.
Once the value of L is determined, the next step is to obtain the test dipole
̂ = 1. As discussed earlier, the purpose is to let
direction 𝛼̂ ∈ ℂ3 , subject to ||𝛼||
D3 (r) ⋅ 𝛼̂ be as close as possible to the space spanned by the first L dominant
singular vectors ui ; that is, we aim at a minimum projection-angle between the
vector D3 (r) ⋅ 𝛼̂ and the space spanned by the singular vectors ui , i = 1, 2, … , L:
∑L
H
|ui ⋅ D3 (r) ⋅ 𝛼|
̂ 2
𝛼̂ max = arg max𝛼̂ i=1
.
(4.57)
|D3 (r) ⋅ 𝛼|
̂ 2
Note that “arg max” means the argument of the maximum, that is, the point at
which the function value is maximized. From the general eigenvalue decomposition, we obtain the solution 𝛼̂ that is given by the eigenvector corresponding
to the maximum eigenvalue of the matrix (D3 (r)H ⋅ D3 (r))−1 ([U ⋅ D3 (r)]H [U ⋅
D3 (r)]), where U = [u1 , u2 , … , uL ]H .
The proposed inversion method is tested through numerical simulations in
two scenarios, the noise free case and noise-contaminated case. We assume
that three small spheres are located at r1 = (0.084 𝜆, 0.196 𝜆, 0.084 𝜆), r2 =
(−0.168 𝜆, −0.056 𝜆, −0.112 𝜆) and r3 = (−0.196 𝜆, −0.084 𝜆, 0.140 𝜆). The
first two are isotropic spheres with permittivity 𝜖1 = 𝜖2 = 2𝜖0 , while the third
is a rotated anisotropic sphere with permittivity tensor 𝜖 3 = diag[𝜖0 , 3𝜖0 , 9𝜖0 ]
and rotation Euler angles (𝜓, 𝜙, 𝜃) = (𝜋∕4, 𝜋∕3, 3𝜋∕8) [20] (section 3.3). These
three spheres are electrically small, with the same radius a = 𝜆∕30. Note that
the smallest distance between the centers of spheres is 0.255 𝜆 (the distance
between the second and the third one) and, for convenience in depicting the
test results, all three spheres are chosen to locate in the y = x + 0.112 𝜆 plane. It
can easily be seen from the constitutive parameters of the scatterers that there
are up to eight independent secondary sources induced at the three scatterers.
93
Electromagnetic Inverse Scattering
There are 16 antenna units employed in this simulation, half of which are
aligned along the y-axis while the other half aligned along the z-axis in the
x = −13 𝜆 plane. The two linear arrays are centered at (−13 𝜆, −9 𝜆, 11 𝜆) with
5 𝜆 separation distance between neighboring units.
For the noise free case, the MSR matrix is calculated by (4.30). The singular
values of the MSR matrix are shown in Fig. 4.8(a), in which we see that the first
eight singular values are much larger than the rest, since they correspond to
the eight singular vectors spanning the signal subspace. Figure 4.8(b), Fig. 4.8(c),
and Fig. 4.8(d) are the pseudospectrum in the y = x + 0.112 𝜆 plane obtained by
the standard MUSIC method using x-, y-, and z-oriented test dipoles, respectively. Not surprisingly, the standard MUSIC algorithm locates only the first
two isotropic spheres and fails to locate the third degenerate anisotropic target. Here, since the value in pseudospectrum is too large at the positions of the
scatterers, we plot the base 10 logarithm of it, and the horizontal and vertical
−5
0.2
−10
0.1
z (λ)
log10(σj)
0
−15
−20
−25
30
20
0
−0.1
0
20
40
Singular Value Number (j)
0.2
0.1
25
0.1
20
0
15
−0.1
10
0.1
z (λ)
30
(c)
0.1
(b)
0.2
–0.2 –0.1 0
x (λ)
10
–0.2 –0.1 0
x (λ)
(a)
z (λ)
94
30
20
0
−0.1
10
–0.2 –0.1 0
x (λ)
0.1
(d)
Figure 4.8 Singular values and the pseudospectrum obtained by the standard MUSIC
algorithm in the noise free case. (a) The base 10 logarithm of the singular values of the MSR
matrix (j = 1, 2, … , 48). (b), (c), and (d) are the base 10 logarithm of the pseudospectrum in
y = x + 0.112 𝜆 plane obtained by the standard MUSIC algorithm with test dipoles in x, y,
and z directions, respectively. Source: Chen 2009, Inverse Problems, 25, 015008. [40]
Reproduced with permission of IOP Publishing.
Inverse Scattering Problems of Small Scatterers
0.2
2.5
2
0
−0.1
z (λ)
z (λ)
0.1
3.5
0.1
3
0
0.1
0.1
10
0
5
−0.1
0.1
15
0.2
z (λ)
0.2
z (λ)
0.1
(b)
15
(c)
2
–0.2 –0.1 0
x (λ)
(a)
–0.2 –0.1 0
x (λ)
2.5
−0.1
1.5
–0.2 –0.1 0
x (λ)
0.2
0.1
10
0
5
−0.1
–0.2 –0.1 0
x (λ)
0.1
(d)
Figure 4.9 Pseudospectrum obtained by the proposed MUSIC algorithm in the noise free
case. (a), (b), (c), and (d) are the base 10 logarithm of the pseudospectrum in the
y = x + 0.112 𝜆 plane obtained by the proposed MUSIC algorithm corresponding to the
L = 4, 5, 6, and 7 cases, respectively. Source: Chen 2009, Inverse Problems, 25, 015008. [40]
Reproduced with permission of IOP Publishing.
axes in Fig. 4.8(b), Fig. 4.8(c), and Fig. 4.8(d) are the x and z coordinate of spatial
points in the y = x + 0.112 𝜆 plane, so do the cases hereafter.
The pseudospectrum obtained by the proposed MUSIC algorithm is shown
in Fig. 4.9 with L = 4, 5, 6, and 7. From these results, we see that, to locate the
first two isotropic spheres, we only need L = 6, but, to locate the third degenerate anisotropic sphere, we need L = 7. This is because when locating either of
the first two isotropic spheres, the remaining two spheres only have five independent induced dipoles, which means that L = 6 is sufficient for (4.56) to have
exact solutions; but, when we locate the third degenerate sphere, the remaining
two isotropic spheres have totally six independent induced dipoles, thus only
when L = 7 can we solve (4.56). For the L = 4 and 5 cases, since the L is not large
enough to solve (4.56), none of the three scatterers can be located precisely. If
L is further increased to 8 and 9, the result will be almost the same as the one
in L = 7 case, which are not presented here.
For the noise-contaminated case, we add white Gaussian noise to the MSR
matrix so that SNR = 30 dB. Figure 4.10(a) shows the singular values of the
95
Electromagnetic Inverse Scattering
× 107
0
3
z (λ)
log10(σj)
0.2
−5
0.1
2
0
1
−0.1
0
20
40
Singular Value Number (j)
(a)
–0.2 –0.1 0
x (λ)
3
0.2
0.1
2
0
1
−0.1
–0.2 –0.1 0
x (λ)
(c)
0.1
0.1
(b)
× 107
× 107
2.5
0.2
z (λ)
−10
z (λ)
96
2
0.1
1.5
0
1
−0.1
0.5
–0.2 –0.1 0
x (λ)
0.1
(d)
Figure 4.10 Singular values and the pseudospectrum obtained by the standard MUSIC
algorithm in a noise-contaminated case (30 dB). (a) The base 10 logarithm of the singular
values of the MSR matrix (j = 1, 2, … , 48). (b), (c), and (d) are the pseudospectra in
y = x + 0.112 𝜆 plane obtained by the standard MUSIC algorithm with test dipoles in x, y,
and z directions, respectively. Source: Chen 2009, Inverse Problems, 25, 015008. [40]
Reproduced with permission of IOP Publishing.
noise-contaminated MSR matrix, in which the singular values corresponding
to the noise subspace are much larger than those in the noise-free case. In
such a case, if we apply the standard MUSIC algorithm to locate the scatterers, the pseudospectrum obtained by the test dipoles in x, y, and z direction are
shown in Fig. 4.10(b), (c), and (d), respectively, which show that all the three test
dipole directions fail to locate any of the three scatterers. By using the proposed
MUSIC algorithm, the pseudospectrum are drawn in Fig. 4.11. In Fig. 4.11, for
the L = 4, 5, 6, and 7 cases, image patterns are somewhat similar to those in
noise-free case shown in Fig. 4.9. However, for the L = 8 and 9 cases, some
unwanted disturbance appear in between the second and the third spheres,
which shows that the singular vector corresponding to the eighth singular value
is contaminated by the noise to such an extent it cannot be regarded as being
in the signal subspace anymore.
Inverse Scattering Problems of Small Scatterers
500
0.2
0.1
400
0.1
300
0
z (λ)
z (λ)
1500
0.2
200
−0.1
0
0.1
–0.2 –0.1 0
x (λ)
(a)
1500
1000
0
0.2
500
−0.1
500
−0.1
0.1
–0.2 –0.1 0
x (λ)
0.1
(d)
1500
0.1
1000
0
500
−0.1
0.1
0.2
1500
0.1
z (λ)
0.2
z (λ)
1000
0
(c)
(e)
1500
0.1
z (λ)
z (λ)
0.1
–0.2 –0.1 0
x (λ)
0.1
(b)
0.2
–0.2 –0.1 0
x (λ)
500
−0.1
100
–0.2 –0.1 0
x (λ)
1000
1000
0
500
−0.1
–0.2 –0.1 0
x (λ)
0.1
(f)
Figure 4.11 Pseudospectra obtained by the proposed MUSIC algorithm in a
noise-contaminated case (30 dB). (a), (b), (c), (d), (e), and (f ) are the pseudospectra in
y = x + 0.112 𝜆 plane obtained by the proposed MUSIC algorithm corresponding to the
L = 4, 5, 6, 7, 8, and 9 cases, respectively. Source: Chen 2009, Inverse Problems, 25, 015008.
[40] Reproduced with permission of IOP Publishing.
4.5 Discussions
A number of examples presented in this chapter have used synthetic data, but
it is worth noting that the MUSIC algorithm has been applied to measured data
experiments for various real world applications. For example, in seismic signal
97
98
Electromagnetic Inverse Scattering
processing, MUSIC has been utilized to perform higher-resolution imaging of
small-scale subsurface structures [22]; [23] has used sensor array to detect the
direction of arrival (DOA); in the framework of inverse synthetic aperture radar
(ISAR), MUSIC is used for radar target identification [24]; in [25], MUSIC algorithm was demonstrated to be suited for two-dimensional radar imaging; in
[26], it is applied to through-the-wall detection of life signs. This algorithm is
also extended to super-resolution fluorescence microscopy [27].
For point-like scatterers, one of the fundamental principles of MUSIC
can be traced to the injectivity of the source-to-field mapping operator. The
key step in proving injectivity lies in the singularity of the Green’s function
as the observation point approaches to the point-like scatterer (or induced
source). Consequently, MUSIC is not limited to the homogeneous background
inverse scattering problem, as long as the singularity of the Green’s function
holds. For example, MUSIC can be applied to locate point-like scatterers in
a half-space background [28, 29] and for scattering data with only intensity
information [30].
The information of point-like scatterers is not necessarily given a priori, but
instead it can be inferred from the singular value distribution of the multistatic response matrix, where a few dominant singular values are much larger
than the rest. If scatterers are larger than or comparable to the wavelength,
that is, referred to as extended scatterers, then the singular values exhibit a
slow-varying distribution. In principle, there is no rigorous theory on the application of MUSIC to extended scatterers due to the loss of injectivity of the
source-to-field mapping operator. The reason can be intuitively understood as
follows. The induced source in an extended scatterer is a spatially continuous
source, rather than a discrete point source. The measured scattered field can be
analytically continued in the background medium all the way to the boundary
of the extended scatterer. Thus, when a test point is inside the extended scatterer, the aforementioned singularity of Green’s function cannot occur since
the observation point that is analytically related to the measured scattered field
is exterior to the scatterer. Due to this reason, if we force to apply MUSIC
algorithm to an extended scatterer, the value of the pseudospectrum will not
reach infinity for a test point inside the scatterer. For a convex scatterer, the
pseudospectrum usually exhibits larger values inside than outside of the scatterer, but this property does not hold for a concave scatterer or two scatterers
very close to each other. Numerical examples of the application of MUSIC to
extended scatterers can be found in [31]. A special shape of scatterer is a thin
line, which can be considered as a state between the point-like and extended
scatterer. Theories and numerical examples of the application of MUSIC to thin
line scatterers can be found in [32, 33].
The MUSIC algorithm can be used to locate point-like scatterers and to
determine the direction of arrival (DOA). There is a great similarity between
these two applications. The correspondence is as follows. For scalar waves, the
Inverse Scattering Problems of Small Scatterers
discrete point-like scatterer in spatial space corresponds to the discrete DOAs
in the angle space. For vector waves, the polarization of induced dipoles inside
point-like scatterers corresponds to the signal polarization of DOA problem.
In Section 4.4.4, fewer singular vectors associated with larger singular values
are used to enhance the resolution of the MUSIC imaging method. This idea can
be further generalized to obtain an even higher resolution image by sampling
pairs of spatial points [34].
This chapter has shown from theoretical and numerical viewpoints that
MUSIC is able to achieve unlimited resolution for noise-free data. Nevertheless, it is not a well-posed question if one asks whether the MUSIC algorithm
has taken scatterers’ multiple scattering effects into account. The reason is
obvious from the definition of the MUSIC pseudospectrum, where the Green’s
function is directly used to process measured data. In addition, even if in a
fictitious situation where multiple scattering is absent in the forward problem
(i.e., assuming that measured data correspond to the fictitious situation),
MUSIC is still able to locate point-like scatterers, since multiple scattering
changes only the amplitude of induced source rather the position of induced
source. Mathematically, as mentioned earlier, the key step lies in the singularity
of the Green’s function as the observation point approaches to the point-like
scatterer (or induced source). Multiple scattering will not change the position
where the Green’s function becomes singular.
This chapter presents the theories and numerical methods of locating
point-like scatterers in a way emphasizing wave physics. Other approaches are
beyond the scope of the chapter. For example, the signal processing approach
has discussed methods like RAP-MUSIC (recursively applied and projected
MUSIC) [35] and ESPRIT (estimation of signal parameters via rotational
invariance techniques) [36]. The mathematical approach to small scatterer
localization can be found in [37, 38].
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103
5
Linear Sampling Method
To fully recover a scatterer’s information, such as the distribution of
permittivity, one has to solve a nonlinear inverse problem, which is often cast
into an optimization problem. This approach is referred to as the quantitative
approach. The disadvantages of this approach are heavy computational cost
and the possibility of being trapped in local minima. However, in many
practical situations, it suffices to provide basic information, such as how many
scatterers are present, their locations, shape, and size. In this case, the qualitative approach is preferred. Qualitative inversion methods seek to provide
approximate information about the shape, size, and positions of scatterers but
in general do not provide information about the material properties of the scatterer, such as the value of permittivity. Qualitative inversion methods choose
an indicator function that is defined on the domain of interest (DOI) and then
determine whether a spatial point lies inside or outside the scatterers. Since
each spatial point in the DOI is sampled, qualitative inversion methods are also
referred to as sampling methods. The time reversal (TR) imaging introduced
in Chapter 3 and multiple signal classification (MUSIC) method introduced in
Chapter 4 are both categorized as qualitative imaging approaches, though both
methods are discussed from other motivations in the previous two chapters.
In qualitative imaging methods, the definition of an indicator function usually exploits only the linearly mapped portion of the forward problem. Further,
since the most computationally demanding part for evaluation of the indicator
function involves only linear operations of matrices, the computational costs
associated with the qualitative methods are significantly smaller than their nonlinear counterparts; that is, quantitative inversion methods. The disadvantage
of qualitative methods is that quantitative information about the scatterer cannot be obtained.
The linear sampling method (LSM) is one of the most frequently used
qualitative inversion methods. It was developed in 1996 in the mathematical
community [1], and has been numerically proven to be a fast and reliable
method in many situations. This chapter mainly discusses the LSM for the
following reasons. The indicator function is easy to calculate and its physical
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
104
Electromagnetic Inverse Scattering
meaning is straightforward. The engineering community has provided several
physical interpretations to the method and consequently provided several
modified inversion algorithms.
The organization of the chapter is as follows. Section 5.1 outlines the LSM,
introducing the indicator function and the corresponding linear equation it
satisfies. Section 5.2 provides physical interpretation to the mechanism of the
LSM. Section 5.3 presents a modified LSM that is based on the multipole expansion of scattered field. Section 5.4 briefly introduces the factorization method
(FM) that is closely related to the LSM and more importantly provides the proof
of the rigor of the LSM. Section 5.5 discusses several topics regarding qualitative inversion methods, such as several other qualitative methods.
5.1 Outline of the Linear Sampling Method
We consider a two-dimensional scalar electromagnetic scattering problem (i.e.,
infinitely extending cylinders under TM illumination where the electric field
is parallel to the cylinder). Note that the LSM applies to the 3D case as well,
and the reason for presenting a 2D problem is that it is easy to understand
and to numerically implement. Scatterers can be either dielectric or perfectly
electrically conducting (PEC). The space occupied by scatterers is referred to
as the support of scatterers Σ and its boundary is denoted by 𝜕Σ. Scatterers
are illuminated by plane waves and the scattered field is measured in far field.
Let the measurement be carried out at Γ, which is a circle centered at the origin and has a sufficiently large radius r compared with the wavelength 𝜆. Let
Es (r, 𝜃 s , 𝜃 i ) be the scattered electric field measured on Γ in the direction 𝜃 s ,
when a unit-amplitude plane wave impinges from the direction 𝜃 i . The far-field
pattern of the scattered field E∞ (𝜃 s , 𝜃 i ) is defined by the following identity
eikr
Es (r, 𝜃 s , 𝜃 i ) = √ E∞ (𝜃 s , 𝜃 i )
r
(5.1)
The two-dimensional Green’s function is denoted by Φ(r, r′ ) = (i∕4)H0(1)
(k|r − r′ |), where k is the wave number. It is obvious that the radiation source
is a monopole, and its field pattern Φ(r, r′ ) is circularly symmetric with respect
to r′ .
The LSM chooses an indicator function that is defined on the domain of interest (DOI), denoted as D, and then determines whether a spatial point lies inside
or outside the scatterers. For any test point rt = (rt cos 𝜃 t , rt sin 𝜃 t ) in D, the LSM
first solves the far-field integral equation for the unknown g(rt , 𝜃 i ),
∫Γ
E∞ (𝜃 s , 𝜃 i )g(rt , 𝜃 i )d𝜃 i = Φ∞ (𝜃 s , rt )
(5.2)
where Φ∞ (𝜃 s , rt ) is the far-field pattern of the Green’s function Φ(r, rt ) when
the source point is at rt and the observation point is in the direction 𝜃 s . Strictly
Linear Sampling Method
speaking, the differential length in (5.2) should be rd𝜃 i , where the constant r
is left off for convenience. The large-argument asymptotic form of the Hankel
function, together with the definition of far-field pattern (5.1), gives
√
i
2 −i𝜋∕4 −ikrt cos(𝜃s −𝜃t )
s t
Φ∞ (𝜃 , r ) ≈
e
(5.3)
e
4 𝜋k
The L2 -norm of the solution g(rt , 𝜃 i ) is defined as
[
]1∕2
||g(rt , 𝜃 i )|| =
|g(rt , 𝜃 i )|2 d𝜃 i
(5.4)
∫Γ
which is often referred to as the energy of g(rt , 𝜃 i ).
The LSM chooses ||g(rt , 𝜃 i )|| as the indicator function. According to [1–3],
the value of ||g(rt , 𝜃 i )|| becomes unbounded if the sampling point rt , which is
also referred to as the test point, does not belong to the scatterer support Σ. In
other words, the support of scatterer is identified as those rt with corresponding
indicator function ||g(rt , 𝜃 i )|| being finite.
In numerical implementation of the LSM, all continuous vector spaces in
(5.2) are discretized into finite-dimension vector spaces. In addition, there is
always noise that contaminates the measured scattered field. Consequently, the
indicator function ||g(rt , 𝜃 i )|| cannot be infinite for a sampling point rt that is
out of the support of scatterer, but instead, the indicator function is much larger
for rt that is outside the support than inside. The following procedure, used to
determine g(rt , 𝜃 i ), has been proven to be numerically quite successful:
1) Select a grid of “sampling points” in a domain of interest D that is known a
priori to contain the support Σ of scatterers.
2) For each rt in the foregoing grid, cast (5.2) into a matrix-vector linear
equation and then use Tikhonov regularization to compute an approximate
solution g(rt , 𝜃 i ) to the discrete linear equation. In discrete form, consider a
system of Ni transmitters and Ns receivers. Equation (5.2) is converted to
E ∞ ⋅ g = Φ∞
(5.5)
where E∞ is a matrix of size Ns × Ni , g is an Ni dimensional vector that
needs to be solved for, and Φ∞ is a vector of size Ns . In terms of the singular value decomposition (SVD), the matrix E∞ has the property of E∞ ⋅ 𝑣s =
𝜎s us , where us and 𝑣s denote the sth left and right singular vectors, and 𝜎s
denotes the sth singular value of the matrix E∞ . The solution of vector g
computed using Tikhonov regularization is denoted as,
∑ 𝜎s
gT =
uH ⋅ Φ∞ 𝑣s
(5.6)
2
2 s
s 𝜎s + 𝛼
where the subscript T denotes the Tikhonov regularization, 𝛼 denotes the
Tikhonov regularization parameter, and the superscript H denotes the
Hermitian operator.
105
106
Electromagnetic Inverse Scattering
3) Choose a cutoff value C, and determine the support of scatterers by the criterion that rt is in scatterer support Σ if and only if ||g(rt , 𝜃 i )|| ≤ C. The choice
of C is heuristic.
5.2 Physical Interpretation
In the LSM, the right-hand side of (5.2) represents a circularly symmetric far
field pattern. However, this pattern is not necessarily achieved by an elementary point source or a circularly symmetric current around the sampling point.
The reason for this argument is discussed next in Section 5.2.1. It highlights
that this argument suggests other physical interpretations should be sought.
Section 5.2.2 interprets the LSM from the angle of the multipole expansion of
the far field, which eventually leads to the modified LSM that is to be discussed
in Section 5.3.
5.2.1
Source Distribution
First, we construct a source distribution in the free space that is able to produce
the far field radiation pattern Φ∞ (𝜃 s , rt ). Let Ω be any smooth domain containing the sampling point rt ; that is, rt ∈ Ω. An arbitrary function u(r) ∈ C 2 (Ω)
that is twice continuously differentiable on the closure Ω of Ω, satisfying the
following boundary conditions, is chosen:
u(r) = Φ(r, rt ) and
𝜕u(r) 𝜕Φ(r, rt )
=
𝜕n
𝜕n
both for r ∈ 𝜕Ω
(5.7)
where 𝜕u
= ∇u ⋅ n is the directional derivative of u in the direction of the out𝜕n
ward pointing normal n to the surface element ds. We choose the source distribution in Ω as
J(r) = −∇2 u(r) − k 2 u(r).
(5.8)
For r ∈ Ω, using the Green’s formula [4] (section 2.2),
[
]
′
′
′ 𝜕u(r )
′ 𝜕Φ(r, r )
u(r) =
Φ(r, r )
− u(r )
ds(r′ )
∫𝜕Ω
𝜕n(r′ )
𝜕n(r′ )
−
∫ ∫Ω
[∇2 u(r′ ) + k 2 u(r′ )]Φ(r, r′ )dr′ .
(5.9)
and substituting (5.7) to it, we obtain
[
]
′
𝜕Φ(r′ , rt )
′ t 𝜕Φ(r, r )
u(r) =
,
r
)
Φ(r, r′ )
−
Φ(r
ds(r′ )
∫𝜕Ω
𝜕n(r′ )
𝜕n(r′ )
+
∫ ∫Ω
J(r′ )Φ(r, r′ )dr′ .
(5.10)
Linear Sampling Method
It is easy to prove that the first integral vanishes by applying Green’s second
theorem [4] (section 2.2),
[
]
′
𝜕Φ(r′ , rt )
′ t 𝜕Φ(r, r )
,
r
)
Φ(r, r′ )
−
Φ(r
ds(r′ )
∫𝜕Ω
𝜕n(r′ )
𝜕n(r′ )
=
∫ ∫Ω
[Φ(r, r′ )∇2 Φ(r′ , rt ) − Φ(r′ , rt )∇2 Φ(r, r′ )]dr′
(5.11)
considering the fact that the right-hand-side integral is equal to zero since both
Φ(r′ , rt ) and Φ(r, r′ ) satisfy the Helmholtz equation. Thus, we have
u(r) =
∫ ∫Ω
J(r′ )Φ(r, r′ )dr′
(5.12)
r ∈ Ω.
1.5
–1
–1
0
1
x axis (λ)
(a)
1
2
0
1.5
–1
–1
0
1
x axis (λ)
(b)
1
1
0
–1
–1
x 10–4
8
6
4
2
0
1
│υn│→
0
1
y axis (λ)
2
y axis (λ)
1
y axis (λ)
This definition of u(r) for r ∈ Ω can be extended to the domain outside Ω; that
is, for r ∈ ℝ2 ∖Ω. Thus, u(r) satisfies the radiation condition, coincides with
Φ(r, rt ) on r ∈ 𝜕Ω, and satisfies the Helmholtz equation in ℝ2 ∖Ω. Considering the fact that the Helmholtz equation has a unique solution in the closed
domain ℝ2 ∖Ω once the boundary values are given at the two boundaries 𝜕Ω
and Γ∞ , it can be concluded that u(r) = Φ(r, rt ) in ℝ2 ∖Ω. Thus, (5.12) means
that the current distribution J(r) generates the field Φ(r, rt ). Consequently, the
current distribution J(r) is able to produce the far field pattern Φ∞ (𝜃 s , rt ). Due
to the arbitrariness of both the domain Ω and the function u(r) ∈ C 2 (Ω), so long
as u(r) satisfies (5.7), the current distribution J(r) that produces the far field pattern Φ∞ (𝜃 s , rt ) is not necessarily a circularly symmetric source centered at rt or
an elementary source (i.e., monopole) located at rt .
Next, we consider a current distribution in the scatterer (r ∈ Σ) that is able to
produce the far field radiation pattern Φ∞ (𝜃 s , rt ). Let the Ω that has a smooth
boundary, as introduced in the previous paragraphs, be infinitely close to Σ.
Then following the previous argument, the induced current distribution J(r ∈
Σ) = −∇2 u(r) − k 2 u(r) is not necessarily a circularly symmetric source centered
in rt or an elementary source located at rt in order to produce Φ∞ (𝜃 s , rt ).
0
1
x axis (λ)
(c)
0.5
0
–20
0
n→
20
(d)
Figure 5.1 Illustration of the current distribution for the sampling point rt = (0, 0) and the
proposed multipole-based interpretation. (a) Scatterer profile (relative permittivity); (b)
Reconstruction LSM; (c) Current distribution (sampling point (0,0)); (d) Plot of 𝑣n for various n
(sampling point (0,0)). Source: Agarwal 2010, Opt. Express, 18, 6366–6381. [6] Reproduced
with permission of The Optical Society.
107
108
Electromagnetic Inverse Scattering
As an example, let us consider the scatterer shown in Fig. 5.1(a), which
is an S-shape dielectric scatterer with uniform relative permittivity 𝜖r = 2.
In numerical simulations, 13 detectors and 13 sources, distributed uniformly
along Γ with a radius 10 𝜆, have been used for reconstruction. The scattered
data is noise free and the LSM reconstruction result thus obtained is shown
in Fig. 5.1(b). Here, for visualization purposes, the indicator −log10 ||g(rt , 𝜃 i )||
has been plotted. It is important to note that the values of this new indicator
function are higher for points inside the support of the scatterer than outside.
Considering the sampling point rt = (0, 0), which LSM detects as belonging
to the scatterer, the distribution of induced current is plotted in Fig. 5.1(c). It
is noticeable that there is neither a monopole source at rt , nor is the current
distribution circularly symmetric.
5.2.2
Multipole Radiation
As shown in the previous section, the monopole radiation pattern is not necessarily produced by a point source or a circularly symmetric source distribution. This indicates that instead of investigating current distribution, we should
investigate the radiation field itself. The scattered fields received at receivers can
be decomposed into various independent terms corresponding to the multipole
radiation from a sampling point rt . In fact, solving (5.2) amounts to finding a
linear superposition of E∞ (𝜃 i , 𝜃 s ), with g(rt , 𝜃 i ) being the coefficient, such that
among all multipoles, the monopole radiation is the only dominant contributor
in the resultant total radiation.
Let the far field pattern E∞ (𝜃 s , 𝜃 i ) be measured at the point r𝜃s = (r cos 𝜃 s ,
r sin 𝜃 s ), and the exact scattered field Es (r, 𝜃 s , 𝜃 i ) at this point is related to the
far field pattern by (5.1). Scattered field can be interpreted as the reradiation of
induced current,
Es (r, 𝜃 s , 𝜃 i ) =
∫ ∫Σ
J(r′ , 𝜃 i )Φ(r𝜃s , r′ )dr′
(5.13)
Using the addition theorem [5] (section 9.1) on Φ(r𝜃s , r′ ), the expression of
Es (r, 𝜃 s , 𝜃 i ) can be rewritten in terms of various multipoles corresponding to
a sampling point rt as below:
Es (r, 𝜃 s , 𝜃 i ) =
∞
∑
𝛼 (n) (rt , 𝜃 i )Φ(n) (r𝜃s , rt )
(5.14)
n=−∞
where
𝛼 (n) (rt , 𝜃 i ) =
t
J(r′ , 𝜃 i )Jn (k|r′ − rt |)e−in arg(r −r ) dr′
∫ ∫Σ
t
i
Φ(n) (r𝜃s , rt ) = Hn(1) (k|r𝜃s − rt |)ein arg(r𝜃s −r )
4
′
(5.15)
(5.16)
Linear Sampling Method
It is evident that 𝛼 (n) (rt , 𝜃 i ) represents the nth effective multipole current at rt .
From (5.14), the fundamental equation of LSM (5.2) is equivalent to:
{
1, if n = 0
(n) t i
t i
i
𝛼 (r , 𝜃 )g(r , 𝜃 )d𝜃 =
(5.17)
∫Γ
0, otherwise
For convenience of further use, we define:
𝑣n =
∫Γ
𝛼 (n) (rt , 𝜃 i )g(rt , 𝜃 i )d𝜃 i
(5.18)
Physically, 𝑣n can be understood as the nth order multipole current induced
by the incident fields that are generated by the primary source distribution
g(rt , 𝜃 i ). This means that as long as the multipole expansion of the induced
current distribution at a sampling point is such that the monopole is the
only prominent contributor, the sampling point will be judged to be inside a
scatterer.
For the example in Section 5.2.1, we use 41 multipoles and consider the
same sampling point as before; that is, rt = (0, 0). The values of 𝛼 (n) (rt , 𝜃 i )
are determined analytically using (5.15), where J(r′ , 𝜃 i ) is obtained by
solving a forward problem with the primary source distribution being
∑20
g(rt , 𝜃 i ). The error in the multipole expansion || n=−20 𝛼 (n) (rt , 𝜃 i )Φ(n) (r𝜃s , rt ) −
−14
s i
s i
Es (r, 𝜃 , 𝜃 )||∕||Es (r, 𝜃 , 𝜃 )|| is approximately 10 . Thus, the multipole expansion is reasonably correct. The values of 𝑣n , which are calculated from (5.18),
are shown in Fig. 5.1(d). It is evident that (5.17) is approximately satisfied, that
is, the monopole radiation dominates the scattered field for the sampling point
rt = (0, 0) that is indeed inside the scatterer.
5.3 Multipole-Based Linear Sampling Method
5.3.1
Description of the Algorithm
Based on the multipole analysis, we present an approach to construct the primary source distribution g(rt , 𝜃 i ). For the ease of reference, especially in figures,
we shall call the proposed method the multipole-based linear sampling method
(MLSM). Although in principle one needs to consider infinite number of multipoles to fully account for the scattered field in (5.14), in practice a sufficiently
large finite number of multipoles is enough to approximate Es (r, 𝜃 s , 𝜃 i ). Considering (2N + 1) number of multipoles, the expression for the far field (5.14) can
be rewritten as:
N
∑
Es (r, 𝜃 s , 𝜃 i ) ≈
𝛼 (n) (rt , 𝜃 i )Φ(n) (r𝜃s , rt )
(5.19)
n=−N
109
110
Electromagnetic Inverse Scattering
Equation (5.19) suggests that the multipole radiation functions Φ(n) (r𝜃s , rt ) can
be understood as a mapping from the effective multipole source 𝛼 (n) (rt , 𝜃 i ) at a
sampling point rt to the measured scattered electric field Es (r, 𝜃 s , 𝜃 i ). In discrete
form, consider a system of Ni transmitters and Ns receivers. For each incidence
𝜃 i , (5.19) can be written as a form of the aforementioned mapping,
E = Φ ⋅ A,
(5.20)
where, E, an Ns -dimensional vector, consists of all the receiver measurements,
Φ, a matrix of dimension Ns × (2N + 1), consists of the multipole radiation
terms Φ(n) (r𝜃s , rt ), and A, a (2N + 1)-dimensional vector, contains the effective
multipole sources 𝛼 (n) (rt , 𝜃 i ).
The value of A can be solved uniquely using the least-squares pseudoinverse
(see Appendix B),
†
A = Φ ⋅ E,
(5.21)
where the superscript † denotes the pseudoinverse of a matrix. If the singular
H
†
′
value decomposition of Φ is U ⋅ Σ ⋅ V , then the pseudoinverse is Φ = V ⋅ Σ ⋅
H
U , where the diagonal element Σ′ii = Σ−1
for Σii ≠ 0 and Σ′ii = 0 for Σii = 0.
ii
After obtaining A for each of the Ni incidences, a discretized version of (5.17)
is written as:
A⋅g =D
(5.22)
where A, of dimension (2N + 1) × Ni , contains the vectors A for all incidences, g
is an Ni -dimensional vector that needs to be determined, and D is a vector with
all elements except the (N + 1)th element being zero. The (N + 1)th element,
which corresponds to a monopole, is equal to one. The value of vector g can be
determined from (5.22) by the least-squares pseudoinverse,
†
gM = A ⋅ D
(5.23)
where the subscript M denotes the multipole-based LSM.
The physical interpretation of the LSM is as follows: for a sampling point
outside the support of the scatterer, the indicator function becomes large in the
(impossible) attempt of synthesizing a primary source that is able to suppress
the radiation of induced dipole and higher-order multipoles.
5.3.2
Choice of the Number of Multipoles
The reconstruction method presented here is mathematically equivalent to
the standard LSM (i.e., the Tikhonov-regularized LSM) if we consider a large
N such that the multipoles of order higher than the considered (2N + 1)
multipoles have a negligible contribution to the scattered field. However, from
numerical point of view, at least (2N + 1) receivers are needed to uniquely
solve (5.20). Further, using a large number of multipoles is expected to increase
the computational cost of the problem. Thus, it is interesting to study the effect
of choosing a small number of multipoles.
Linear Sampling Method
To this end, it is worth trying N = 1, which is one plus the order of monopole.
Such a choice is based on the fact that monopole term is dominant among
all multipole terms in most scattering problems. The use of N = 1 implies the
following. To solve (5.20) for A would mean that we seek an optimal combination of the monopole and dipole sources such that the resultant radiation fields
match the measured scattered fields as closely as possible. To solve (5.22) for g M
means that a linear superposition of monopoles and dipoles leads to a negligible
dipole term. We mention in passing that although the monopole term is dominant in most scattering problems, cases indeed exist where other higher order
multipoles are dominant. For such cases, a suitable choice of N will be one plus
the order of the dominant multipole. Such cases rarely happen in the monofrequency scenario since the operation frequency of the wave has to be carefully
chosen so that it is close to the resonance frequencies of higher-order modes.
For the sake of illustration and further discussion, we present the reconstruction results for the geometry shown in Fig. 5.1(a). The measurement setup used
is as before. We plot the indicator function −log10 ||gM (rt , 𝜃 i )||. Figure 5.2(a)
and (b) present the reconstruction results for N = 20 and N = 1, respectively. It is evident that N = 20 gives a reconstruction similar to the result of
Tikhonov-regularization LSM. Compared with the case of N = 20, it is noticeable that N = 1 produces a higher value of −log10 ||gM (rt , 𝜃 i )||. This is expected
because when N = 1, the strict requirement on gM (rt , 𝜃 i ) of suppressing all the
higher order multipoles (N = 2, 3, … , 20) is now significantly eased. The case
of N = 1 requires suppression of only the dipole term.
0
1
x axis (λ)
(a)
y axis (λ)
1
2
1.8
1.6
0
–1
–1
0
1
x axis (λ)
(e)
1.4
1.2
–1
–1
2
0
–0.5
–1
–1
1.9
0
1
x axis (λ)
(f)
–1
–1
1
0
1
x axis (λ)
(g)
2
0
1
0
0
1
x axis (λ)
(d)
1
2
0
1
–1
–1
0
1
x axis (λ)
(c)
1
2.1
0.5
1
0.5
–1
–1
0
1
x axis (λ)
(b)
1
0
y axis (λ)
1.9
1.5
y axis (λ)
2
0
2
y axis (λ)
–1
–1
1.4
1.2
1
2.1
y axis (λ)
0
1
y axis (λ)
2
1.8
1.6
y axis (λ)
y axis (λ)
1
2
0.5
1.5
0
1
–0.5
–1
–1
0
0.5
0
1
x axis (λ)
(h)
0
Figure 5.2 Comparison of MLSM (N = 20) and MLSM (N = 1) for noise-free and noisy (10%
additive Gaussian noise) scenarios. (a) Reconstruction based on N = 20 multipoles
(noise-free); (b) Reconstruction based on N = 1 multipoles (noise-free); (c) Support
estimated using (a) (noise-free); (d) Support estimated using (b) (noise-free); (e)
Reconstruction based on N = 20 multipoles (10% noise); (f ) Reconstruction based on N = 1
multipoles (10% noise); (g) Support estimated using (e) (10% noise); (h) Support estimated
using (f ) (10% noise). Source: Agarwal 2010, Opt. Express, 18, 6366–6381. [6] Reproduced
with permission of The Optical Society.
111
Electromagnetic Inverse Scattering
An estimate of the scatterer support, denoted as Σ′ , can be determined as
below:
Σ′ = {rt ∶ −log10 ||gM (rt , 𝜃 i )|| > Min + 𝛽(Max − Min)}
(5.24)
where Min and Max are minimum and maximum of the value of −log10
||gM (rt , 𝜃 i )|| for all the test points rt , respectively, and 𝛽 is a user-defined
threshold used for estimating the scatterer support. The effect of threshold
shall be discussed later. Presently we use 𝛽 = 0.8. The scatterer’s supports
estimated for N = 20 and the proposed method (N = 1) are presented in
Fig. 5.2(c) and (d), respectively. The results presented are for the noise-free
scenario. Results for the noisy situation (10% additive white Gaussian noise)
are presented in Fig. 5.2(e)–(h). It is evident that the proposed method (N = 1)
yields a better estimation of the scatterer’s shape than the case of N = 20.
The choice of N = 1 is also supported by a cross checking via the
Tikhonov-regularization LSM. First, we still calculate the value of A from
(5.21) for N = 1 and N = 20, respectively. Instead of solving (5.22) for g M , we
adopt gT (rt , 𝜃 i ) that is obtained by the Tikhonov regularization method and
then compare whether the left-hand sides of the equations, that is, A ⋅ g, are
close to each other for N = 1 and N = 20. Since Eq. (5.22) is a discretized
version of (5.17), it is clear that the rows of A ⋅ g correspond to 𝑣n , n = −N to
N, defined in (5.18). The absolute value of the difference between the effective
multipole sources 𝑣n (n = −1 to 1) for N = 20 and N = 1 is plotted in Fig. 5.3.
Since 𝑣−1 = 𝑣1 = 0 and 𝑣0 = 1 in the ideal case (N = +∞) for points inside the
support of the scatterer, it can be seen that the difference is very small (of the
order 10−4 ) for the three multipoles over the complete DOI. Thus, the choice
of N = 1 is reasonable.
In addition, for the scatterer shown in Fig. 5.1, it is observed in [6], where
hundreds of simulations are performed that, compared with other multipoles,
the monopole is least affected by noise no matter whether a sampling point is
y axis (λ)
y axis (λ)
∣υ0(N = 20) – υ0(N = 1)∣
∣υ1(N = 20) – υ1(N = 1)∣
∣υ–1(N = 20) – υ–1(N = 1)∣
×10–4
×10–4
×10–4
1
1
1
12
4
4
10
3
8
3
0
0
0
6
2
2
4
1
1
2
–1
–1
–1
1
0
1
–1
0
–1
–1
0
1
x axis (λ)
x axis (λ)
x axis (λ)
Plot of difference between υn(N = 20) and υn(N = 1)
y axis (λ)
112
Figure 5.3 Cross checking the effect of reduction of multipoles. The values of 𝑣n are
obtained for N = 20 and N = 1, respectively, and the difference between the two sets of 𝑣n is
computed. The first, second, and third columns show this absolute value of the difference
for n = −1, 0, and 1, respectively. Source: Agarwal 2010, Opt. Express, 18, 6366–6381. [6]
Reproduced with permission of The Optical Society.
Linear Sampling Method
inside or outside the support of the scatterer. For a sampling point inside the
support of scatterer, since the monopole radiation is the only dominant multiple and in addition the monopole term is most stable in presence of noise, we
conclude that the value of N does not have to be large. This argument partially
supports the choice of N = 1.
5.3.3
Comparison with Tikhonov Regularization
In the multipole-based LSM, the value of N, the highest order of multipole,
can be considered a regularization parameter. On one hand, the larger the value
of N, the better the data fitting in (5.20), but the more unstable the recovered g M
in presence of noise. On the other hand, the smaller the value of N, the poorer
the data fitting but the more stable the solution. Since the monopole term is
dominant among all multipole terms in most scattering problems, we have
chosen N = 1, which means truncating all higher order multipoles except the
dipole and monopole terms. For the uncommon case where a higher order multipole dominates the radiation, a suitable choice of N will be one plus the order
of the dominant multipole.
Tikhonov regularization chooses the real-valued 𝛼 as the regularization
parameter. The smaller the value of 𝛼, the better the data fitting but the more
unstable the solution. For a high value of 𝛼, the effect is the opposite. The
value of 𝛼 is usually computed using the discrepancy principle for each pixel
in the DOI if the level of noise is known a priori [2]. Otherwise, it is chosen
empirically.
From a practical point of view, the Tikhonov-regularization parameter
𝛼 is not easy to determine since it is a real number. In comparison, the
multipole-based regularization parameter N is much easier to choose since it
is an integer. In most of numerical simulations the author has implemented,
the choice of N = 1 works well. For N = 2, the reconstruction results do not
significantly differ from the case of N = 1, though there are some minor differences. If N gradually increases with step one to a large number, say N = 20,
then the reconstructed result steadily evolves to a pattern that is comparable
to a reconstructed result obtained by standard LSM with a relatively small
Tikhonov regularization parameter.
To compare the performance of the standard LSM and the multipole-based
LSM, we apply them to reconstruct the scatterer shown in Fig. 5.1. The reconstruction for the multipole-based LSM with N = 1 has been shown in Fig. 5.2.
For the standard LSM, we choose various values of the regularization parameter
𝛼 = a𝜎1 , where a takes values {0.001, 0.01, 0.1, 1} and 𝜎1 is the largest singular
value of the matrix E∞ that is defined in (5.5). The reconstruction results are
presented in Fig. 5.4(a).
We define an error parameter that can be used to evaluate the quality of
reconstruction. The error measure is defined on the convex hull (also known as
the “convex envelope”) of the scatterer. The threshold 𝛽 classifies the sampling
points as scatterers or nonscatterers. If the number of sampling points inside
113
0
–1
0
–1
1
x axis (λ)
1.2
0
1.5
–1
–1
1
0
x axis (λ)
LSM α = 0.1 σ1
1
0
–1
0
–1
1
x axis (λ)
2
1.8
1.6
y axis (λ)
1.6
LSM α = 0.01 σ1
1
2
y axis (λ)
LSM α = 0.001 σ1
1
2
y axis (λ)
Electromagnetic Inverse Scattering
y axis (λ)
1
LSM α = 1 σ1
0
–1
0
–1
1
x axis (λ)
2.15
2.1
2.05
(a)
0.3
0.25
0.2
Error
114
0.15
LSM: (α = 0.001 σ1)
LSM: (α = 0.01 σ1)
LSM: (α = 0.1 σ1)
LSM: (α = 1 σ1)
MLSM
0.1
0.05
0
0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
β→
(b)
Figure 5.4 Comparison of LSM and MLSM. The results are obtained in the presence of 10%
noise. (a) Result of the conventional LSM for various values of 𝛼; (b) Plot of Error for various
values of threshold. Source: Agarwal 2010, Opt. Express, 18, 6366–6381. [6] Reproduced with
permission of The Optical Society.
the convex hull that get classified wrongly is Merr(𝛽) , then the error parameter
is defined as:
Error(𝛽) = Merr(𝛽) ∕M,
(5.25)
where M is the total number of sampling points inside the convex hull. This
error measure is useful to evaluate the accuracy of the reconstruction methods
in reconstructing scatterers with complicated supports, such as an annulus. In
this numerical example, we vary the threshold 𝛽 in the range [0.6, 0.9] and plot
the error measure Error (𝛽) for the conventional LSM (for various values of 𝛼)
and the proposed method in Fig. 5.4(b). It should be noted that, in practice, a
suitable value of 𝛽 is not known a priori and 𝛽 is chosen heuristically in most
cases. It is evident that the proposed method has lower values of Error(𝛽) for
most values of 𝛽.
5.3.4
Numerical Examples
We consider various examples and compare the performance of the standard
LSM and the proposed method. Each of the examples considered here is
difficult to reconstruct qualitatively using the conventional LSM on various
2
0
1.5
–1
–1
0
1
x axis (λ)
1
0
–1
–1
1
0
1
x axis (λ)
3.8
3.6
3.4
3.2
3
2.8
1
y axis (λ)
1
y axis (λ)
y axis (λ)
Linear Sampling Method
4
0
3.9
–1
–1
3.8
0
1
x axis (λ)
2
0
1.5
–1
–1
0
1
x axis (λ)
1
1
0
–1
–1
0
1
x axis (λ)
3.8
3.6
3.4
3.2
y axis (λ)
1
y axis (λ)
y axis (λ)
(a)
1
4
0
3.9
–1
–1
0
1
x axis (λ)
3.8
(b)
–1
–1
1.5
0
1
x axis (λ)
1
y axis (λ)
y axis (λ)
2
0
1
3.9
3.8
3.7
3.6
3.5
0
–1
–1
0
1
x axis (λ)
1
y axis (λ)
2.5
1
4
3.9
0
–1
–1
3.8
0
1
x axis (λ)
(c)
Figure 5.5 Examples of reconstruction of dielectric cylinders. The first column shows the
scatterer profile (relative permittivity), the second column shows the reconstruction using
the conventional LSM and the third column shows the reconstruction using the proposed
MLSM. The results are obtained in the presence of 10% noise. (a) Example 1: Austria profile
[31]; (b) Example 2: Obstructed circular cylinders; (c) Example 3: Enclosed circular cylinders.
Source: Agarwal 2010, Opt. Express, 18, 6366–6381. [6] Reproduced with permission of The
Optical Society.
accounts. These include annulus-shape scatterers and multiple scatterers in
close proximity (distance being less than half wavelength). All the examples
consider a square DOI of size 2 × 2 m2 , where the scatterers are placed in
free space. The frequency of the incident wave is 300 MHz (wavelength
𝜆 = 1 m). The measurement setup is the same as described in Section 5.2.1. The
measured data is corrupted by 10% additive Gaussian noise. Three examples of
dielectric scatterers (Fig. 5.5) and two examples of PEC scatterers (Fig. 5.6) are
considered. In Fig. 5.5 and Fig. 5.6, the first column shows the scatterer profile
(relative permittivity for dielectric cylinders, the contours for PEC cylinders),
the second column shows the reconstruction using the conventional LSM and
the third column shows the reconstruction using MLSM. The regularization
parameter used in the standard LSM has been computed using the discrepancy
principle. The proposed MLSM chooses N = 1.
115
Electromagnetic Inverse Scattering
y axis (λ)
y axis (λ)
0
–5
0
5
x axis (λ)
2
0
1.8
–1
–1
1.6
0
x axis (λ)
1
1
2
y axis (λ)
1
5
–5
0
1.8
–1
–1
0
x axis (λ)
1
(a)
1
0
–5
–5
0
5
x axis (λ)
0
–1
–1
1
2
1.8
0
x axis (λ)
1
1.6
y axis (λ)
y axis (λ)
5
y axis (λ)
116
2
0
–1
–1
1.9
0
x axis (λ)
1
1.8
(b)
Figure 5.6 Same as Fig. 5.5 except that scatterers are perfectly conducting cylinders.
(a) Example 1: Three cylinders; (b) Example 2: Nine circular cylinders. Source: Agarwal 2010,
Opt. Express, 18, 6366–6381. [6] Reproduced with permission of The Optical Society.
The reconstruction results presented in Fig. 5.5 and Fig. 5.6 show that the
multipole-based LSM outperforms the standard LSM. In particular, for the
scatterer in Fig. 5.5(a), there is a central hollow area inside the annulus. As
expected from the theory of the LSM, the middle panel in Fig. 5.5(a) shows
that the standard LSM cannot detect the presence of the central hollow
area. This is because the annulus separates the inner and outer spaces into
completely disconnected regions and consequently the scattered field inside
the annulus cannot be analytically continued all the way to the far field where
measurement is conducted. In comparison, the right panel in Fig. 5.5(a) shows
that the multipole-based LSM roughly detects the presence of the central
hollow area, although the shape of the reconstructed hollow area is somewhat
different from the exact one.
5.4 Factorization Method
Although the LSM numerically works well in many numerical examples, it
lacks a rigorous justification regarding the regularized solution of the far-field
equation. In general, a solution to the far-field equation (5.2) does not exist
for noise-free data [3]. Nevertheless, a theory has proven that there exists
an “approximate solution” ga to (5.2), such that the corresponding incident
i
i
field onto the scatterer Ei (x, y, rt ) = ∫Γ ei(kx cos 𝜃 +ky sin 𝜃 ) ga (rt , 𝜃 i )d𝜃 i (known as
Linear Sampling Method
the Herglotz wave function in the mathematical community) converges if and
only if rt is inside the support Σ of the scatterer [7] (chapter 7). However, there
are two problems associated with the LSM: (1) the convergence property of
Ei (x, y, rt ) depends on the support Σ of the scatterer, and unfortunately such
a convergence claim cannot be made about the “approximate solution” ga to
(5.2); and (2) it is not clear yet whether the Tikhonov regularized solution of
(5.2) inherits the same behavior as the “approximate solution” ga of (5.2).
The factorization method (FM) as developed by Kirsch [8–10] overcomes
the aforementioned questions. Simply speaking, the factorization method is an
extension of the MUSIC imaging algorithm to the case of extended objects
and infinite-dimensional scattering operators [10, 11]. We first recall some
main results of the MUSIC for point-like scatterers that have been presented
in Chapter 4. The expression of the multistatic response (MSR) matrix K is
K = GS ⋅ Λ ⋅ (I M − Φ ⋅ Λ)−1 ⋅ GI ,
(5.26)
where GS is the scattering matrix, physically representing the radiation of
induced source. When primary sources and receivers are both placed in far
field, the MSR matrix K is rewritten as F, following the convention adopted
in the mathematical community. The matrix F is understood to be the discretized version of the far-field operator F, which is defined as the integral
operator in (5.2) with the kernel E∞ (𝜃 s , 𝜃 i ). Standard linear algebra yields that
if the the numbers of incidents Ni and measurements Ns are large enough
so that the rank of the far-field MSR matrix F is equal to the number M of
point-like scatterers, the the ranges (F) and (GS ) of F and GS , respectively,
coincide. When applying MUSIC to the far-field MSR matrix F, we have:
For any position r, the Ns -dimensional vector Gs (r) = [G(r′1 , r), … ,
G(r′N , r)]T , which is referred to as the background Green’s function vector
s
evaluated at r, is in the range (F) of F if and only if r ∈ {r1 , r2 , … , rM }.
Next, for the case of extended penetrable scatterers and infinite-dimensional
incidences and measurements, the continuous counterparts of GS and F are
denoted as Gs and F, respectively. The following theorem is proven in [10].
For any position rt , the far field expression of Green’s function Φ∞ (𝜃 s , rt ) evaluated at rt , which is shown in ( 5.3), is in the range (Gs ) of Gs if and only if
rt ∈ Σ.
The sufficient condition in fact has been proven in Section 5.2.1, noting that
Gs is an integral operator with the kernel Φ(r, r′ ). The necessary condition can
be proven by contradiction, which is outlined as follows. The scattered field is
an analytical function in the space exterior to the support of scatterer, and thus
the far-field information can be analytically continued all the way to the external
boundary of the scatterer. For a testing point rt outside the support of scatterer,
if its far field expression of Green’s function Φ∞ (𝜃 s , rt ) is in the range of (Gs ) of
Gs , say Φ∞ (𝜃 s , rt ) = Gs (𝑤) for a certain function 𝑤 of source distribution, then
117
118
Electromagnetic Inverse Scattering
Φ(r, rt ) is singular at rt but Gs (𝑤) is a continuous function at rt . Thus, proving
by contradiction, we conclude that if rt is outside the support of scatterer, its
Φ∞ (𝜃 s , rt ) cannot be in the range of (Gs ) of Gs . From the proof of the sufficient
and necessary conditions, we easily see that for an annulus scatterer, the space
interior to the inner boundary is completely separated from that exterior to
the outer boundary, so that the far-field information cannot be analytically
continued all the way to the space interior to the inner boundary. This is why
the LSM and factorization method cannot deal with annulus-shape scatterers.
It is important to note that in the infinite-dimensional case, the ranges
of Gs and F do not coincide, which is a distinct difference from their
finite-dimensional counterparts. This fact can be seen from the physical
meaning of Gs and F. The Gs maps the source distribution on the support
of scatterer Σ to the scattered field in far-field, whereas the F maps the
amplitudes of incident plane waves to the scattered field in far-field. For Gs ,
the source distribution can be an arbitrary square-integrable function in Σ;
that is, the domain of Gs is L2 (Σ). For F, however, the source on Σ is induced
and its distribution, obtained by solving the Lippmann–Schwinger equation,
is only a subset of L2 (Σ).
The factorization method (FM) replaces the far-field operator F in the
far-field Equation (5.2) by the operator (F ∗ F)1∕4 [3, 12], where F ∗ is the adjoint
operator of F. Since the range ((F ∗ F)1∕4 ) of (F ∗ F)1∕4 coincides with the range
(Gs ) of Gs , we conclude that (F ∗ F)1∕4 g = Φ∞ (𝜃 s , rt ) has a solution if and only
if rt ∈ Σ. Despite considerable efforts in mathematical community, the FM
is still limited to a restricted class of scattering problems. In particular, the
method has not been established for the case of limited aperture data, that
is, limited angle of incidences or measurements, which is, however, common
in real-world applications, such as subsurface sensing [13]. However, when
applicable, the factorization method provides a mathematical justification for
using the regularized solution of an appropriate far-field equation to determine
the support of scatterer. Other versions of the factorization method are beyond
the scope of this chapter, and interested readers are referred to [10, 12].
5.5 Discussions
It is worth mentioning that although the LSM involves solving sets of linear
equations when calculating the indicator function, it does not assume any linearizing approximation to the original nonlinear scattering operation, such as
the Born approximation. The reason why the method is named linear sampling
can be found in [3].
The following three comments on the LSM are from the theoretical point
of view. First, the mathematical justification of LSM and FM requires that k 2
(the square of the wavenumber) is not an eigenvalue of the interior Dirichlet
Linear Sampling Method
problem when reconstructing PEC scatterers and k is not a transmission
eigen-value of an interior transmission problem when reconstructing dielectric scatterers [3]. Second, the LSM is not limited to far-field data and it can
be implemented for near-field data as well (see [14] and section 4.6 of [7]).
Third, the LSM is independent of the boundary conditions; that is, the material
properties of the scatterer. It works for both PEC and dielectric scatterers, and
even a mixture of them.
Some practical issues are worth discussing. For limited-aperture data,
numerical simulations have shown that the performances of LSM and FM
degrade with decreasing aperture angle [15, 16]. The criterion of choosing
the cutoff value C to determine the support of scatterers is heuristic, which
is even more difficult in presence of noise. Usually, the resolution of the
reconstruction image provided by the LSM is limited, especially for concave
scatterers. The sampling methods are usually highly sensitive to the level of
noise and their performances degrade fast when the noise becomes high.
Nevertheless, the information provided by LSM, such as the estimation of the
support of the scatterer, helps in improving the performance of other inversion
methods [14, 17].
The application of LSM to vectorial wave scattering problems is straightforward [13, 18]. At the same time, due to the fact that the lowest order multipole
in vectorial wave case is dipole, choosing an appropriate direction of the test
dipole significantly influences the performance of linear sampling [19].
There is an interpretation of the LSM from the angle of induced current [14],
which motivates the development of an improved sampling methods in [13].
Along the same line, LSM has been exploited to devise an effective approximation of the scattering phenomenon, which leads to a new quantitative inversion
method [20].
There are many other qualitative inversion algorithms, for example, the point
source method and the method of singular sources [21–23], the probe method
and the enclosure method [24–27]. A direct sampling method that does not
perform any matrix operations is proposed in [28], and a direct imaging algorithm based on a physical factorization is proposed in [29]. Other qualitative
methods can be found in the survey [30].
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Probl., 13 (6), 1477.
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120
Electromagnetic Inverse Scattering
3 Colton, D. and Kress, R. (2006) Using fundamental solutions in inverse scat-
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6
Reconstructing Dielectric Scatterers
This chapter is devoted to quantitative reconstruction algorithms that are
aimed at retrieving permittivities of dielectric scatterers from the knowledge
of measured scattering data. The scatterers are assumed to be nonmagnetic
and have dimensions comparable to or larger than the wavelength. This
inverse scattering problem is a typical parameter-identification problem,
where the coefficients in an integral or differential equation are to be estimated
from the measured data. In the framework of the exact scattering model,
where the multiple scattering effect exists, to invert the integral equation is
highly nonlinear so the nonlinearity becomes the most difficult issue, even
compared with instability. Usually, iterative approaches are used to solve
such a nonlinear problem, no matter whether one attempts to directly solve
the original nonlinear equation or to solve an optimization problem into
which the original problem is cast. When compared with qualitative methods,
quantitative methods are usually characterized by quite a heavy computational
cost, but have the advantage of providing the most complete information
on the inspected scatterers. Under some conditions, for example Born-type
approximations, quantitative methods can be implemented without iteration
and thus are able to provide reconstruction results in a fairly short time period.
The organization of the chapter is as follows. Section 6.1 introduces the
fundamentals of quantitative reconstruction algorithms. First, the properties
of uniqueness, stability, and nonlinearity are discussed. Then the formulation
of the forward method is presented, which is followed by casting the inverse
problem to an optimization problem. Section 6.2 briefly reviews noniterative
inversion methods, of which there are two types. In particular, one type is
the well-known Born-type inversion algorithm, which involves a linearization of the original problem. Section 6.3 reviews several iterative inversion
methods. In particular, for the inversion methods, their implementation
steps are presented and their properties and characteristics are commented
on. Section 6.4 details a particular iterative inversion method, that is, the
subspace-based optimization method (SOM), for which the implementation
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
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Electromagnetic Inverse Scattering
steps and numerical reconstruction results are presented. Section 6.5 discusses
several important issues in the theories and implementations of quantitative
reconstruction algorithms.
6.1 Introduction
6.1.1
Uniqueness, Stability, and Nonlinearity
The first question to ask about the inverse scattering problem involving
dielectric scatterers regards uniqueness. For scalar waves, that is, the acoustic
wave, the uniqueness of the inverse scattering problem has been obtained by
[1, 2]. Under certain conditions, the refractive index is uniquely determined
by a knowledge of the far field pattern in all directions for all incidence angles
at a fixed wavenumber. For vectorial waves, that is, electromagnetic waves, [3]
proved that under certain conditions, for a fixed wavenumber, all directions
of incidence, and all polarizations of the incident electric field, the knowledge
about the electric far field pattern for all angles uniquely determines the permittivity. If the magnetic permeability is also a spatially varying unknown, the
uniqueness theorem has been given in [4]. The proof of the uniqueness of the
inverse scattering problem involving dielectric scatterers is closely related to
and strongly motivated by the electric impedance tomography (EIT) problem,
also known as Calderón’s problem, which consists of determining the electric
conductivity of a medium by measuring voltage and current at the boundary of
the medium. Calderón’s pioneering work in 1980, reprinted in [5], motivated
many developments in inverse problems, in particular, the construction of
complex geometrical optics (CGO) solutions for partial differential equations
that are widely used to prove the uniqueness of several inverse problems [6].
It is important to recall what has been presented in Chapter 1: this book concentrates mainly on numerical methods that solve inverse scattering problems
that are known a priori to have unique solutions. Inverse scattering problems
that do not have unique solution are not considered in this book. In fact, the
conditions of nonuniqueness are rather stringent and thus in practice it is rare
to encounter such inverse scattering problems. For example, for anisotropic
scatterers, if the permittivity and permeability are allowed to be zero or infinite,
then it is possible to have infinitely many solutions to an inverse scattering problem. One of the applications of such nonuniqueness is invisibility and cloaking,
and the idea of designing such anisotropic scatterers is referred to as transformation optics [7, 8]. The same transformation results for electrostatics was
first proposed in [9, 10]. In addition, many inverse scattering methods cannot
work reliably when an isotropic scatterer does not scatter off a certain incident wave for certain wavenumbers. In the mathematical community, such a
wavenumber is referred to as the transmission eigenvalue of the interior transmission problem [11]. To be specific, this chapter considers dielectric scatterers
Reconstructing Dielectric Scatterers
that are nonmagnetic, that is, the relative permeability 𝜇r = 1, and have relative
permittivity with real and imaginary parts satisfying 1 ≤ (𝜖r ) < ∞ and (𝜖r ) ≥
0 and in addition all discrete numerical resonance frequencies are avoided [12].
Next, we turn to the question of stability. Inverse scattering problems involving dielectric scatterers cannot be stably solved. In fact, even if the amount of
data collected is sufficient to guarantee uniqueness, the unknowns usually do
not depend on the measured data in a stable way (mathematically, it is referred
to as continuous). An obvious question to ask is how large the error of the solution could be in the worst case if the error in the measured data is at most 𝜏.
For an ill-posed problem, the error in solution could be arbitrarily large, which
really means instability. In order to recover some kind of stability, we need to
restrict the space of admissible unknowns by assuming that they satisfy some
a priori conditions. With this a priori information, it is possible to prove that
the unknowns depend in a continuous way on the measured data. To determine
the modulus of this continuity is referred to as the stability estimate (section 2.2
of [13]). For an inverse scattering problem involving dielectric scatterers, it has
been proven that the stability is of the logarithmic type [14, 15]. Roughly speaking, if the error in the measured data is at most 𝜏, then the error of solution in
the worst case is on the order of | ln 𝜏|−s (where 0 < s < 1). By the L’Hôpital’s
rule, as 𝜏 approaches zero, we see that a small error in measured data leads to a
much larger error in the solution. In the mathematical community, an inverse
problem is said to be exponentially ill-posed, or severely ill-posed, if stability
estimate is of the logarithmic type.
It is important to remark that the inverse scattering problem is nonlinear, so
it is not appropriate to prove its ill-posedness using methods that are devised
for linear problems. For example, the total amount of available data, that is,
the product of the number of transmitters and receivers, is less than the number of unknowns or, along the same line, the number of essentially independent equations (i.e., the product of the number of transmitters and the degrees
of freedom of the scattering operator) is less than the degrees of freedom of
unknowns. It is easy to see from the following numerical example why the
aforementioned methods cannot be applied to nonlinear problems. We aim to
∑99
solve a single equation (1 − x1 )2 + i=1 (xi − xi+1 )2 = 0 for 100 real unknowns
xi , i = 1, 2, … , 100. Although the number of unknowns is much larger than the
number of equation, there is only one solution; xi = 1, i = 1, 2, … , 100.
We make two remarks on the ill-posedness of inverse scattering problems.
First, its nature of severe ill-posedness cannot be changed no matter what or
how many measurements we take. The proof of logarithmic-type stability estimate in fact deals with the ideal case of performing all possible measurements,
which in practice means that performing more measurements or different ones
does not solve the problem of ill-posedness. Secondly, it has been observed
numerically that the stability increases if one increases the frequency [16]. Several rigorous justifications of increasing the stability can be found in [17] and
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Electromagnetic Inverse Scattering
[18] and references therein. In other words, the static problem (EIT problem)
is most unstable and the optical scattering problem is relatively more stable.
In addition to instability, the second main difficulty of inverse scattering problems is the fact that inverse problems are nonlinear, even if the corresponding forward problems are linear ones. As presented in Chapter 1, an inverse
scattering problem deals with the relationship between a scatterer’s parameter
and scattered field, whereas the forward problem deals with the relationship
between the incident and scattered fields. The nonlinearity of inverse problem
is obvious by the fact that the scattered field isn’t doubled when the scatterer’s
permittivity is doubled. In addition, the nonlinearity is not a convex one. The
intrinsic nonlinearity of inverse scattering problem makes the development of
effective algorithms a difficult task because a solution procedure can be trapped
into false solutions that are different from the exact one.
6.1.2
Formulation of the Forward Problem
For convenience of presentation, we consider the two-dimensional TM case,
where scalar wave equations are involved. In a homogeneous medium background that has permittivity 𝜖0 and permeability 𝜇0 , nonmagnetic scatterers,
characterized by 𝜖r (r), are located in the domain of interest D ⊂ R2 and illuminated from different angles by time-harmonic electromagnetic waves. The
incident waves are expressed as Epi (r), p = 1, 2, … , Ni , r ∈ D. For each incidence, the scattered field is measured at a surface S that is usually in the far
zone. The forward problem consists of the following two equations. The first
one is the Lippmann–Schwinger equation,
Et (r) = Ei (r) + i𝜔𝜇0
∫D
g(r, r′ )[−i𝜔𝜖0 (𝜖r (r′ ) − 1)Et (r′ )]dr′
for r ∈ D,
(6.1)
which describes the wave-scatterer interaction, and the second one describes
the scattered field as a re-radiation of the induced contrast current,
Es (r) = i𝜔𝜇0
∫D
g(r, r′ )[−i𝜔𝜖0 (𝜖r (r′ ) − 1)Et (r′ )]dr′
for r ∈ S
(6.2)
The physical meaning of −i𝜔𝜖0 (𝜖r (r) − 1)Et (r) is an induced contrast current
density, but in practice it is more convenient to define a normalized contrast
current density as J(r) = (𝜖r (r) − 1)Et (r). Then, (6.1) and (6.2) can be rewritten
as
Et (r) = Ei (r) + k02
∫D
g(r, r′ )J(r′ )dr′
for r ∈ D,
(6.3)
and
Es (r) = k02
∫D
g(r, r′ )J(r′ )dr′
for r ∈ S
(6.4)
Reconstructing Dielectric Scatterers
√
respectively, where k0 = 𝜔 𝜇0 𝜖0 is the wavenumber of the homogeneous
medium background.
For convenience, we denote 𝜖r (r) − 1 as the contrast 𝜒(r) and introduce the
operators GS (⋅) and GD (⋅) as
{
for r ∈ S
GS (J),
2
′
′
′
k0 g(r, r )J(r )dr =
(6.5)
∫D
for r ∈ D
GD (J),
Then the governing equations can be written in two different kinds of compact
form. The first kind are referred to as field-type equations,
Et (r) = Ei (r) + GD (𝜒Et ) for r ∈ D,
(6.6)
Es (r) = GS (𝜒Et )
(6.7)
for r ∈ S
where electric field is involved in both equations, and the second type are
source-type equations,
J(r) = 𝜒(r)[Ei (r) + GD (J)] for r ∈ D,
(6.8)
s
(6.9)
E (r) = GS (J) for r ∈ S
where induced source is involved in both equations. For convenience, (6.6) and
(6.8) are referred to as the state equations, and (6.7) and (6.9) are referred to as
data equations.
6.1.3
Optimization Approach to the Inverse Problem
In an inverse problem, the scattered fields Eps (r), r ∈ S for all Ni incidences are
measured, and then the task is to reconstruct 𝜒(r) for r ∈ D. It is important
to clarify that while the notation D means the domain of scatterer in the forward problem, it means the domain of interest (DOI) (i.e., a region somewhere
in which the scatterer resides) in the inverse problem. This is because we do
not know a priori the boundary of scatterer in inverse problem. Nevertheless,
Equation (6.6)–(6.9) can be applied to both forward and inverse problems. The
reason is that for a point r ∈ D that is not inside the scatterer, its 𝜒(r) is equal
to zero.
In the presence of noise, which is unavoidable, the nonlinear equations
Eps = F(𝜖r , Epi )
(6.10)
do not have a solution of 𝜖r , where F denotes the operator of solving the forward
problem. Consequently, instead of solving (6.10), we seek the solution to the
optimization problem,
Min: f (𝜖r ) =
Ni
∑
p=1
||F(𝜖r , Epi ) − Eps ||2S
(6.11)
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Electromagnetic Inverse Scattering
where || ⋅ ||S denotes the Euclidian length of a function defined on S. In order
to obtain a stable solution, a regularization strategy has to be used so that the
objective function becomes, for example,
Min: f (𝜖r ) =
Ni
∑
||F(𝜖r , Epi ) − Eps ||2S + 𝛼R(𝜖r ),
(6.12)
p=1
where R(𝜖r ) is a nonnegative regularization functional. The regularizing term is
usually chosen on the basis of some a priori assumptions (or knowledge) of the
unknown function. The details of regularization can be found in Appendix A.
After discretization, the unknown 𝜖r appears in the diagonal of a matrix, and
the objective function involves the inverse of this matrix. Such an objective
function is obviously nonlinear and nonconvex, which is known to be difficult
to solve due to the presence of local minima. Roughly speaking, algorithms
proposed for solving nonlinear optimization problems can be classified into
deterministic algorithms and heuristic (or stochastic) ones. Among popular
optimization algorithms, steepest descent, conjugate gradient, Gauss–Newton,
quasi-Newton methods, Levenberg–Marquardt, and so on, belong to deterministic algorithms [19–23], whereas simulated annealing, genetic algorithms,
differential evolution, particle swarm optimization, and so on, belong to heuristic or stochastic algorithms [24–31]. A large number of optimization algorithms
are not capable of making a distinction between locally optimal solutions and
globally optimal solutions. In practice, no optimization algorithm is guaranteed
to reach a global minimum for the discussed nonlinear and nonconvex optimization problem. To evaluate the performance of an optimization algorithm,
it is more meaningful to consider this question: out of one hundred simulations
starting from different initial guesses, how many times will the optimization
algorithm reach a solution that is in the neighborhood of the global minimum?
It is important to determine the computational complexity of solving (6.12).
Let the domain of interest be discretized into M pixels, and each pixel is so
small so that 𝜖r therein is a constant. If the forward problem is evaluated by an
iterative solver in each iteration step of an optimization algorithm, the overall
computational complexity is
Nopt Ninc Nfor M log2 M,
(6.13)
where Ninc is the number of incidences, Nfor is the number of iterations for
solving the forward problem, and Nopt denotes the number of iterations during
the optimization procedure. M log2 M is the major computational cost of the
forward problem solver; that is, the matrix-vector multiplication by the fast
Fourier transform (FFT). To improve the overall computational complexity as
formulated in (6.13), three factors should be taken into account. The first factor
Nfor M log2 M is determined by the state-of-the-art in computational electromagnetics. The second factor Ninc can be as small as the degrees of freedom of
Reconstructing Dielectric Scatterers
the radiation operator GS [32–34]. While there is barely room to further reduce
the first two factors, the third factor Nopt becomes the bottleneck of inverse
scattering problems. Two phenomena are often observed due to nonlinearity
of the objective function; that is, either a large Nopt is needed to reach a global
minimum or optimization algorithms pre-converge to a local minimum. Thus,
the key to solve inverse scattering problems is to reduce the number of iterations Nopt needed for the optimization algorithm to reach a global minimum.
One way of reducing Nopt is to try different optimization algorithms, such as the
aforementioned deterministic algorithms and heuristic (or stochastic) ones.
No matter which optimization algorithm is chosen, the difficulty of the nonlinear problem does not change since the objective function remains the same. As
an analogy, when climbing a hill, changing climbing equipment might increase
the speed of climbing, but it does not change the steepness of the hill. Thus,
in the author’s opinion, a more effective approach to reduce Nopt is to rewrite
the objective function such that it depends in a much less nonlinear way on
unknowns. A well rewritten objective function increases the chance of all optimization algorithms to find a solution that is close to the exact solution. For this
reason, it is not our primary concern to discuss the advantages and limitations
of individual optimization algorithms. Instead, the chapter concerns mainly on
the modelling of the inversion; that is, the way of rewriting and approximating
the original objective function. The theories and implementation details of
deterministic and heuristic (or stochastic) optimization algorithms can be
found in [19–21, 24–30]. This book mainly applies the conjugate gradient (CG)
optimization algorithm and its variants. Appendix C briefly introduces the
implementation steps of the conjugate gradient optimization algorithm.
6.2 Noniterative Inversion Methods
To solve an inverse scattering problem based on an exact scattering model, one
needs to solve it iteratively by optimization methods. Under some conditions,
inverse scattering problems can be solved without iteration. Such noniterative
inversion algorithms can be either linear or nonlinear in terms of the dependence of the scattered field on unknowns. In the first type, based on certain
approximations in scattering models (e.g., those based on Born and Rytov type
approximations), the original nonlinear problem becomes a linear one [35]. If
we know a priori that the inspected objects fulfill the conditions that make the
approximate models valid, then linear inversion methods are able to provide
reconstruction results in a fairly short time period. For example, in order to
apply the Born approximation, one needs to know that the object is a weak
scatterer and its size is not much larger than the wavelength. In the second
type, although measured scattered field eventually depends nonlinearly on
unknowns, the inverse problem decomposes into several linear equations
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Electromagnetic Inverse Scattering
and each linear equation can be solved without iteration. The extended Born
approximation method and back-propagation method are examples of this
type. Despite their limited applicability, noniterative inversion methods are
of significant interest because of their computational efficiency. In addition,
reconstruction results provided by noniterative inversion methods can often
be used as initial guesses for nonlinear inversion methods.
Since there is an abundance of literature on noniterative inversion methods,
this section briefly introduces some of those algorithms.
6.2.1
Born Approximation Inversion Method
When scatterers are weak ones, that is, when the permittivity of scatterers
differs only slightly from that of the background medium, the inverse scattering problem can be solved by the first-order Born approximation or simply
referred to as the Born approximation (BA) method. Since the scattering field
is very weak compared with the incident field, it is plausible to assume that
one will obtain a good approximation if the total field Et (r) for r ∈ D, that is,
inside the scatterer, is replaced by the incident field Ei (r). Consequently, (6.7)
is approximated by
Es (r) ≈ GS (𝜒Ei )
for r ∈ S
(6.14)
To recover 𝜒 is an ill-posed linear problem. Since in practice the measured
Es is definitely noise contaminated, a direct inverse of the linear operator
yields meaningless results. In order to balance the accuracy in data fitting
and the stability of solution, regularization has to be used. For example, when
Tikhonov regularization is used, the original linear equation is cast into an
optimization problem
Min: f (𝜒) =
Ni
∑
||GS (𝜒Epi ) − Eps ||2S + 𝛼||𝜒||2D ,
(6.15)
p=1
The minimization problem has an analytical solution, which can be found
in Appendix B. If the incident wave is a plane wave and the scattered field
is measured in the far zone, then BA reconstruction can be interpreted as a
low pass filter of contrast in the reciprocal space, which will be discussed in
detail in Chapter 10. In addition to weak scattering, the validity of BA requires
that the size of scatterer cannot be much larger than the wavelength [36, 37].
We mention in passing that the BA-based inversion method for near-field
microscopy is presented in [38].
6.2.2
Rytov Approximation Inversion Method
For weak scatterers, another linearized version of inverse scattering is based on
the Rytov approximation (RA). The Rytov approximation is closely related to
the Born approximation, but the linearized inverse scattering formula based
Reconstructing Dielectric Scatterers
on the Rytov approximation employs the complex-valued phase of the total
field rather than the scattered field as the measured field quantity. The Rytov
procedure expresses the total field Et (r) in the form
Et (r) = e𝜓(r) ,
(6.16)
where 𝜓(r) is generally complex-valued. In fact, −i𝜓(r) can be considered as
the phase of the total field. Then 𝜓(r) satisfies a nonlinear partial differential
equation, referred to as the Riccati equation. The Rytov expansion expands 𝜓(r)
in a perturbation series. The first-order Rytov approximation reads
Et (r) ≈ E1t(R) (r) = Ei (r)e𝜓1 (r) ,
(6.17)
where the first-order phase 𝜓1 (r) relates to the first-order Born approximation
of the scattered field E1s(B) (r) by
𝜓1 (r) = [Ei (r)]−1 E1s(B) (r).
(6.18)
In inverse scattering problems, the RA treats 𝜓1 (r) = ln[Et (r)∕Ei (r)] as measured data, which is different from the case of the BA in which the scattered
field Es (r) is treated as measured data. Since E1s(B) (r) linearly depends on the
contrast, (6.18) shows that 𝜓1 (r) consequently depends linearly on the contrast
[36, 37].
If the incident wave is a plane wave and the scattered field is measured
in the far zone, inverse scattering is regarded as one of the most frequently
encountered diffraction tomography (DT) problems. The theory of diffraction
tomography resembles the classical theory of computed tomography (CT)
that is modelled by Radon transform and reduces to the latter theory in limits
where the wavelength approaches zero (section 8.7 of [39]). In solving DT
problems, the RA is also able to reconstruct contrast by performing an inverse
Fourier transform. This widely used reconstruction procedure can be done in
quasi real-time.
Rytov and Born approximations are only applicable to weak scatterers, but
they have different domains of validity. The Rytov approximation is generally
more accurate than the Born approximation if scatterer is large compared with
wavelength [36, 37].
6.2.3
Extended Born Approximation Inversion Method
To increase the range of validity of BA, the extended Born approximation
(EBA) method is proposed in [40]. To derive the EBA, we first rewrite the state
equation (6.1) as
Et (r) =Ei (r) + k02
+ k02
∫D
∫D
g(r, r′ )𝜒(r′ )Et (r)dr′
g(r, r′ )𝜒(r′ )[Et (r′ ) − Et (r)]dr′
(6.19)
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Electromagnetic Inverse Scattering
Based on the fact that the Green’s function g(r, r′ ) is singular when r approaches
r′ , the EBA method drops off the second integral in (6.19) via the argument that
its integrand is much smaller than the counterpart in the first integral when r
approaches r′ . Consequently, the resultant equation yields a localized electric
field; that is, an electric field in one position is independent of electric fields at
other positions. Indeed, the electric field in the EBA is formulated as
Et (r) = M−1 (r)Ei (r)
(6.20)
M(r) = 1 − k02
g(r, r′ )𝜒(r′ )dr′
(6.21)
From (6.7) and the definition of GD , the scattered field is then
{
}
Es = GS 𝜒[1 − GD (𝜒)]−1 Ei
(6.22)
where
∫D
which is a nonlinear equation of 𝜒. To solve the inverse problem, a twostep noniterative procedure has been proposed in [41]. The nonlinear
equation (6.22) is rewritten as the following two linear equations via the
introduction of an intermediate parameter 𝑤(r) = 𝜒(r)M−1 (r),
Es = GS (𝑤Ei )
(6.23)
𝑤[1 − GD (𝜒)] = 𝜒
(6.24)
and
Equation (6.23) is applicable to all Ni incidences, and the resultant equation
system is still linear for the unknown 𝑤. This linear problem is ill-conditioned,
so a regularization method should be used. Once 𝑤 is obtained, (6.24) is a
well-posed linear equation for the unknown 𝜒.
Inspired by the EBA, the quasi-analytical (QA) approximation [42] and diagonal tensor approximation (DTA) [43] models are proposed that have wider
ranges of validity than the BA. In these approximations, the total electric field
at a position is assumed to be proportional to the incident field at the same
position, similar to (6.20) in the EBA case, but the ratio of them depends on
the incident wave, which is different from the field-independent ratio M−1 (r)
in the EBA. For inverse problems under these two approximations, the original
nonlinear equation can also be decomposed into two linear equations by introducing an intermediate parameter. After some modifications, we can apply the
EBA inversion procedure outlined in the previous paragraph to QA and DTA
inversions as well. However, since the ratio of total field to incident field at a
position depends on incidences in QA and DTA, the noniterative approach by
solving two linear equations in sequence is seldom explored by the developers
of these two approximation models.
The EBA can also be interpreted as approximating the Et (r′ ) in the integral
of (6.1) by the first order term of a Taylor expansion about the field point r.
Reconstructing Dielectric Scatterers
We mention in passing inversion methods that are based on a higher-order
extended Born approximation will be intrinsically nonlinear.
6.2.4
Back-Propagation Scheme
The noniterative inversion algorithm that is based on back-propagation (BP)
consists of three steps [44]. The first step is to determine induced current by
a BP scheme, where induced current is assumed to be proportional to the
BP field.
J = 𝛾GS† (Es )
(6.25)
where GS† denotes the adjoint of operator GS and maps scattered field measured
in the domain S to the induced
current in the domain of interest D. In discrete
H
s
† s
form, GS (E ) is written as GS ⋅ E , where the superscript H denotes the Hermitian operator. The complex parameter 𝛾 is chosen to minimize the cost function
defined as the quadratic error in the scattered field,
F(𝛾) = ||Es − GS (𝛾GS† (Es ))||2S .
(6.26)
The minimum of F(𝛾) requires the derivative with respect to 𝛾 to be zero, which
leads to an analytical solution of 𝛾,
𝛾=
⟨Es , GS (GS† (Es ))⟩S
||GS (GS† (Es ))||S2
,
(6.27)
where ⟨a, b⟩S denotes the inner project of function a and b in the domain S, and
∗
T
its discrete form reads a ⋅ b , where the superscripts T and * denote the transpose and complex conjugate, respectively. Equation (6.25) shows that induced
current J is obtained once 𝛾 is determined.
The second step calculates the total field in the domain D,
Et = Ei + GD (J).
(6.28)
The first and the second steps are applied to each of Ni incidences. The third
step obtains the contrast 𝜒(r) by taking all incident waves into account. For the
pth incidence, the definition of 𝜒(r) requires
Jp (r) = 𝜒(r)Ept (r).
(6.29)
The enforcement of (6.29) to all incidents leads to a least-squares problem and
the solution of 𝜒(r) can be obtained analytically,
∑Ni
t
∗
p=1 Jp (r) ⋅ [Ep (r)]
𝜒(r) =
(6.30)
∑Ni
t
2
p=1 |Ep (r)|
If the scatterer is lossless, then the real part of the right-hand side of (6.30) is
chosen.
133
Electromagnetic Inverse Scattering
This BP scheme works for arbitrary incident fields and for both near and far
field measurements, which is different from the filtered back-propagation algorithm of DT [45].
The noniterative inversion algorithms discussed in Sections 6.2.1–6.2.4
have made some approximations or assumptions and thus have somewhat
limited ranges of validation. Here, we do not intend to discuss the ranges of
their validity, nor to compare their inversion results. Instead, these noniterative inversion algorithms might be useful in providing initial guesses for
iterative inversion algorithms that will be discussed in Section 6.3 where no
approximation is made in the modeling of ISP.
6.2.5
Numerical Examples
This section provides some numerical examples to illustrate the performance
of the noniterative inversion algorithms discussed in Sections 6.2.1–6.2.4. The
purposes of these numerical examples are threefold. The first is to provide a
visualization of the reconstruction results obtained by the four noniterative
inversion algorithms. The second is to demonstrate the effect of the Tikhonov
regularization parameter on the solution of inverse problems, the theoretical
description of which is given in Appendix A. The third is to demonstrate
that the four noniterative inversion algorithms fail to reconstruct scatterers
that are not weak-scattering.
We choose the scatterer to be the “Austria” profile [117]. As shown in Fig. 6.1,
it consists of two discs and one ring. The discs of radius 0.2 m are centered at
(0.3, 0.6) m and (−0.3, 0.6) m. The ring has an exterior radius 0.6 m and an
inner radius 0.3 m, and is centered at (0, −0.2) m. The background is air and
the contrast between the scatterers and the background is 1 (i.e., 𝜖r = 2). The
domain of interest (DOI) is a 2 × 2 m2 square centered at the origin. Sixteen line
1
2
1.8
0.5
1.6
y(m)
134
0
1.4
–0.5
–1
–1
1.2
–0.5
0
x(m)
0.5
1
1
Figure 6.1 Inverse scattering
experiment of the Austria profile:
exact profile. The shaded bar
shows the value of relative
permittivity.
Reconstructing Dielectric Scatterers
sources and 32 line receivers are evenly placed on circles with radius 6 m and
3 m, respectively, centered at the origin. The operating frequency is 400 MHz,
corresponding to the wavelength 𝜆 = 0.75 m in the background medium of air.
The scattering data are generated by the full-wave solver in the forward process and are recorded in the format of the multistatic response (MSR) matrix
K whose size is Ns × Ni . Then additive white Gaussian noise 𝜅 is added to the
MSR matrix, the resultant noisy matrix K + 𝜅 is treated as the measured MSR
matrix and is used to reconstruct scatterers. The noise level is quantified as
||𝜅 ||F
× 100%, where || ⋅ ||F denotes the Frobenius norm of a matrix, defined as
||K ||F
the square root of the sum of the absolute squares of all its elements. In solving
the inverse problem, the DOI is discretized into 64 × 64 grid meshes.
In order to study a weak-scattering scatterer, we reduce the the contrast
between the scatterers and the background to be 0.1 (i.e., 𝜖r = 1.1). When 5%
additive white Gaussian noise is present, the reconstruction results by the
four noniterative inversion algorithms are shown in Fig. 6.2, where subfigures
(a)–(d) correspond to the BP method, BA, EBA, and RA methods, respectively.
The results show that the BA, EBA, and RA methods yield comparable results
(for this scatterer) that are close the to exact profile. In comparison, the reconstruction result obtained by the BP method exhibits far lower permittivity
than the exact one.
Next, we instigate the effect of the Tikhonov regularization parameter 𝛼
appearing in (6.15) on the solutions of the BA-based inversion method. The
singular values of the discretized version of the GS Epi operator appearing
in (6.15) is plotted in Fig. 6.3. The condition number, which is defined as
the ratio of the largest to smallest singular values, is larger than 100 in this
problem, indicating that the linear equation (6.14) is ill conditioned. We use
the Tikhonov regularization method (6.15) to obtain a stable and reasonable
approximate solution.
As discussed in Appendix A, the role of the regularization parameter 𝛼 is to
keep a balance between the accuracy of data-fitting and the stability of the solution. As 𝛼 tends to zero, the data-fitting reaches better accuracy, but the solution
becomes more unstable. When 𝛼 becomes large, the effect is just the opposite.
In presence of 5% white Gaussian noise, the reconstruction results under different values of 𝛼 (108 , 109 , 1010 , 1012 , 1014 , and 1018 ) are shown in Fig. 6.4.
From the visual point of view, 𝛼 = 1012 performs the best among the six results;
that is, reaching a better balance between the accuracy of data-fitting and the
stability of the solution compared with other values of 𝛼.
We define the relative error Re as follows to quantify the reconstruction
result,
Re =
||𝜖 − 𝜖 R ||
||𝜖||
(6.31)
135
Electromagnetic Inverse Scattering
1
1
1.045
1.04
0.5
1.1
0.5
1.025
–0.5
1.02
–1
–0.5
0
x(m)
(a)
1
0.5
–1
1.08
1.06
0
1.04
1.02
–0.5
1
–0.5
0
x(m)
0.5
0
x(m)
(b)
0.5
1
1.1
0.5
1.05
0
1
–0.5
0.98
–1
–1
–0.5
1
1.12
1.1
1
–1
1
0.5
1.05
0
–0.5
1.015
–1
y(m)
y(m)
1.03
0
y(m)
y(m)
1.035
–1
–1
1
–0.5
(c)
0
x(m)
0.5
1
(d)
Figure 6.2 Application of noniterative inversion algorithms to the reconstruction of a
weak-scattering “Austria” profile with 𝜖r = 1.1. (a) BP method; (b) BA method; (c) EBA
method; and (d) RA method.
6.5
6
5.5
log10(σj)
136
5
4.5
4
3.5
0
5
10
15
20
25
30
35
Singular value number, j
Figure 6.3 The distribution of singular values of the operator GS Epi , where the base 10
logarithm of the singular values is plotted.
0.95
Reconstructing Dielectric Scatterers
1.2
0
1
–0.5
–1
–1
–0.5
0.8
–0.5
1
0.5
0
x(m)
(a)
–1
–1
1
0.5
0
1.04
–0.5
0
0.5
x(m)
(d)
1
0.5
0
x(m)
(b)
1
1
1.005
0
1.004
–0.5
–1
–1
–0.5
1.002
1
0.5
0
x(m)
(c)
1
1.0000007
1.0000006
1.0000005
0
1
0
0.5
x(m)
(e)
1
0.5
1.003 –0.5
–0.5
1.05
0
1.02 –0.5
–1
–1
1.1
0.5
1
1.006
0.5
y(m)
1.06
–0.5
–0.5
1
1.08
–1
–1
0
1.25
1.2
1.15
1.1
1.05
1
0.95
0.9
y(m)
0.5
1.4
y(m)
y(m)
0.5
y(m)
1
1.6
y(m)
1
–1
–1
1.0000004
1.0000003
–0.5
0
0.5
x(m)
(f)
1
1.0000002
Figure 6.4 BA reconstruction results as a function of regularization parameter 𝛼: (a) 108 ;
(b) 109 ; (c) 1010 ; (d) 1012 ; (e) 1014 ; and (f ) 1018 .
0.16
0.14
0.12
Re
0.1
0.08
0.06
0.04
0.02
0
5
10
15
log 10α
20
25
Figure 6.5 The relative error Re of the BA reconstruction as a function of the regularization
parameter 𝛼.
where 𝜖 R is the reconstructed permittivity. The relative errors Re for the reconstruction results displayed in Fig. 6.4 are shown in Fig. 6.5, where we see that
Re reaches its minimum somewhere between 𝛼 = 1010 and 1012 . In practice,
any value between 1010 and 1012 is a good candidate for 𝛼 and there is no
137
Electromagnetic Inverse Scattering
need to seek the “best” value. As discussed in Appendix A, finding such an
interval is often done by trial and error. Other methods used to determine the
optimal value of the regularization parameter, such as the discrepancy principle, the generalized cross validation, and the L-curve method, can be found in
Appendix A and references therein.
Finally, we test the performance of the four noniterative inversion algorithms
in reconstructing scatterers that are not weak-scattering. The original “Austria” profile, with the contrast at 1 (i.e., 𝜖r = 2), is considered. The reconstruction results are shown in Fig. 6.6, where none of the four algorithms is able
to satisfactorily reconstruct the scatterer. Due to the limitation of the scope of
the applicability of noniterative inversion algorithms, it is mandatory to apply
full-wave iterative inversion methods to solve such a problem.
1
1
2
1.2
1
0
0.8
–0.5
0.6
–1
–1
0.5
y(m)
y(m)
0.5
–0.5
0
x(m)
(a)
0.5
–1
–1
0.5
4
0.5
0
2
–0.5
0
0
x(m)
(c)
0.5
1
y(m)
1
–0.5
0.5
0
–0.5
1
–2
1
0
6
–1
–1
1.5
–0.5
0.4
1
y(m)
138
–0.5
0
x(m)
(b)
0.5
1
–1
5
4
3
2
0
1
0
–0.5
–1
–2
–1
–1
–0.5
0
x(m)
(d)
0.5
1
Figure 6.6 Application of noniterative inversion algorithms to the reconstruction of
“Austria” profile with 𝜖r = 2. (a) BP method; (b) BA method; (c) EBA method; and (d) RA
method. All methods fail to reconstruct a scatterer that is not weak-scattering.
Reconstructing Dielectric Scatterers
6.3 Full-Wave Iterative Inversion Methods
This section is devoted to full-wave iterative inversion methods that are based
on “exact” models, which are theoretically valid for any scatterers, even those
with high contrast with respect to the background and large in comparison with
wavelength.
6.3.1
Distorted Born Iterative Method
The distorted Born iterative method (DBIM) iteratively implements the
distorted-wave Born approximation (DWBA). The DBIM starts with the
field-type integral equations (6.6) and (6.7). The scattered field is a nonlinear
function of contrast,
Es = GS 𝜒(I − GD 𝜒)−1 Ei
(6.32)
We first introduce the distorted-wave Born approximation. Consider the
contrast 𝜒(r) to be a small perturbation 𝛿𝜒(r) with respect to background
contrast 𝜒0 (r); that is, 𝜒(r) = 𝜒0 (r) + 𝛿𝜒(r). The inhomogeneous Green’s
function g 𝜒0 (r, r′ ) for the background medium 𝜒0 (r) is
g 𝜒0 (r, r′ ) = (I − GD 𝜒0 )−1 g(r, r′ )
(6.33)
The physical meaning is that the total field g 𝜒0 (r, r′ ) consists of the original field
g(r, r′ ) in absence of 𝜒0 (r) and the scattered field GD (𝜒0 g 𝜒0 ) due to the presence
of 𝜒0 (r). For convenience, we introduce the notation
𝜒
GS 0 (J) = k02
∫D
g 𝜒0 (r, r′ )J(r′ )dr′
for r ∈ S
(6.34)
The scattered field Es (r) due to the presence of 𝜒(r) consists of two parts, that
is, the scattered field E0s (r) due to the background scatterer 𝜒0 (r), and the perturbation field due to the presence of 𝛿𝜒(r) embedded in the inhomogeneous
background medium 𝜒0 (r),
𝜒
Es = E0s + GS 0 𝛿𝜒Et
(6.35)
The data equation (6.35) is exact, but it is a nonlinear integral equation for 𝜒
since the total field Et depends on 𝜒 as well. The DWBA replaces the Et with
E0t , which is the total field in the inhomogeneous background medium 𝜒0 (r),
𝜒
Es ≈ E0s + GS 0 𝛿𝜒E0t ,
(6.36)
which introduces an error in the order of (𝛿𝜒)2 , which vanishes as 𝛿𝜒 goes to
zero [46].
139
140
Electromagnetic Inverse Scattering
Next, we introduce the DBIM to solve inverse scattering problems. The Es on
the left-hand side of (6.35) is the measured data. The DBIM was proposed by
Chew and Wang in [47] and is implemented as follows:
Step 1: Choose initial guess 𝜒0 , which is usually chosen as zero or the results
obtained by noniterative inversion algorithms. Initialize the iteration
step n = 0.
𝜒
Step 2: Calculate GS n for the background medium contrast 𝜒n at the nth iteration. For each incident wave Epi , p = 1, 2, ..., Ni , solve the forward probt
in the domain of interest (DOI) and
lem to determine the total field Ep,n
s
the scattered field Ep,n at receivers. Solve for 𝛿𝜒 that simultaneously
satisfies
𝜒
s
t
Eps = Ep,n
+ GS n 𝛿𝜒Ep,n
,
(6.37)
for all incidences. This is a linear equation for 𝛿𝜒. The regularization method should be used in solving this linear problem due to
ill-posedness. Update the solution 𝜒n+1 = 𝜒n + 𝛿𝜒.
Step 3: If termination condition is satisfied, stop the iteration. Otherwise,
update n = n + 1 and go to Step 2.
𝜒
Note that g 𝜒n (r, r′ ) that appears in GS n in Step 2 is obtained by (6.33). In
numerical simulations, it is more convenient to use reciprocity theorem
g 𝜒n (r, r′ ) = g 𝜒n (r′ , r) since the latter can be efficiently obtained by solving a forward problem using the conjugate-gradient fast Fourier transform (CG-FFT)
algorithm, where a point source located at the position of receiver r illuminates
the scatterer with contrast 𝜒n [48].
It has been proven that the DBIM method is equivalent to the Newton–
Kantorovich (NK) method [49]. The NK method is similar to the DBIM except
that it requires that 𝛿𝜒 simultaneously satisfies
s
t
Eps = Ep,n
+ GS (I − 𝜒n GD )−1 𝛿𝜒Ep,n
,
(6.38)
𝜒
GS n J
for all incidences. In fact, it can be easily seen that the formula
= GS J +
𝜒
GS n 𝜒n GD J holds for an arbitrary source J, which means that the radiated field
in presence of scatterer 𝜒n consists of the direct radiating field in absence of
the scatterer and the perturbation field due to the presence of the scatterer.
𝜒
Thus, we have GS (I − 𝜒n GD )−1 = GS n , which consequently means that (6.37)
and (6.38) are same equations. That is, it proves the equivalence of the DBIM
method and the NK method.
It is worth discussing the equivalence of the DBIM method and the NK
method from a different perspective. In 1948, L. V. Kantorovich extended the
Newton method for solving nonlinear equations to functional spaces [50].
Consider the nonlinear functional
F(𝜒) = Es − GS 𝜒(I − GD 𝜒)−1 Ei
(6.39)
Reconstructing Dielectric Scatterers
which is the residual version of (6.32). The Kantorovich method solves the
nonlinear equation F(𝜒) = 0 by iterative methods, 𝜒n+1 = 𝜒n + 𝛿𝜒, where 𝛿𝜒
satisfies F ′ (𝜒n )𝛿𝜒 = −F(𝜒n ) and F ′ (⋅) is the Fréchet derivative, which is the
generalization of the concept of gradient to the case of infinite dimensional
spaces [51]. Equation (6.38) shows that the NK method explicitly derives the
the Fréchet derivative, the details of which can be found in [49], whereas
(6.37) shows that the DBIM implicity obtains the Fréchet derivative using the
physical concept of an inhomogeneous Green’s function, that is, the perturbation of the scattered field due to a small perturbation 𝛿𝜒 of the contrast 𝜒n
t
is just the radiation field by the contrast source 𝛿𝜒Ep,n
in the inhomogeneous
background medium 𝜒n .
The NK method (equivalently the DBIM method) solves the inverse scattering problem by finding the roots of the nonlinear functional. A closely related
approach is to cast the root-finding problem to an optimization problem,
Min: f (𝜒) =
Ni
∑
||Eps − GS 𝜒(I − GD 𝜒)−1 Epi ||2S ,
(6.40)
p=1
which is a nonlinear least-squares problem. To numerically solve (6.40) (or
F(𝜒) = 0 for (6.39)), we have to discretize it so that the dimension of unknowns
is finite. Consequently, since the NK method deals with functional spaces that
are continuous, from here onward we use the term “Newton’s method for solving equations” instead of the NK method. A necessary condition for minimization is Fermat’s condition,
∇f (𝜒) = 0,
(6.41)
that is, we should find the roots of the nonlinear equation ∇f (𝜒). Using Newton’s method for solving equations, we obtain the iteration scheme,
𝜒n+1 = 𝜒n − [H(f (𝜒n ))]−1 ∇f (𝜒n ),
(6.42)
where H(⋅) is the Hessian matrix operator that is a square matrix of
second-order partial derivatives of a function. The updated scheme (6.42)
is referred to as “Newton’s method in optimization.” Since it is computationally costly for calculating the Hessian matrix (or its inverse directly),
some modified versions of Newton’s method are often used in practice, such
as the Gauss–Newton algorithm, quasi-Newton methods, and Levenberg–
Marquardt algorithm.
This section deals with solving F(𝜒) = 0 for (6.39) (or equivalently (6.40))
by using the DBIM (equivalently Newton’s method and its variants). Although
these methods differ in updating schemes, they share the same property in that
they do not change the nonlinearity relationship of the original problem. For
example, for (6.40), the dependance of the objective function on the unknown 𝜒
is regardless of the optimization algorithm that one adopts, no matter whether
141
142
Electromagnetic Inverse Scattering
it is the Newton’s method, Gauss–Newton algorithm, quasi-Newton methods,
or Levenberg–Marquardt. Since the contrast 𝜒 appears in the diagonal of an
inversion matrix, (6.40) is not a polynomial function of 𝜒. Though there are
barely any theories on such a nonlinear dependance, substantial numerical simulations have shown that there are many local minima in (6.40) and it is quite
challenging to obtain the global minimum. The next two subsections will introduce some algorithms that have changed the nonlinear dependance by rewriting the objective function in different but equivalent forms.
6.3.2
Contrast Source Inversion Method
The contrast source inversion (CSI) method deals with the source-type integral
equations,
J(r) = 𝜒(r)[Ei (r) + GD (J)] for r ∈ D,
(6.43)
s
(6.44)
E (r) = GS (J) for r ∈ S
and treats the contrast source J as an independent parameter instead of eliminating it by solving it from (6.43). The CSI method recasts the inverse problem
as a minimization of an objective function that is a linear combination of normalized mismatches in the data equation and the state equation,
∑Ni
2
s
p=1 ||Ep − GS (Jp )||S
Min: f (J1 , J2 , … , JNi , 𝜒) =
∑Ni
s 2
p=1 ||Ep ||S
∑Ni
2
i
p=1 ||𝜒Ep + 𝜒GD (Jp ) − Jp ||D
+
.
(6.45)
∑Ni
i 2
||𝜒E
||
p
p=1
D
The first term measures the mismatch in the data equation and the second term
measures the mismatch in the state equation. This is a quadratic functional in J,
but nonlinear in 𝜒. The objective function does not involve solving the forward
problem. The CSI method proposes an iterative minimization scheme using an
alternating method that first updates Jp and then updates 𝜒. In [52], the CSI
method is implemented as follows:
Step 1: Calculate GS and GD .
Step 2: Initial iteration step, n = 0: Choose initial guesses for J0 and 𝜒0 , for
example, using the back propagation inversion method; Initialize the
search directions, 𝑣Jp,0 = 0 for the contrast current and 𝑣𝜒0 = 0 for the
contrast.
Step 3: n = n + 1.
Step 3.1: Update Jp,n : Calculate the gradient (Fréchet derivative)
J
gp,n
= ∇Jp f evaluated at Jp,n−1 and 𝜒n−1 ; Determine the
Polak–Ribière conjugate gradient search directions
Reconstructing Dielectric Scatterers
J
J
J
Re[⟨gp,n
, gp,n
−gp,n−1
⟩D ] J
J
𝑣p,n−1 . Define the scalar dp,n
J
||gp,n−1 ||2D
J
by Jp,n = Jp,n−1 + dp,n
𝑣Jp,n . The objective function becomes
J
J
quadratic in terms of parameter dp,n
, and dp,n
can be easily
J
𝑣Jp,n = gp,n
+
obtained as done in [52, 53]. (In comparison with the stanJ
𝑣Jp,n should
dard CG method presented in Appendix C, dp,n
J
)(−𝑣Jp,n ))
be understood as (−dp,n
Step 3.2: Update 𝜒n : Calculate the gradient (Fréchet derivative)
gn𝜒 = ∇𝜒 f evaluated at Jp,n and 𝜒n−1 ; Determine the
Polak–Ribière 𝜒 conjugate
gradient search directions
𝜒
Re[⟨gn , gn𝜒 −gn−1
⟩D ] 𝜒
𝑣
.
Define the scalar dn𝜒 by
𝑣𝜒n = gn𝜒 +
n−1
||g 𝜒 ||2
n−1 D
𝜒n = 𝜒n−1 + dn𝜒 𝑣𝜒n . The value of dn𝜒 can be explicitly obtained
by minimizing the objective function, but the result is rather
tedious and its details can be found in [52].
Step 4: If a pre-determined termination condition (e.g., no obvious change in
objective function or reaching a maximum iteration count) is satisfied,
stop iteration. Otherwise, go to Step 3.
The CSI method is proposed on the base of both the modified gradient
method [54] and the source-type integral equation (STIE) method [55]. The
modified gradient method is based on the field-type equations and treats both
the contrast (𝜒) and the electric field (Ept ) as independent unknowns. The
objective function also contains two normalized mismatches, one for the data
equation and the other for the state equation. The two types of unknowns
are simultaneously updated in each iteration. The modified gradient method
is one of the first few inversion models for solving the ISP that do not solve
the corresponding forward problem. One of the advantages of the modified
gradient method is that the objective function is a quadratic in both Ept and 𝜒
individually, and consequently it is a quartic function of unknowns Ept and 𝜒
simultaneously. This is because the normalization factors do not involve any
unknowns, which is significantly different from the objective function (6.45)
of the CSI. To the best knowledge of the author, the modified gradient method
is the first inversion model for solving the ISP in which the objective function
depends on unknowns in a quartic manner. The STIE method proposed
in [55] is closer to the CSI method since it is also based on the source-type
integral equation and consequently it treats the contrast and the contrast
current as independent unknowns. A key difference is that the STIE method
decomposes the contrast source to radiating and nonradiating sources. While
the radiating source can be easily obtained, the unknown nonradiating source
is expanded with respect to some bases. Two independent objective functions
are considered, one for the mismatch in the state equation and the other for the
mismatch in the constitutive relationship J = 𝜒Et . The two types of unknown,
that is, the contrast and the expansion-coefficient of nonradiating source, are
143
144
Electromagnetic Inverse Scattering
updated alternatively by minimizing the two objective functions. One of the
advantages of the STIE method is that the analytical expression of both types
of unknown can be easily obtained by solving linear equations in the sense of
least squares during each iteration.
The first version of the CSI method was proposed in [53] in 1997 and then
the modified version was developed in [52] in 1999, the latter being more popular nowadays. One of the main improvements is the method of updating 𝜒n
in Step 3.2. The first version finds an analytical formula to update 𝜒n by minimizing the numerator of the second mismatch term in the objective function
(6.45). However, due to the presence of the unknown 𝜒 in the denominator,
the updating scheme may not reduce the objective function. The second version remedies this problem by minimizing the second mismatch term using
the conjugate gradient method. As a side note, the scheme of alternately updating two types of unknowns presented in the CSI is what an earlier paper [56]
calls an alternating direction implicit method. We mention in passing that there
are several variants of the CSI method. For example, [57] uses the EBA-based
two-step inversion result rather than BP as the initial solution to speed up the
convergence.
6.3.3
Contrast Source Extended Born Method
While both the DBIM and the CSI method are based on the original field-type
or source-type integral equations, the contrast source extend Born (CS-EB)
method, proposed by Isernia et al. [58], deals with a different source-type
equation that is nevertheless equivalent to the original source-type equation.
The original source-type state equation can be rewritten as
{
J(r) = 𝜒(r) Ei (r) + k02 g(r, r′ )J(r)dr′
∫D
}
+ k02 g(r, r′ )[J(r′ ) − J(r)]dr′
(6.46)
∫D
If we define the second integral term as a modified integral operator, it can be
written as
GDM (J) = GD (J) − fD (r)J(r)
(6.47)
where
fD (r) = k02
∫D
g(r, r′ )dr′ .
(6.48)
If we further define a new parameter of material
p(r) =
𝜒(r)
,
1 − 𝜒(r)fD (r)
(6.49)
Reconstructing Dielectric Scatterers
then the state equation in the CS-EB reads
J(r) = p(r)[Ei (r) + GDM (J)] for r ∈ D.
(6.50)
The inverse problem of reconstructing 𝜒 is to first reconstruct the new material
parameter p from the state equation (6.50) and the data equation (6.9). For
example, we can follow the procedure of the CSI model to alternatively update
p and J in each iteration of optimization. Once p is reconstructed, the value of
𝜒 is subsequently obtained from (6.49).
Some comments on the CS-EB are now in order. First, the state
equation (6.50) is exact; that is, no approximation is made therein. In
addition, it has the same structure as the original state equation (6.8), where
𝜒 is replaced by p and GD is replaced by GDM . Since the new defined material
parameter p and operator GDM lead to different convergence speeds in both
forward and inverse problems compared with the case of the original 𝜒 and
GD , the CS-EB model provides an important alternative effective tool for
scattering problems. The comparison of the original contrast-source model
and the CS-EB model is made in [59].
Second, if the second term on the right-hand side of (6.50) is dropped off,
based on the same argument made in the EBA method (see Section 6.2.3), then
such an approximation exactly coincides with the result one would achieve
within the EBA when the scatterers are homogeneous. This is why (6.50) is
referred to as the “CS-EB model”: that is, the contrast-source version of the
extended Born model.
Third, partially motivated by the method of contracting integral equation that
is used to solve the forward scattering problem with high contrast [60, 61],
Zhong et al. have proposed a new integral equation (NIE) in [62],
𝛽(r)J(r) = R(r)[Ei (r) + GDN (J)] for r ∈ D,
(6.51)
where 𝛽(r) is an arbitrary function, R(r) = 𝛽(r)𝜒(r)[1 + 𝛽(r)𝜒(r)]
parameter of material, and the new operator GDN (⋅) is defined as
−1
GDN (J) = GD (J) + 𝛽(r)J(r).
is the new
(6.52)
It is important to note that the NIE (6.51) is derived without approximation.
Since the function 𝛽(r) can be arbitrary, it is desirable to choose one so that it
helps to reduce the degree of nonlinearity of the inverse problem. Some comments are made on 𝛽(r).
1) Since 𝜒(r) has a positive real part and a nonnegative imaginary part, the
magnitude of the new parameter of material R(r) will be definitely smaller
than 1 if 𝛽(r) has a positive real part and a nonpositive imaginary part. This
is one of the advantages of the NIE model.
2) In [62], R(r) and J(r) are treated as independent parameters in solving the
inverse problem and they are updated alternatively. When updating R(r),
what is inside the square bracket in (6.51) contains the local-effect term
145
146
Electromagnetic Inverse Scattering
Ei (r) + 𝛽(r)J(r) and the global-effect term GD (J) due to multiple scattering.
Based on the argument that nonlinearity of the inverse scattering problem
is mainly due to the multiple scattering effect, [62] has chosen large values
of 𝛽(r) so that the local-effect term dominates the global-effect term, which
to a certain degree reduces the degree of nonlinearity of the problem.
3) When 𝛽(r) is chosen as −fD (r), the NIE model reduces to a structure very
close to the CS-EB model.
Fourth, it is straightforward to see that fD (r) can be chosen as an arbitrary
function, not necessarily restricted to the right-hand side of (6.48). The derivation of (6.48) is a consequence of (6.46), with the motivation that the second
integral term in (6.46) is much smaller than the first integral term; that is, ideally
GDM (J) is negligible. With this motivation, the state equation (6.50) becomes a
localized equation if GDM (J) is dropped off. To summarize, the fact that p(r) is
chosen as (6.49), with fD (r) being the right-hand side of (6.48), can be understood as an analytical approach to approximately reach a localized effect. In
comparison, the diagonalized contrast source inversion (DCSI) presented in
[63] can be understood as a numerical approach to obtain p(r) so that the localized equation J(r) = p(r)Ei (r) approximately holds.
6.3.4
Other Iterative Models
This section briefly introduces some other iterative inversion methods, all of
which involve rewriting the original state equation in alternative forms.
The dual space method [64] breaks the inverse scattering problem into two
parts: the first part deals with the linear ill-posed problem of looking for the
superposition coefficients of incidence waves coming from different angles so
that the resulting scattered fields match certain multipole radiation patterns in
the far field, and the second part deals with the nonlinear problem of looking
for the total electric field Et (r) and the contrast 𝜒(r) inside the domain of
interest D so that they satisfy both the field-type state equation (6.6) and the
transmission condition on the boundary 𝜕D of the D. The details can be found
in [64] and section 10.3 of [65]. In practice, the dual space method for solving
the ISP is formulated as an optimization problem, where the objective function
is an addition of the mismatches in each of the aforementioned two parts
and the unknowns consist of three types of parameter; that is, the contrast,
the total electric field, and the superposition coefficient of incident waves
coming from different angles. The dual space method is mathematically well
justified and it has the advantage of avoiding solving the forward problem.
Nevertheless, the method requires full aperture incidence and reception,
which is too demanding in practice. Numerical simulations shows that the
reconstruction results obtained by this method are not stable in the presence
of noise. The research and application of the dual space method are mainly
carried out in applied mathematics.
Reconstructing Dielectric Scatterers
The coupled dipole method (CDM), which is also known as the discrete dipole approximation (DDA), is an alternative method for solving
electromagnetic scattering problems [66]. In this method, the volume of
the scatterer is divided into small cells and scattered field is considered as
the superposition of reradiation of induced dipoles. The dipole moment is
derived from the original field-type state equation. It is important to note
that the CDM is based on the concept of an exciting field, rather than the
actual field. The exciting field for a cell refers to the total field incident
upon the cell, consisting of the original incident field coming directly from
transmitters and the scattered field coming from all other cells. The exciting
field is derived from the original electric-field integral equation by moving the
self-contribution term for a cell to the other side of the equation. The dipole
moment induced within a cell is the product of the polarization tensor (also
known as the depolarization tensor) and the exciting field. To summarize, the
state equation for the CDM presents a relationship between the exciting field
and the polarization tensor, which is different from the original state equation
that presents a relationship between the actual field and the contrast. For
an extensive review of the CDM (or equivalently the DDA), including both
theoretical and computational aspects, the reader is referred to [67, 68] and
references therein. For inverse scattering problems that are based on the CDM
model, the task is to first recover the polarization tensor of each cell and then
analytically derive the contrast. Numerical inversion based on the CDM model
has been implemented to solve ISPs in both two-dimensional (e.g., [69]) and
three-dimensional (e.g., [70, 71]) scenarios. It is worth mentioning that [72]
has adopted the T-matrix model to solve a two-dimensional problem. Since
the first order multipole happens to be the dipole term, the T-matrix model
for small circular or spherical cells is in essence the same as the CDM.
In another model, [73] has rewritten the field-type state equation by extracting the singularity of the Green’s function, arriving at a so-called the integral
equation for strong permittivity fluctuation. This new state equation expresses
the relationship between a scaled electric field and a new defined parameter
of material. The contrast is an analytical function of the new defined parameter of material. Numerical simulations are performed and the results show that
the DBIM for the new state equation converges faster and can obtain better
reconstructions for scatterers with larger dimensions and higher contrasts in
comparison with the DBIM for the original state equation.
Instead of using the traditional complex number formulation, that is, real
and imaginary values, [74] proposes a reconstruction algorithm that directly
incorporates log-magnitude and phase of the measured electric field data.
The process minimizes squared differences between measured and computed
electric field log-magnitude and phase by iteratively adjusting the spatial
distribution of electromagnetic parameters within the DOI through a regularized least-squares approach. Simulation studies and microwave imaging
147
148
Electromagnetic Inverse Scattering
experiments demonstrate that significant image quality enhancements occur
with this approach for large high-contrast scatterers.
An important approach of rewriting the original state equation is to expand
the contrast, electric field, or contrast source in certain orthogonal bases. For
example, [75] expands the contrast and electric field in Fourier bases and the
state equation is transformed to express the relationship between the Fourier
coefficients of the contrast and the electric field. Wavelet bases are used in [76]
to expand the contrast and the contrast source. In practice, only a finite number
of orthogonal bases are employed so the number of unknowns in the inversion
model is usually much less than that when traditional pixel bases are used.
In solving the forward problem, many alternative forms of the electric field
integral equation have been proposed aimed at accelerating the convergent
rate of iterative solvers. For example, [60] proposes the contraction integral
equation (CIE) motivated by an alternative form of the electromagnetic integral equation that is based on the modified Green’s operator with a norm less
than 1, which has been presented in earlier series papers that can be found
in the introduction of [60]. The CIE can be treated as a preconditioned conventional integral equation, where the preconditioners are diagonal operators
determined by the contrast distribution. Though these preconditioners have
been beneficial to forward problems, their effect on inverse problems is not
guaranteed since the preconditioners themselves depend on the contrast that
is, however, the sought-after unknown. This is a distinct difference between the
preconditioner approach and all other iterative inversion models introduced so
far in this chapter.
The state equations in various iterative inversion models discussed so far
are either exact in the continuous form or of high precision in the discrete
form, and thus they have wider ranges of applicability. In comparison, another
type of iterative inversion model is based on inexact state equations. For
example, there have been attempts to approximate the state equation by
local-effect equations. For the field-type state equation, the electric field at a
position is assumed to depend on the incident field only in the same position,
which means a localized effect. This is a practice used in the quasi-analytical
(QA) approximation [42], diagonal tensor approximation (DTA) [43], and
quasi-linear (QL) approximation (see [42] and references therein) methods.
For the source-type state equation, the contrast source at a position is assumed
to be linearly proportional to the incident field at the same position, which
is used in the diagonalized contrast source inversion (DCSI) [63]. Due to the
approximation made, the inversion methods based on these models are much
less challenging compared with the original inverse problem. For a practical
inverse scattering problem, if the conditions of those approximations are well
satisfied, then the so-obtained reconstruction results are regarded as the final
reconstruction results. Otherwise, the obtained results can be treated as initial
guesses for other inversion models that are based on the exact state equation.
Reconstructing Dielectric Scatterers
In addition to the aforementioned models that rewrite the objective function such that it depends on unknowns in a less nonlinear way, there are other
models as well. It is not the intention of this section to provide a complete
list. Another model, the so-called subspace-based optimization method, is purposely left in the next section, Section 6.4, where the theory, algorithm, implementation issues, and numerical examples are provided in detail.
6.4 Subspace-Based Optimization Method (SOM)
As shown in Chapter 4, for discrete point-like scatterers, the inverse scattering
problem can be solved by subspace methods in such a way that the corresponding forward problem is not iteratively evaluated. To be specific, the multiple
signal classification (MUSIC) algorithm has been applied to locate the positions of scatterers and the two-step least-squares method has been applied to
retrieve their scattering strengths. The scattered field is in a subspace spanned
by the the singular vectors associated with nonzero singular values of the multistatic response (MSR) matrix, which turns out to be equivalent to the subspace
spanned by Green’s functions with sources located at point-like scatterers. As
discussed in Chapter 4, the key principle behind the subspace methods lies in
the fact that the operator GS , defined in (4.6), which maps the complex amplitude of source induced at point-like scatterers to the scattered fields, is injective.
It is natural to think of applying the subspace methods to extended scatterers
as well. However, since the induced source inside an extended scatterer is continuous instead of discrete, the corresponding GS operator exhibits different
properties from its discrete counterpart. The inversion model developed along
this line is referred to as the subspace-based optimization method (SOM). For
ease of presentation, the SOM in the case of the two-dimensional scalar wave
equation is considered in this section. The SOM for 2D or 3D vector waves can
be similarly constructed.
6.4.1
Gs-SOM
The early stage of SOM algorithms is directly motivated by the subspace
methods developed for discrete point-like scatterers. Since they decompose
the space of induced source into subspaces by decomposing the GS operator,
they are referred to as the Gs-SOM. All equations presented in this section will
be written in discrete form to facilitate presenting the implementation details
of the Gs-SOM.
The configuration of the inverse scattering problem is first introduced. We
consider a two-dimensional setting (the ẑ being the longitudinal direction)
under the transverse magnetic (TM) incidence. In a free space background,
nonmagnetic scatterers are located in the domain of interest (DOI) D ⊂ R2
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150
Electromagnetic Inverse Scattering
and illuminated by time-harmonic electromagnetic waves. Both the electric
field and induced contrast source are in the z direction, and the subscript z
is omitted for convenience. A total number of Ni incidences are due to line
sources that are located at rip , p = 1, 2, ..., Ni . For each incidence, the scattered
electric field is measured by an array of Ns antennas, which are located at rsq ,
q = 1, 2, ..., Ns . The inverse scattering problem consists of determining the
permittivity 𝜖(r), r ∈ D, given a set of Ni Ns scattering data, Eps (rsq ). In practice,
the domain D is discretized into a total number of M small square cells whose
sizes are much smaller than the wavelength and whose centers are located at
r1 , r2 , … , rM . The inverse scattering problem reduces to determining 𝜖(rm ), or
equivalently the contrast 𝜒(rm ) = 𝜖(rm )∕𝜖0 − 1, m = 1, 2, ..., M. It is noted that
M usually is much larger than Ni and Ns .
We apply the method of moment (MOM) to discretize the source-type state
equation (6.8), using the pulse basis function and the delta test function. By
approximating every square
√ cell as a small circle of the same area, that is, with
an equivalent radius a = S∕𝜋 where S is the area of the cell, we obtain the
following the discrete state equation,
i
J = 𝜒 ⋅ (E + GD ⋅ J),
(6.53)
i
where J = [J(r1 ), J(r2 ), ..., J(rM )]T is the normalized current density, E = [Ei (r1 ),
Ei (r2 ), ..., Ei (rM )]T , and the superscript T denotes the transpose operator. The
diagonal matrix 𝜒 consists of 𝜒m and the M × M matrix GD is given by
⎧ ik0 𝜋a
(1)
if m ≠ m′
⎪ 2 J1 (k0 a)H0 (k0 |rm − rm′ |),
′
GD (m, m ) = ⎨
(6.54)
′
⎪ ik0 𝜋a H (1) (k0 a) − 1,
if
m
=
m
1
⎩ 2
where k0 is the wave number in the background medium. Similarly, the discretized data equation (6.9) reads
s
E = GS ⋅ J,
(6.55)
s
where E = [Es (rs1 ), Es (rs2 ), ..., Es (rsN )]T and the Ns × M matrix GS is given by
s
GS (q, m) =
ik0 𝜋a
J (k a)H0(1) (k0 |rsq − rm |).
2 1 0
(6.56)
For discrete point-like scatterers, the mapping GS , from the space of current
s
J to the space of scattered field E , is proven to be injective [77–79]. Following
this fundamental principle, Chapter 4 retrieves the current vector J from the
data equation in the first step, and then retrieves the scattering strength 𝜏 from
the state equation in the second step. For extend scatterers, however, the case
is quite different. It is well known that the operator, GS in (6.9) in continuous
Reconstructing Dielectric Scatterers
form, mapping from the space of induced current in the domain D to the space
of scattered field, is compact (simply speaking, the operator has infinitely
many small singular values accumulating at zero), and the inverse problem
of determining the induced current from the measurement of scattered field
is ill-conditioned [55, 77]. As the discrete version of this current-to-field
mapping operator, GS in (6.55) shares similar properties, and in practice the
induced current J cannot be uniquely determined solely from (6.55). The
ambiguity of induced current, due to the presence of non-radiating current (or
invisible current [55]), is well known in inverse source problems [55, 80, 81].
Despite the impossibility of recovering the current J from the data
equation (6.55), a subspace of J can be obtained thanks to the salient
properties of the singular value decomposition (SVD) of GS . The SVD of GS
H
is represented as GS = U ⋅ Σ ⋅ V , where U is of size Ns × Ns and is composed
of orthonormal left singular vectors uq , V is of size M × M and is composed
of orthonormal right singular vectors 𝑣m , Σ is of size Ns × M with the diagonal
terms being singular values 𝜎m that are placed in nonincreasing order 𝜎1 ≥ 𝜎2 ≥
· · · = 𝜎M = 0, and the superscript H denotes the Hermitian. A basic property of
s
the SVD is GS ⋅ 𝑣m = 𝜎m um . The vector of scattered field E can be represented
as a span of the left singular vectors; that is, the set of Ns unit vectors uq form
the orthonormal bases in C Ns . The vector of the induce current J can be written
as a span of the right singular vectors; that is, the set of M unit vectors 𝑣m form
the orthonormal bases in C M . The salient feature of the singular-vector bases
is that orthogonal inputs to GS yield orthogonal outputs; that is, there is no
crosstalk among different bases for both the input space and output space of
GS . This feature is absent for other bases; for example, Fourier bases.
To proceed, we write J = V ⋅ 𝛼, where 𝛼 is an M-dimensional vector. If there is
no noise in measured scattered field, the coefficients 𝛼j , j ≤ Ns , can be uniquely
determined from (6.55),
H
𝛼j =
uj ⋅ E
𝜎j
s
,
j ≤ Ns .
(6.57)
Since noise is inevitable, small singular values lead to tremendously large error
in 𝛼j . Since singular values are in nonincreasing order, there must be a certain
integer index L0 so that 𝛼j is considered to be within the range of acceptable
error for j ≤ L0 . Due to the salient feature of the SVD, the 𝛼j is independent
of each other, and consequently we can decompose current J into two comple+
−
+
−
mentary and orthogonal parts, J = J + J . Alternatively, J = V ⋅ 𝛼 + + V ⋅ 𝛼 − ,
+
−
where V and V are composed of the first L and the remaining M − L columns
of V , respectively, and L is an arbitrary integer no larger than L0 . The superscripts + and − denote major and minor parts, respectively.
151
152
Electromagnetic Inverse Scattering
Whereas the 𝛼j+ for the major part of current is reliably determined from
(6.57), the 𝛼j− for minor part is unknown yet and has to be obtained in other
ways. Due to the truncation of the singular values, a residual in the data
equation (6.55) appears and is defined as
dat
−
+
s
Δdat = ||𝛿 ||2 = ||GS ⋅ V ⋅ 𝛼 − + GS ⋅ J − E ||2 ,
(6.58)
Similarly, the residual in the state equation (6.53) is defined to be
sta
Δsta = ||𝛿 ||2 = ||A ⋅ 𝛼 − − B||2 .
(6.59)
where
−
−
A = V − 𝜒 ⋅ (GD ⋅ V ),
(6.60)
and
i
+
+
B = 𝜒 ⋅ (E + GD ⋅ J ) − J .
(6.61)
The total relative residual is defined to be
s
+
Δtot = Δdat ∕||E ||2 + Δsta ∕||J ||2 .
(6.62)
For each incidence, the total relative residual can be calculated as Δtot
p , p =
1, 2, ..., Ni . The contrast 𝜒 is obtained by minimizing the total relative residuals. It is important to note that 𝛼j− is still undetermined yet. There are two
approaches to calculate 𝛼j− , and their objective functions are defined here.
−
The first approach treats 𝛼 as an intermediate parameter, which is the
−
practice in the earliest version of Gs-SOM [82]. The value of 𝛼 is chosen
H
H
as (A ⋅ A)−1 ⋅ (A ⋅ B) that minimizes (6.59) in the least-squares sense (in
−
practice, it is preferable to obtain approximate values of 𝛼 by minimizing
(6.59) using iterative solvers since there is no need to waste time inverting
a matrix that depends on unknowns being sought). Thus, the total relative
residuals depend solely on the contrast 𝜒. Then, a nonlinear least-squares
2
objective function is defined from (Δtot
p ) , explicitly
(
)2
−
s 2
+
Ni
−
||A ⋅ 𝛼 −p − Bp ||2
1 ∑ ||GS ⋅ V ⋅ 𝛼 p + GS ⋅ J p − Ep ||
+
,
f (𝜒) =
s
+
2 p=1
||Ep ||2
||J p ||2
(6.63)
where the dependence on the incidence is denoted by the subscript p.
Many optimization algorithms can be adopted to minimize (6.63). The
Levenberg–Marquardt (LM) algorithm [83], which is a mixture of the
Gauss–Newton algorithm and the method of gradient descent, is used in [82].
−
The second approach treats 𝛼 as an independent parameter, which is the
practice in the second version of Gs-SOM [84]. The objective function is
Reconstructing Dielectric Scatterers
defined from Δtot
p as follows,
−
−
−
f (𝛼 1 , 𝛼 2 , … , 𝛼 Ni , 𝜒) =
(
−
s
+
Ni
∑
||GS ⋅ V ⋅ 𝛼 −p + GS ⋅ J p − Ep ||2
+
s
||Ep ||2
p=1
||A ⋅ 𝛼 −p − Bp ||2
+
||J p ||2
)
,
(6.64)
−
where both 𝛼 p and 𝜒 are unknowns.
The two approaches are compared in [85], and it turns out that the second
approach outperforms the first one. Although the first approach converges
in fewer iterations, the computational burden in each iteration is so high due
−
to the matrix inversion in obtaining 𝛼 that its overall running time is much
longer than the second approach needs. In addition, numerical simulations
show that the second approach stands a higher chance in converging to the
global minimum. For this reason, we will focus the discussion of Gs-SOM
mainly on the second approach.
Reference [84] adopts the optimization method used in the CSI method
−
to minimize (6.64); that is, alternatively updating the coefficients 𝛼 p and the
contrast 𝜒. The implementation steps are as follows:
+
Step 1: Calculate GS , GD , and the SVD of GS . Obtain J p from (6.57),
p = 1, 2, ..., Ni .
−
Step 2: Initial step, n = 0: 𝜒 0 is obtained from the BP [52, 53]; 𝛼 p,0 = 0; Initialize
the search direction 𝜌p,0 = 0.
Step 3: n = n + 1.
−
Step 3.1: Update 𝛼 p,n : Calculate gradient (Fréchet derivative) g p,n =
−
∇𝛼−p f evaluated at 𝛼 p,n−1 and 𝜒 n−1 ; Determine the Polak–
Ribière conjugate gradient search directions 𝜌p,n =
g p,n +
Re[(g p,n −g p,n−1 )H ⋅g p,n ]
−
𝛼 p,n
||g p,n−1 ||2
−
𝛼 p,n−1 +
𝜌p,n−1 [52]. Define the scalar dp,n
by
=
dp,n 𝜌p,n . The objective function becomes
quadratic in terms of the parameter dp,n , and it is easy
to obtain dp,n = Num∕Den, where the numerator and
denominator are, respectively,
−
Num = −
dat
(GS ⋅ V ⋅ 𝜌p,n )H ⋅ 𝛿 p,n−1
s
||Ep ||2
sta
−
(An ⋅ 𝜌p,n )H ⋅ 𝛿 p,n−1
−
Den =
dat
sta
(6.65)
+
||J p ||2
||GS ⋅ V ⋅ 𝜌p,n ||2
s
||Ep ||2
+
||An ⋅ 𝜌p,n ||2
+
||J p ||2
(6.66)
where 𝛿 , 𝛿 , and A have been defined in (6.58)–(6.60).
153
154
Electromagnetic Inverse Scattering
Step 3.2: Update 𝜒 n : For the mth cell, m = 1, 2, ..., M, update the
−
+
induced current (J p,n )m = (J p )m + (V ⋅ 𝛼 −p,n )m . Then update
t
i
the total field in the mth cell, (Ep,n )m = (Ep )m + (GD ⋅ J p,n )m .
The objective function becomes quadratic in terms of (𝜒 n )m ,
and the solution is straightforwardly given by
t
/ Ni | t
|2
Ni
⎡∑
(Ep,n )∗m (J p,n )m ⎤ ⎡∑
| (Ep,n )m | ⎤
⎥ ⎢ |
| ⎥.
(𝜒 n )m = ⎢
⋅
+
⎢ p=1 ||J + ||
⎥ ⎢ p=1 || ||J + || || ⎥
||J
||
p
p
⎣
⎦ ⎣ | p |⎦
(6.67)
Step 4: If a pre-determined termination condition (e.g., no obvious change in
objective function or reaching a maximum iteration count) is satisfied,
stop iteration. Otherwise, go to Step 3.
Note that in comparison with the standard CG method presented in
Appendix C, dp,n 𝜌p,n appearing in the Step 3.1 should be understood as
(−dp,n )(−𝜌p,n ). Some comments on the Gs-SOM are made as follows.
First, the two versions of objective functions do not make approximations.
In particular, the second version, that is, (6.64), does not need to solve the
forward problem; that is, no matrix inversion is involved. What is more, it
is a quadratic in both 𝛼p− and 𝜒 individually, and consequently it is a quartic
function of unknowns 𝛼p− and 𝜒 simultaneously. This quartic-polynomial minimization problem is equivalent to a constrained quadratic minimization problem, where (6.58) is the objective function and the quadratic function A ⋅ 𝛼 − −
B = 0 is the constraints. This is a standard quadratically constrained quadratic
problem (QCQP). It is well known that solving the general case of a QCQP is
NP-hard (non-deterministic polynomial-time hard), which is at least as hard
as a large number of other problems that have been proven to be hard in the
sense that all known algorithms for solving them have a complexity that grows
at a rate higher than polynomial (e.g., exponentially) with problem dimensions.
In the optimization community, there has been a handful of methods to solve
QCQPs, such as semidefinite relaxation (SDR) [86] and PhaseLift [87]. The SDR
technique is a computationally efficient approximation approach in the sense
that its complexity is polynomial in problem dimensions. However, there are
some difficulties in applying the SDR to solving inverse scattering QCQP. On
one hand, the solution provided by the SDR is not compatible with the variables
of the original QCQP problem since SDR has relaxed the original problem.
Consequently, one needs to extract, from the SDR solution, a feasible solution
that is compatible to the original QCQP. On the other hand, the dimension
of the SDR is significantly larger than the original problem, which makes it
impractical to solve any medium or large scale inverse scattering problems.
Second, the total number of unknowns in (6.64) is Ni (M − L) + M. The
predetermination of the Ni L-dimensional major part of induced current
Reconstructing Dielectric Scatterers
significantly speeds up the convergence of optimization iterations, which is
due to not only the reduction of the number of unknowns but also the fact
that the already-known major part of induced current significantly reduces the
relative residual in the data equation.
Third, the value of parameter L can be any integral between one and L0 , due
to the salient mathematical feature of the singular-vector bases that orthogonal inputs to GS yield orthogonal outputs. In the extreme case of L = 1, the
Gs-SOM (6.64) is close in spirit to the CSI model. It worth mentioning that the
case of L = 0 is in fact even closer to the CSI model, but this case is not considered here since the denominator is zero in (6.64). The other extreme case L = L0
is close in spirit to a particular STIE method [55, 88]. The STIE method extracts
the radiating current by solving the data equation in the minimum norm sense,
and then reconstructs nonradiating current (or more precisely speaking, “invisible current” with respect to specific discrete receivers) by minimizing the
residual in the state equation. In absence of noise, this method works well.
However, in presence of noise, the method performs poorly due to the fact the
minimum-norm solution to the data equation produces large errors in the state
equation. Even if the radiating current is redefined as the solution to the data
equation that satisfies the discrepancy principle, which is widely adopted in
inverse problems (see Appendix A), that is, the residual of the data equation is
comparable to the noise level, great care has to be paid to choose the best value
of L0 . In comparison, the Gs-SOM does not face the problem of choosing the
best value of L since any integer between one and L0 is feasible. If the value of L0
cannot be accurately estimated, then the suggestion is to conservatively choose
L to be small integers, such as 1 or 2 in extreme cases, so that the noise will not
be amplified in (6.57).
Fourth, regarding the terminologies, “major and minor parts of induced
current” are used here, and they are denoted with the superscripts + and
−, respectively. Such terminologies are different from those adopted in the
author’s previous publications [82, 84], where “deterministic and ambiguous
parts of induced current” are used, which are denoted as superscript the “s” and
“n,” respectively, following the “signal subspace” and “noise subspace” widely
adopted in the MUSIC algorithm. The change of terminologies is due to some
feedback that previous terminologies might be confusing to some readers.
Here we present some numerical results to evaluate the performance of the
Gs-SOM model. The “Austria” profile, as illustrated in Fig. 6.1, is considered in
numerical simulations. All data about the configuration of inverse scattering are
the same as those provided for the “Austria” profile in Section 6.2.5, except that
the 16 line sources are now evenly placed on a circle with a radius of 3 m. In the
forward scattering problem, the data are generated numerically using the MoM
method with a 100 × 100 grid mesh, which is much finer than the one used in
the inverse process (64 × 64) in order to avoid the inverse crime (section 5.3
of [65]). During the inversion, A priori information is used that the scatterers
155
Electromagnetic Inverse Scattering
are lossless and have nonnegative contrasts [53]. Although all numerical results
reported in this section are for the “Austria” profile, the proposed algorithm has
been tested on various other profiles and all conclusions drawn are the same as
the one reported here.
One of the key components of the proposed subspace-based optimization
model is spectrum analysis. In particular, the value of L is determined from the
spectrum. The spectrum of GS for the aforementioned simulation configuration is shown in Fig. 6.7. First, in the absence of noise, the convergence of the
objective function is compared for different values of L in the first 50 iterations
and the results are shown in Fig. 6.8. It is observed that the increase of the value
of L results in a faster convergence. The trajectory of the corresponding relative
error of reconstruction Re is shown in Fig. 6.9, where
R
Re =
||𝜖 − 𝜖 ||F
(6.68)
||𝜖||F
R
and 𝜖 is the reconstructed permittivity. The reconstructed relative permittivity
profiles are shown in Fig. 6.10. The cases of L = 10, 15, 20, and 25 produce
successful reconstruction results. Numerical simulations show that the cases
of L = 1 and 5 also produce successful reconstruction results, however, at
the expense of more iterations, especially in the case of L = 1. Figure 6.11
illustrates the result for L = 1 at the 800th iteration. All three results in
Figs 6.8–6.10 indicate that there is no noticeable difference between the results
for L = 15, 20, and 25.
4.5
4
3.5
log10(σj)
156
3
2.5
2
1.5
0
5
10
15
20
25
Singular value number, j
30
35
Figure 6.7 The spectrum of the operator GS , where the base 10 logarithm of the singular
values is plotted. Source: Chen 2010, IEEE Trans. Geosci. Remote Sens., 48, 42–49. [84]
Reproduced with permission of IEEE.
Reconstructing Dielectric Scatterers
2
1.5
log10f
1
0.5
0
L=1
L=5
L = 10
L = 15
L = 20
L = 25
–0.5
–1
–1.5
0
10
20
30
Number of Iterations
40
50
Figure 6.8 The comparison of convergence trajectories in the first 50 iterations for different
values of L, where the base 10 logarithm of the objective function value is plotted. Source:
Chen 2010, IEEE Trans. Geosci. Remote Sens., 48, 42–49. [84] Reproduced with permission of
IEEE.
0.4
0.35
Re
0.3
0.25
0.2
L=1
L=5
L = 10
L = 15
L = 20
L = 25
0.15
0.1
0
10
20
30
Number of Iterations
40
50
Figure 6.9 The comparison of trajectories of relative error Re in the first 50 iterations for
different values of L.
It is worth discussing the method of determining the value of integer L. The
value of L balances the relative residuals in the data equation and in the state
equation. The larger the value of L, the smaller the relative residual in the
data equation. However, if L is so large that the relative residual in the data
157
158
Electromagnetic Inverse Scattering
1
2
0.5
1.8
1.6
0
1
2
0.5
1.8
1.6
0
1.4
–0.5
–1
–1
1.2
–0.5
1
0
(a)
0.5
1
1
2.2
2
0.5
1.4
–0.5
–1
–1
1.2
–0.5
1
0
(d)
0.5
1
2
0.5
1.8
1.8
0
1.6
1.4
–0.5
1.6
0
1.4
–0.5
1.2
1.2
–1
–1
–0.5
1
0
(b)
0.5
1
2
1.8
1.6
0
1.4
–0.5
1.2
–0.5
0
(c)
–1
–1
–0.5
1
0.5
–1
–1
1
0.5
1
1
1
0
(e)
0.5
1
2
0.5
1.8
1.6
0
1.4
–0.5
–1
–1
1
1.2
–0.5
0
(f)
0.5
1
1
Figure 6.10 Reconstructed relative permittivity profiles at the 50th iteration for different
values of L. (a) L = 1. (b) L = 5. (c) L = 10. (d) L = 15. (e) L = 20. (f ) L = 25. Source: Chen 2010,
IEEE Trans. Geosci. Remote Sens., 48, 42–49. [84] Reproduced with permission of IEEE.
equation is smaller than the noise level, the relative residual in the state
equation will be large and cannot be remedied in the subsequent optimization
process. On the other hand, a small value of L does not produce a nonremediable large relative residual in the state equation, but the simultaneous
minimization of both relative residuals in the data equation and in the state
equation takes longer time to converge. The following criteria are used to
determine the value of L:
Reconstructing Dielectric Scatterers
1
2.2
2
0.5
1.8
y(m)
Figure 6.11 Reconstructed
relative permittivity profiles at
the 800th iteration for L = 1.
Source: Chen 2010, IEEE Trans.
Geosci. Remote Sens., 48, 42–49.
[84] Reproduced with
permission of IEEE.
0
1.6
1.4
–0.5
1.2
–1
–1
–0.5
0
x(m)
0.5
1
1
1) First of all and the most important of all, there is a consecutive range of
integer L, instead of a single value, that can be chosen, with different L values
resulting in different convergence speeds.
2) The noise level affects the value of L. Generally speaking, the value of L in
low-noise case is chosen to be larger than that in the high-noise case.
3) An empirical method is that a good candidate for L takes the value where
singular values noticeably change the slope in the spectrum; for example,
L = 15 in Fig. 6.7. Numerical simulations for various profiles and noise levels
show that this empirical method works well, unless the noise is very high. It
worth mentioning that this method is practically reliable since the spectrum
of GS itself does not depend on the level of additive white Gaussian noise. In
addition, this empirical criterion is supported by the spatial band-limitation
properties of the field radiated (or scattered) by bounded sources and the
details of this systematic work can be found in [32, 33, 89].
Next, the proposed algorithm is tested for noise-contaminated data. The
reconstruction results for 10, 30, and 50% additive white Gaussian noise
are shown in Fig. 6.12, where L is equal to 15. The optimization iteration is
terminated when there is no significant improvement in the objective function
for two consecutive iterations. For three noise levels, the numbers of iterations
are 59, 31, and 21, respectively. The reconstruction is successful in case of 10%
noise. The result in the case of 30% noise is also satisfying, except that more
artifacts appear. In a case of 50% noise, artifacts are more prominent, which
degrades the reconstruction results. Nevertheless, the positions of the disks
and the annulus are correctly determined. In particular, the hole inside the
annulus can be easily identified.
It is important to realize that the various advantages of Gs-SOM that have
been shown in numerical simulations are also accompanied with additional
159
160
Electromagnetic Inverse Scattering
1
2.2
2
0.5
1
2.2
2
0.5
1.8
0
1.6
1.8
0
1.6
1.4
1.4
–0.5
–0.5
1.2
–1
–1
–0.5
0
(a)
0.5
1
1
1.2
–1
–1
–0.5
1
0
(b)
0.5
1
1
2.2
2
0.5
1.8
0
1.6
1.4
–0.5
1.2
–1
–1
–0.5
0
(c)
0.5
1
1
Figure 6.12 Reconstructed relative permittivity profiles for L = 15 at different noise levels.
The optimization is terminated when there is no significant improvement in the objective
function for two consecutive iterations, which are 59, 31, and 21, respectively, for the three
noise levels. (a) 10% noise; (b) 30% noise; and (c) 50% noise. Adapted from: Chen 2010, IEEE
Trans. Geosci. Remote Sens., 48, 42–49. [84] Reproduced with permission of IEEE.
computational cost, which is a main drawback of early-stage Gs-SOMs. The
overhead computational cost consists of the following two aspects:
1) The SVD of GS in Step 1 of the algorithm: The computational complexity is
O(MNs2 ) for obtaining the matrix of singular values Σ, and it is O(M2 Ns )
for simultaneously obtaining the Ns × Ns matrix U and M × M matrix V
(section 8.6 of [90]).
2) The construction of the minor part of induced current in Step 3.2 of the algo−
rithm: The computational complexity is O(M(M − L)) for evaluating V ⋅
𝛼 −p,n .
If SVD bases are not adopted, the main computational cost is the
matrix-vector multiplication GD ⋅ J in each iteration, which has a computational complexity of O(M log M) when the FFT is used. Since it is usually
Reconstructing Dielectric Scatterers
M ≫ Ns , the computation complexity of the two overhead calculations are on
the order of O(M2 ), which is higher than the O(M log M).
Motivated by circumventing the above two computational burdens, an
improved Gs-SOM model is proposed in [70]. The main idea is to compute
only the first L right singular vectors, instead of all M ones. Considering the
+
−
notation V = [V , V ] and the fact that the columns of the unitary matrix
H
+
+H
−
−H
V are orthonormal, we have I M = V ⋅ V = V ⋅ V + V ⋅ V , where I M
denotes the M-dimensional identity matrix. Such a property allows us to
rewrite the minor part of induced current in a different form,
−
−
J = V ⋅ 𝛼−
−
=V ⋅V
−H
⋅𝛽
+
= (I M − V ⋅ V
+
=𝛽−V ⋅V
+H
+H
)⋅𝛽
⋅𝛽
(6.69)
The new form of minor part of induced current has two important consequences:
1) A truncated SVD of GS is calculated, where only the first L singular vectors
need to be obtained. Since L ≤ Ns , it is sufficient to check the computational
complexity for L = Ns . It is O(MNs2 ) for obtaining the matrix of singular values Σ, and O(MNs2 ) for simultaneously obtaining U and the first Ns columns
of V (section 8.6 of [90]).
2) The construction of the minor part of induced current becomes O(ML).
To summarize, the improved Gs-SOM that is based on the new scheme of
updating minor part of induced current has an overall overhead with computational complexity O(M), considering that fact that M ≫ Ns , which is lower than
the original computational complexity O(M log M). In addition, the memory is
−
saved by avoiding saving the M × (M − L) matrix V . The details of implementing the improved Gs-SOM can be found in [70].
6.4.2
Twofold SOM
The inverse scattering problem is based on two equations; that is, the data
equation (6.55) and the state equation (6.53). It is important to realize that
both matrixes GS and GD are independent of scatterers, which makes it possible to analyze these two matrixes before reconstructing scatterers. Whereas
the Gs-SOM analyzes the GS matrix by using the subspace of the SVD singular
vectors, the same idea can be applied the GD as well, which is referred to as the
twofold subspace-based optimization method (TSOM).
In Gs-SOM, the vector space C M , where the induced current J lies, is decom+
posed into two complementary and orthogonal subspaces. That is, J = J +
161
162
Electromagnetic Inverse Scattering
−
+
−
+
−
J = V S ⋅ 𝛼 + + V S ⋅ 𝛼 − , where V S and V S are composed of the first L and the
remaining M − L columns of the matrix V S of the right singular vectors of
GS . The C M can also be decomposed into two complementary and orthogonal
+
−
subspaces, that is, the major subspace V D and minor subspace V D that consist of the first M0 and the remaining M − M0 right singular vectors of GD ,
respectively. Then the whole space C M is decomposed into three orthogonal
+
subspaces, SS+ , SS−D+ , and SS−D− , which correspond to V S , the intersection of
−
+
−
−
V S and V D , and the intersection of V S and V D , respectively.
In TSOM, it is reasonably assumed that the J that lies in the subspace SS−D−
contribute negligibly to the state equation. Since this subspace by its definition
−
is a subset of V S , the J that lies in the subspace SS−D− automatically contributes
little to the data equation. Consequently, the J that lies in the subspace SS−D− ,
which barely contributes to state equation and data equation, can be dropped
off. Thus, we have
+
J = V S ⋅ 𝛼 + + BS− D+ ⋅ 𝛽 S− D+ ,
(6.70)
where BS− D+ of size M × (M0 − L) is the matrix consisting of the bases of the
−
+
intersection of V S and V D . In solving inverse problems, the coefficient 𝛽 S− D+ is
unknown, and its dimension M0 − L is much smaller than the dimension M − L
−
of the unknown 𝛼 in Gs-SOM. For the purpose of illustrating the concept of
TSOM, we consider a special case where M = 3, L = 1, and M0 = 2. As shown
in Fig. 6.13, the intersection of two planes is a line. It is obvious that to search
a solution along a line is much easier than to search it in a plane, which means
that the TSOM outperforms the Gs-SOM.
After presenting the general idea of the TSOM, we next discuss some implementation issues.
=+
Vs
=– =+
Vs ∩ VD
=+
VD
=–
Vs
Figure 6.13 Illustration
the concept of TSOM,
where the special case
M = 3, L = 1, and M0 = 2 is
considered. The subspace
+
of V S is a straight line
−
perpendicular to the V S
plane. The intersection of
−
+
two planes V S and V D is a
straight line.
Reconstructing Dielectric Scatterers
First, the method of constructing the matrix of bases BS− D+ can be found
in section 4.2.3 of [91]. The method is straightforward, but there is additional
heavy computational cost.
Second, due to the aforementioned heavy computational cost in constructing
the exact bases BS− D+ , an approximate model is proposed in [92] as follows,
−
−H
+
BS− D+ ⋅ 𝛽 S− D+ ≈ V S ⋅ V S ⋅ V D ⋅ 𝛽
(6.71)
where 𝛽 is an M0 -dimensional vector. It is important to note that this approx+
−
imation makes a projection of the subspace V D onto V S , whereas the exact
model requires an intersection of these subspaces. Using Fig. 6.13 as an
+
illustration, this approximation means a projection of the plane V D onto the
−
plane V S , resulting a plane region that is smaller in size, instead of a line. This
is why the vector 𝛽 is of dimension M0 instead of M0 − L. From a theoretical
+
point of view, this approximation becomes exact when V D is orthogonal to
−
V S . In practice, numerical simulations show that this approximation performs
well in solving inverse scattering problem, especially when receivers cover a
full or very large aperture.
Third, the value of integer M0 has a different property from the case of L.
While L can be chosen as any integer between one and L0 , the value of M0
should be sufficiently large. The reason for this difference is that the Gs-SOM
deals with the full vector space C M , that is, it determines the major and minor
parts of induced current using analytical and optimization ways, respectively,
whereas the TSOM deals with only the major part of induced current by dropping off the minor part. In absence of noise, the larger the value of M0 , the
more accurate the model. In presence of noise, since there is noise in the data
equation anyway, there is no point in requesting an exact state equation. A useful practical way of choosing M0 is a nested scheme where the result that is
obtained for a lower value of M0 is treated as the initial guess for the problem with a higher value of M0 [92]. The highest value of M0 can be chosen as
when the M0 th singular value reaches a certain percentage, say 1 or 0.5%, of the
largest singular value of GD . It is worth mentioning that although the integral
operators GS and GD are both compact, the kernel of GS is smooth whereas the
kennel of GD is weakly singular. Consequently, in its discrete form, the singular
values of GD decrease much slower than its GS counterpart does. This is why
M0 is much larger than L.
Fourth, the advantages of the TSOM are accompanied with additional computational cost, which is the main drawback of the first version TSOM. The
overhead computational cost consists of two parts: (1) the SVD of GD that has
+
a computational complexity O(M3 ) and (2) the calculation of V D ⋅ 𝛽 in (6.71)
that has a computational complexity O(MM0 ). An improved TSOM, referred
to as the FFT-TSOM, is proposed in [93] based on the conclusion drawn in [94]
that the Fourier functions present similar properties to the singular functions
163
164
Electromagnetic Inverse Scattering
of an integral operator in the sense that the low-frequency Fourier functions
correspond to those singular functions with large singular values, while the
high-frequency Fourier functions to the singular functions with small singular
values. Their discrete forms also follow the same rule. Thus, it is reasonable to
+
use discrete Fourier bases to approximate the subspace spanned by V D ; that is,
+
V D ⋅ 𝛽 = [𝑣D1 , 𝑣D2 , … , 𝑣DM0 ] ⋅ 𝛽 ≈ [F 1 , F 2 , … , F M0 ] ⋅ 𝛽
(6.72)
It is easy to recognize that the right-hand side of (6.72) is simply the inverse
discrete Fourier transform (IDFT) of a new vector that pads 𝛽 with zero, which
can be performed by the 2D FFT algorithm. The computational complexity of
the FFT is O(M log M), which is much less than O(MM0 ) since log M is much
smaller than M0 . Thus, the need to implement the SVD of GD is eliminated.
Fifth, two important comments on discrete Fourier bases are worth
highlighting.
1) The approximation sign in (6.72) does not mean an individual approximation, 𝑣D1 ≈ F 1 , 𝑣D2 ≈ F 2 , …, but instead it means the subspace spanned by
the former set is approximately equal to the subspace spanned by the latter.
Roughly speaking, as the singular value decreases, the corresponding singular vector exhibits a trend of fast oscillation. To illustrate this trend, we
consider a domain D that is a 2 m × 2 m square and is meshed to 64 × 64
cells. For an operating frequency 400 MHz, the real and imaginary parts of
certain singular vectors of GD are shown in Fig. 6.14. We find that Fig. 6.14
provides a visual evidence that low-frequency Fourier bases resemble singular vectors with large singular values and high-frequency Fourier bases
resemble singular vectors with small singular values.
2) The matrix GD can be considered a low-pass filter since low-frequency
Fourier bases correspond to larger singular values. In signal processing, it
is natural to approximate a vector by a span of low frequency Fourier bases.
For example, such a low-frequency approximation can be well applied
to the induced current J that appears in the left-hand side of the state
equation (6.53). However, such a low-frequency approximation deserves a
careful check when it is applied to the J that appears in the right-hand side
of (6.53) since the it is an input of the GD matrix. Fortunately, the GD works
as a low-pass rather than high-pass filter, which validates the drop off of
high frequency components.
6.4.3
New Fast Fourier Transform SOM
In the Gs-SOM and the improved Gs-SOM, the major part of induced current
+
+
J is first obtained from the subspace spanned by V , and then the minor part
−
−
of induced current J is searched within the subspace spanned by V . There
is an opinion in the mathematical community that, whereas it is important to
Reconstructing Dielectric Scatterers
0.01
0.02
0
0
–0.02
–0.01
–0.04
(a)
0.02
0.02
0.01
0
0
–0.02
–0.01
–0.02
(b)
×10–3
0.02
5
0
0
–0.02
–5
(c)
0.02
0.01
0.02
0
0
–0.01
–0.02
(d)
–0.02
×10–4
0.02
1
0
0
–0.02
–1
(e)
Figure 6.14 The left and right panels, respectively, show the real and imaginary parts of
right singular vectors of GD corresponding to: (a) 1st ; (b) 100th ; (c) 200th ; (d) 500th ; and
(e) 4096th order. Adapted from Xu, K. et al. (2014) Singular value decomposition of the
current-to-field operator in solving inverse scattering problems, IEEE Antennas and
Propagation Society International Symposium, Memphis, TN, 659–660. Reproduced with
permission of IEEE.
165
166
Electromagnetic Inverse Scattering
+
+
extract the J from the subspace spanned by V , it is not always beneficial to
−
−
restrict the search for J within the subspace spanned by V . On one hand,
+
when the noise of measured data is high, the J is consequently more or less
incorrect that, however, cannot be remedied by later optimization process. On
−
the other hand, the dimension of the subspace spanned by V is M − L, which
is slightly smaller than the full dimension M. The reduction of L dimensions
in the searching space indeed somewhat helps to accelerate the convergence
of optimization process, but it cannot be a critical factor since L ≪ M. These
observations motivate the proposal of new inversion algorithms that recon−
−
struct the J in the whole space C M , instead of the subspace spanned by V .
A new fast Fourier transform SOM (NFFT-SOM) is proposed to reconstruct
−
the J in the whole space C M with Fourier bases. The NFFT-SOM outperforms
the Gs-SOM and the improved Gs-SOM in the high noise case and at the same
time it has a computational complexity as low as that of the improved Gs-SOM.
The NFFT-SOM also exhibits much similarity to the FFT-TSOM, but has the
advantage of ease in implementation. In the NFFT-SOM, the induced current
is written in the form
+
J =J +F ⋅𝛼
+
(6.73)
where J is still obtained by the first L singular vectors of GS , as done in the
Gs-SOM, F of size M × M is the complete Fourier bases, consisting of elements
F(m, m′ ) = exp[−i2𝜋(m − 1)(m′ − 1)∕M], and 𝛼 is an M-dimensional vector.
In fact, it is more accurate to understand F ⋅ 𝛼 as a residual current, instead
of minor part of current. This is because F ⋅ 𝛼 deals with the complete bases,
+
thus able to compensate the error in J , especially in the high noise case. The
objective function is similar to that of Gs-SOM, that is, (6.64), except that the
unknowns are now 𝜒 and 𝛼 p , p = 1, 2, … , Ni . The overhead computation cost is
O(Ns2 M) for thin-SVD of GS and O(M log M) for the FFT implementation of F ⋅
𝛼, which are low and comparable to the counterpart of the improved Gs-SOM.
The numerical example with the “Austria ring” that appears in Section 6.4.1
is used to compare the performances of the NFFT-SOM and the Gs-SOM. For
as high as 30% additive white Gaussian noise, the trajectories of relative error
of reconstruction for the NFFT-SOM and the Gs-SOM are shown in Fig. 6.15.
Different values of L are chosen, that is, L = 5, 15, and 25, where L = 15 is
at the knee of the logarithmic scale of singular values, as shown in Fig. 6.7.
We observe that for each L the Gs-SOM always converges slightly faster than
the NFFT-SOM at the early stage of iterations, whereas the relative error of
reconstruction of NFFT-SOM becomes smaller than that of Gs-SOM when
more iterations are implemented. In particular, for L = 25, the relative error of
reconstruction of Gs-SOM decreases fast in initial iterations, but it gradually
increases after about the 50th iteration, which is a typical phenomenon in
Reconstructing Dielectric Scatterers
–0.4
SOM:L = 5
SOM:L = 15
SOM:L = 25
NFFT-SOM:L = 5
NFFT-SOM:L = 15
NFFT-SOM:L = 25
log10 (Re)
–0.5
–0.6
–0.7
–0.8
–0.9
0
200
400
600
Number of iterations
800
Figure 6.15 The trajectories of the relative error of reconstruction for the NFFT-SOM and
the Gs-SOM are compared.
inverse problems that the error of reconstruction increases after a certain
number of iterations although the objective function monotonically decreases
(see Appendix A). In comparison, this phenomenon occurs slightly to the
NFFT-SOM after about the 450th iteration. This observation implies that the
NFFT-SOM is more robust in choosing the value of L in presence of high noise.
Luckily, when scattered field is measured in the far zone, there is a knee in the
logarithmic scale of singular values, that is, L = 15 in Fig. 6.7, which provides a
good solution of L in the high noise case. However, when the scattered field is
measured in the near zone, the aforementioned knee does not exist and, consequently, the NFFT-SOM outperforms the Gs-SOM by providing a wider range
of candidate L. It is obvious that if the noise level is extremely high, even if there
is a knee of the logarithmic scale of singular values for far-zone measurements,
the NFFT-SOM outperforms the Gs-SOM since the latter cannot remedy the
−
+
−
severely incorrect J by searching J within the subspace V .
The reconstruction results obtained by the NFFT-SOM and the Gs-SOM at
the 300th iteration for different choices of L are given in Fig. 6.16. It is obvious as expected that the NFFT-SOM outperforms the Gs-SOM for L = 25. We
also notice that the NFFT-SOM seems to obtain visually better reconstruction
results than the Gs-SOM does for both L = 5 and L = 15, although Fig. 6.15
shows that the relative errors of reconstruction, defined as (6.68), of the two
inversion methods are comparable at the 300th iteration. There might be other
definitions of the relative error of reconstruction than (6.68), but discussion of
this topic is beyond the scope of this chapter.
167
Electromagnetic Inverse Scattering
1
1
2.5
1
2.5
3.5
1.5
–0.5
–1
–1
–0.5
1
0
x(m)
(a)
0.5
1
1
2
y(m)
0
0
1.5
–0.5
–1
–1
–0.5
1
0
x(m)
(b)
0.5
1
1.8
0
1.6
–1
–1
0.5
2
1.8
0
1.6
1.4 –0.5
1.2
–0.5
–0.5
0
x(m)
(d)
0.5
1
1
2.5
2
–0.5
–1
–1
1.5
–0.5
0
x(m)
(c)
0.5
1
–1
–1
2.2
0.5
2
1.8
0
1.6
1.4 –0.5
1.2
–0.5
0
x(m)
(e)
0.5
1
1
1
2.4
2.2
y(m)
2
1
3
0
1
2.2
0.5
0.5
y(m)
2
0.5
y(m)
y(m)
0.5
y(m)
168
–1
–1
1.4
1.2
–0.5
0
x(m)
(f)
0.5
1
1
Figure 6.16 Reconstruction results for the Gs-SOM (upper row) and the NFFT-SOM (lower
row) at the 300th iteration for different choices of L: L = 5 for (a) and (d); L = 15 for (b) and
(e); L = 25 for (c) and (f ).
If only the first M0 Fourier bases, instead of the complete M ones, are used
−
−
to represent J , then the NFFT-SOM is similar to the FFT-TSOM where J
is constructed from (6.71) and− (6.72), with the difference that the NFFT-SOM
does not project onto the V S subspace. In implementing such a version of
NFFT-SOM, we can follow the aforementioned nested scheme; that is, the
result that is obtained for a lower value of M0 is treated as the initial guess
for the problem with a higher value of M0 . The nested scheme significantly
improves the ability of the NFFT-SOM to reconstruct stronger scatterers.
For the previous numerical example, that is, the Austria ring, when the relative permittivity is increased to 2.2 with all other parameters unchanged, the
comparison is made between the original NFFT-SOM that simultaneously uses
the complete Fourier bases and the nested-scheme NFFT-SOM. Figure 6.17(a)
shows that the original NFFT-SOM is unable to provide a satisfactory reconstruction result under 30% noise even if the L = 15 is chosen at the knee of the
of the logarithmic scale of singular values. The nested-scheme NFFT-SOM
gradually increases the number of Fourier bases all the way to complete bases,
M0 = 600, 1000, 2000, 4096 at iteration numbers 1, 101, 201, 301, respectively,
and it provides quite a good reconstruction result, as shown in Fig. 6.17(b). The
trajectories of the relative errors of reconstruction for the original NFFT-SOM
and the gradual bases-expansion NFFT-SOM are shown in Fig. 6.17(c).
To summarize, the NFFT-SOM has the following two advantages:
Reconstructing Dielectric Scatterers
y(m)
3
0
2.5
2
–0.5
1.2
–0.5
0
x(m)
(a)
0.5
1
1
0.5
y(m)
3.5
0.5
–1
–1
1
4
0
–0.5
–1
–1
–0.5
0
x(m)
(b)
0.5
1
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
–0.4
–0.45
–0.5
–0.55
–0.6
–0.65
–0.7
–0.75
–0.8
–0.85
NFFTSOM
NFFTSOMg
log10 (Re)
1
0
100
200
300
Number of iterations
(c)
400
Figure 6.17 Comparisons of performances of the original NFFT-SOM and a gradual
bases-expansion NFFT-SOM for 𝜖r = 2.2, 30% noise, and L = 15. (a) Reconstruction results
obtained by the original NFFT-SOM at the 400th iteration steps; (b) Reconstruction results
obtained by the gradual bases-expansion NFFT-SOM at the 400th iteration steps; and (c) The
trajectories of the relative errors of reconstruction for both inversion methods. In the
legend, the suffix “g” denotes the “gradual bases expansion”.
Firstly, it is much easier to implement. The first reason is that the projection
−
onto the V S subspace is avoided. The second reason is that, 1D Fourier bases are
adopted in NFFT-SOM so that the first M0 bases can be directly determined,
whereas the 2D Fourier bases adopted in FFT-TSOM have to be sorted in order
to determine the first M0 bases. In addition, when the domain of interest D is
not a rectangle, the application of 2D Fourier bases requires the extra work of
extending the DOI by zero padding to a rectangle that fully covers D. For the
NFFT-SOM, there is no need to extend the DOI to a rectangle.
The second advantage is that it performs more robustly with respect to the
variation of L in presence of high noise. For the NFFT-SOM, there is a wide
range of consecutive integers L that can be chosen no matter the noise is high or
low. However, it is admitted that, for the low noise case, this advantage is absent
since there is also a wide range of candidate integer L in the Gs-SOM as well.
The drawback of the NFFT-SOM, compared with the Gs-SOM, is that the
relative error of reconstruction decreases slightly more slowly at the initial stage
of the optimization process.
6.4.4
SOM for the Vector Wave
The Gs-SOM, TSOM, NFFT-SOM, and their variants can be applied to
vector-wave ISPs with no or minor modifications. For example, [69] has
applied the Gs-SOM to 2D TE inverse scattering problems. The forward
problem is based on the coupled dipole method (CDM). Both transmitters
and receivers are placed uniformly in a circle in the far field, with Ni = 20
and Ns = 30 . The domain D is a rectangle of 1.4 𝜆 × 0.7 𝜆 and is discretized
into a grid of 40 × 20 cells. Scatterers are chosen as a set of digit patterns with
relative permittivity 2, as shown in Fig. 6.18. In the numerical experiment, 10%
white Gaussian noise is added to the scattering data. Figure 6.19 shows that
169
170
Electromagnetic Inverse Scattering
2.2
2
Figure 6.18 Exact profile of the
relative permittivity of digit patterns.
Source: Pan 2009, J. Opt. Soc. Am. A,
26, 1932–1937. [69] Reproduced
with permission of The Optical
Society.
1.8
1.6
1.4
1.2
1
0.8
the original digits can be easily identified from the reconstructed patterns that
are obtained after 20 iterations.
For 3D inverse scattering problems, the FFT-TSOM has been used in [93],
where the integral equation equation is discretized by a finite difference
technique presented in [95]. In the numerical simulation, the scatterer is a
coated cube centered at the origin with its inner edge length a = 1 m and outer
edge length b = 2 m, as shown in Fig. 6.20. The relative permittivity of the inner
layer is 𝜖r1 = 2 + 0.8i while the relative permittivity of the outer layer is 𝜖r2 =
1.5 + 0.3i. The coated cube is illuminated by 60 electric dipole antennas operated at 300 MHz (wavelength in air is 1 m), which are distributed along three
circles (with 20 dipole antennas evenly distributed on each circle) with the same
radius 3 m. Scattered fields are collected by 60 detectors, which are located at
the same positions as the 60 dipole sources. 10% AWGN is added to the exact
scattered field. In the inverse problem, the domain D is a cube of length 3 m
concentric to the layered cubic scatterer and is discretized into 30 × 30 × 30
cells. Since a cell is parameterized by its real and imaginary parts of relative
Reconstructing Dielectric Scatterers
Figure 6.19 Reconstructed profile
of relative permittivity of digit
patterns under 2D TE incidences.
Source: Pan 2009, J. Opt. Soc. Am. A,
26, 1932–1937. [69] Reproduced
with permission of The Optical
Society.
2.2
2
1.8
1.6
1.4
1.2
1
0.8
permittivity, there are in total 54,000 unknowns in the optimization problem.
Figure 6.21 shows the reconstruction result by a net-scheme implementation
of the FFT-TSOM. The first and second columns are the real and imaginary
parts of the reconstruction result after 122 iterations. The first, second, and
third rows correspond to the cross sections at z = −0.05 m, y = −0.05 m, and
x = −0.05 m, respectively. The concentric squares clearly displayed in three
slices indicate that the FFT-TSOM obtains a very satisfactory reconstruction
result.
6.5 Discussions
Full-wave nonlinear inversion algorithms have been tested by several
experimental test databases. For example, the Ipswich database includes three
sets of scattering data, the first of which was introduced in [96]. The Institut
Fresnel database includes 2D TE and TM data [97, 98] and 3D data [99].
171
172
Electromagnetic Inverse Scattering
εr1
εr2
Figure 6.20 The scatterer is a coated cube with
its inner edge length a = 1 m and outer edge
length b = 2 m, The relative permittivity of the
inner and outer layer is 𝜖r1 = 2 + 0.8i and
𝜖r2 = 1.5 + 0.3i, respectively. Source: Zhong
2011, IEEE Trans. Antennas Propag. 59, 914–927.
[93] Reproduced with permission of IEEE.
a
b
1
1
0
0
–1
–1
–1
0
1
1
1
0
0
–1
0
1
–1
0
1
–1
0
1
–1
–1
–1
0
1
1
1
0
0
–1
–1
–1
0
1
1 1.2 1.4 1.6 1.8 2 2.2
Figure 6.21 Reconstruction
results for the coated cube by
the FFT-TSOM. The first and
second columns are the real
and imaginary parts of the
reconstruction result after 122
iterations. The first, second, and
third rows correspond to the
cross sections at z = −0.05 m,
y = −0.05 m, and x = −0.05 m,
respectively. Source: Zhong
2011, IEEE Trans. Antennas
Propag. 59, 914–927. [93]
Reproduced with permission of
IEEE.
0 0.2 0.4 0.6 0.8
These test databases give researchers an opportunity to test and validate their
inversion algorithms against reliable experimental data. In addition, many
research groups have built experimental systems to test the performance of
full-wave nonlinear inversion algorithms [100–102].
Full-wave nonlinear inversion algorithms have been successfully applied to
solve many real world problems. For example, validated by experimental data or
anatomically realistic numerical models, they turn out to be effective in breast
cancer detection and biological tissue imaging [103–106]. They are widely used
in nondestructive evaluation [107] and in geoscience applications [108].
Inverse scattering problems can be categorized into two types, as far as how
scatterers are represented is concerned. The first type is a pixel (or voxel) based
model, where the DOI is discretized into pixels and the material property at
Reconstructing Dielectric Scatterers
each pixel will be reconstructed. The second type is a parameterized model,
where scatterers are known a priori to be represented by a few parameters
describing geometry, location, material, and so on. The full-wave nonlinear
inversion algorithms introduced in this chapter apply to a pixel (or voxel) based
model. The parameterized model has far fewer unknowns, but it needs a priori
information. For example, the distribution of relative permittivity is required
to be piecewise constant.
The subspace concept used in the SOM can be applied to other inversion
methods as well to improve their performances in solving ISPs. For example, it
has been applied to the distorted Born iterative method (DBIM) in [109] and to
the contrast source extended Born (CSEB) method in [110]. This concept can
be also applied to inverse problems that are based on other physical principles,
such as the radiative transport equation in [111] and the electric impedance
tomography (EIT) problem in [112].
The concept of the degrees of freedom proposed in [32–34, 113] turns out
be very helpful in solving ISPs. By analyzing the property of the source-to-field
operator (6.5), Bucci et al. proved that electromagnetic fields radiated (or scattered) by bounded sources (or scatterers) can be accurately, up to a prescribed
approximation error, represented over a substantially arbitrary surface by a
finite and nonredundant number of samples. The concept of the degrees of freedom helps to determine the minimum number of receivers. By the principle of
reciprocity, the concept of degrees of freedom also helps in determining the
minimum number of required incident waves.
When multiple-frequency data are available, the frequency hopping
approach is powerful in improving reconstruction results [16, 114, 115]. In this
approach, low-frequency data are first used to perform the reconstruction, and
the resultant scatterer is used as an initial guess for the next higher-frequency
reconstruction. Substantial numerical and experimental results validate that
the frequency-hopping approach mitigates the effect of nonlinearity in the
optimization procedure so that the chance for an algorithm to be trapped
in local minima is significantly reduced. For low frequencies, the incident
wave with longer wavelengths interact with scatterers, and consequently
the corresponding scattered field mainly contains large-scale information
of scatterers; that is, low spatial-frequency information. The nonlinearity
of inverse scattering problems is significantly reduced at low frequencies.
When frequencies become higher, the scattered field contains more and more
higher spatial-frequency information. From this perspective, the frequency
hopping approach shows similarity to the nested-scheme implementation of
the NFFT-SOM that is presented in Section 6.4.4, with the difference that the
former is a physical (or experimental) implementation of the hopping and the
latter is a numerical implementation. We mention in passing a relevant but
different hopping scheme, that is, the iterative multiscaling method presented
in [116], where the hopping is directly applied to the physical spatial space.
173
174
Electromagnetic Inverse Scattering
This chapter has not applied or discussed, in an explicit way, regularization
techniques yet. Roughly speaking, in order to induce stability and to incorporate a priori information about the desired solution, a regularization functional
(also known as the penalty functional) should be added to the original objective function. In a certain sense, the determination of the number L of the
leading singular values in the SOM can be considered to be a kind of regularization. If a priori information about the scatterer is available, such as being
piece-wise homogenous or lossy, then the total variation (TV) penalty term or
the nonnegativeness constraint should be used together with the original objective function. Appendix A and references therein provide detailed theoretical
and practical issues on regularization.
When numerical synthetic data are used for inversion, the inverse crime
should be avoided. When the same numerical procedure is used to generate
synthetic data and to reconstruct unknowns, it often yields overoptimistic
results, which is called the inverse crime. For example, for an ill-conditioned
inverse problem with continuous unknowns, if the discretization mesh is not
fine enough so that the nature of ill-condition has not yet appeared, and if
the same numerical procedure is used in both the forward solver and inverse
solver, then the inversion results are often trivially satisfying. One possibility
to avoid the inverse crime is to use different mathematical/numerical models
for the forward and the inverse procedures. The other possibility is to use
different meshes in the forward and the inverse procedures, although the same
numerical model is used for both the forward and the inverse procedures.
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7
Reconstructing Perfect Electric Conductors
This chapter deals with reconstructing perfectly electrically conducting
(PEC) scatterers where the boundary of PEC scatterers is reconstructed
from measured scattered data. Since the boundary condition requires that
the tangential electric field vanishes at the boundary of PEC scatterers, the
electric-field integral equation (EFIE) is only applicable to the boundary
of the PEC scatterer, which is, however, unknown in an inverse problem.
Consequently, the ISP problem involving PEC scatterers is quite different from
its dielectric counterpart discussed in Chapter 6.
The organization of this chapter is as follows. Section 7.1 first briefly introduces the forward problem, followed by a review of fundamental properties
of the inverse problem. Section 7.2 provides a survey of inversion models that
require knowledge of prior information about PEC scatterers. Section 7.3
focuses on inversion models that do not require prior information. Both TM
and TE modes are dealt with. Section 7.4 presents a method to reconstruct a
mixture of PEC and dielectric scatterers. Section 7.5 discusses a few issues on
the reconstruction of PEC scatterers.
7.1 Introduction
7.1.1
Formulation of the Forward Problem
The inverse scattering problem (ISP) under investigation is in a two-dimensional
setting with time-harmonic illuminations. In other words, the whole domain
of interest, including the unknown scatterers, as well as the incident electrical
field, is invariant along the z-axis. The background medium is free space and its
permittivity and permeability are denoted as 𝜖0 and 𝜇0 , respectively. Consider
a PEC scatterer, occupying a space D, with the boundary 𝜕D.
For the transverse magnetic (TM) mode, both the electric field E and
the induced electric current density J are in the z-axis direction, where
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
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Electromagnetic Inverse Scattering
the subscript z is omitted for convenience. The scalar electric-field integral
equation (EFIE) reads
Ei (r) + i𝜔𝜇0
∫𝜕D
g(r, r′ )J(r′ )dr′ = 0,
r ∈ 𝜕D
(7.1)
For the transverse electric (TE) mode, both the electric field E and the
induced electric current density J are in the transverse plane, and the vector
EFIE reads
{
}
i
′
′
′
̂ × E (r) + i𝜔𝜇0
n(r)
G(r, r ) ⋅ J(r )dr = 0, r ∈ 𝜕D
(7.2)
∫𝜕D
where the two-dimensional dyadic Green’s function is
]
[
1
′
G(r, r ) = I 2 + 2 ∇t ∇t g(r, r′ )
k0
where the subscript t means “transverse,” and k0 is the wavenumber.
In the forward scattering problem, the induced surface current density J and
J can be solved from (7.1) and (7.2), respectively. The scattered field is then
calculated as the convolution of the induced surface current density and the
Green’s function,
Es (rs ) = i𝜔𝜇0
∫𝜕D
g(rs , r′ )J(r′ )dr′
(7.3)
G(rs , r′ ) ⋅ J(r′ )dr′
(7.4)
for the TM case and
Es (rs ) = i𝜔𝜇0
∫𝜕D
for the TE case, where rs is the position of measurement antenna.
7.1.2
Uniqueness and Stability
The first question to ask about the inverse scattering problem regards uniqueness. Since the proof is mainly given by mathematicians, this section introduces
the terminologies that are used in applied mathematical communities. Impenetrable scatterers are referred to as obstacles. For a scalar wave, impenetrable scatterers that require the total field to be equal to zero at the boundaries
are known as sound-soft scatterers. For ISPs involving sound-soft scatterers,
the first proof, due to Schiffer, was described in the book of Lax and Phillips [1].
For a fixed wavenumber, measurement of scattered far field in all directions
for all incidence directions uniquely determines the sound-soft scatterer. For
polyhedral sound-soft scatterers, [2, 3] established uniqueness for only a single incident plane wave. For electromagnetic waves, the uniqueness theorem
can be found in section 7.1 of [4], which states the following two theorems.
First, for a fixed wavenumber, the electric far field pattern for all incident directions and all polarizations uniquely determines the PEC scatterer. Second, for
one fixed incidence direction and polarization, the electric far field pattern
Reconstructing Perfect Electric Conductors
for all wavenumbers contained in some interval uniquely determines the PEC
scatterer. For polyhedral-type PEC scatterers, [5] established uniqueness for
only two incident plane waves given at a fixed wavenumber and a fixed incident
direction, along with two different polarizations.
The next question is whether the solution of inverse scattering problem is stable. The stability estimates for the determination of a sound-soft scatterer turns
out to be of logarithmic type [6–8], that is, as introduced in Section 6.1.1, if the
error in the measured data is at most 𝜏, then the error of solution in the worst
case is on the order of | ln 𝜏|−s (where 0 < s < 1), which is severely ill-posed.
This issue of ill-posedness will be handled using regularization techniques.
This chapter mainly presents numerical methods to solve two-dimensional
ISPs involving PEC scatterers. Here, we categorize inversion models by whether
or not they need to know a priori the number of scatterers and their topologies.
7.2 Inversion Models Requiring Prior Information
Many inversion models require parameterization of the boundary of PEC using
local polar coordinates. Usually, a point inside a scatterer is chosen as the origin
of the local polar coordinate system and then the boundary 𝜕Ω of scatterer is
parameterized as an angular-dependant distance function. Spline functions or
Fourier series under local coordinate are most commonly used [9, 10]. Obviously, this parameterization method requires a priori information about the
number and the approximate locations of the scatterers.
The ISP is then cast into an optimization problem, where the parameters
of shape function are looked for so that the mismatch between the calculated
and measured scattered fields is minimized. An optimization algorithm can
be either deterministic, such as the Newton–Kantorovich (NK) method in
[11], where the Fréchet derivative is calculated, or stochastic, such as the
differential evolution (DE) in [10] and the genetic algorithm (GA) in [9], where
there is no need to calculate the Fréchet derivative. Such methods require
the solution of the corresponding forward problem at each iteration step,
which is often time consuming. It is worth mentioning that two inversion
models developed in applied mathematics do not need the solution of the
forward problem ([12] and sections 5.4 and 5.5 of [4]). The principal idea of
these methods is to break the ISP into a linear ill-posed part and a nonlinear
part [13]. The first algorithm, referred to as the dual space method, adopts
an approach of linear superposition of incident fields [14, 15]. The first step
looks for superpositions of incident fields with different directions so that
the resulting scattered fields match certain multipole radiation patterns in
far field and the second step solves a nonlinear minimization problem where
the unknown boundary 𝜕Ω of the scatterer is determined as the locations of
the zeros of the total electric field. The second algorithm first solves a linear
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Electromagnetic Inverse Scattering
ill-posed problem of constructing a near-field scattered field from measured
far-field counterpart and then deals with the nonlinearity by determining the
unknown boundary 𝜕Ω of the scatterer as the location of the zeros of the total
field [16]. Though the two algorithms are mathematically elegant, they have
not been widely used in the engineering community yet due to some practical
constraints. For example, the first algorithm requires full-aperture incidences
and receptions and the second algorithm requires a reasonable auxiliary closed
contour contained inside the unknown scatterer.
When a priori information about the number and the approximate locations
of the scatterers is absent, the parameterization method cannot work. In addition, for scatterers with complex shapes, such as 𝔽 and 𝕊 that exhibit several
concave parts, it is impossible to parameterize them using local polar coordinates. Thus, it is necessary to develop inversion models that do not need the
aforementioned a priori information.
7.3 Inversion Models Without Prior Information
The absence of a priori knowledge of the number and the approximate locations of the scatterers poses two significant difficulties; that is, representing the
geometry of scatterers and constructing the objective function. While the first
is obvious, the second deserves further discussion. Due to the boundary condition on the PEC scatterer, the methods for solving the PEC inverse scattering
problems are significantly different from those for dielectric scatterers. In the
case of dielectric scatterers, both the background medium and the scatterers
can be parameterized by the permittivity such that the EFIE can be applied to
the whole domain of interest (DOI). Therefore, the objective function can be
constructed as a function of the unknown permittivity. In the case of PEC, the
EFIE is only applicable to the boundary of the PEC scatterer, which is, however,
unknown in inverse problem. A physical parameter to represent both the PEC
scatterers and the background medium does not exist. Therefore, the objective
function for the PEC case is quite different from the one for the dielectric case.
In the following, we briefly review four inversion models that do not require
the aforementioned a priori information.
Qualitative methods, such as the linear sampling method [17], multiple signal
classification (MUSIC) algorithm [18], and factorization method [19], are able
to locate PEC scatters in the absence of a priori knowledge of their number and
approximate locations. Nevertheless, no exact criteria exist in identifying the
boundaries of scatterers from the indicator of qualitative methods. In addition,
the reconstruction results are very sensitive to measurement noise.
The local shape function (LSF) imaging seeks to reconstruct arbitrary number of PEC scatterers distributed in the DOI [20–22]. LSF imaging provides
a complete basis for PEC scatterers by discretizing the DOI into small cells
(which are usually squares) and then assigning to each cell a binary local shape
Reconstructing Perfect Electric Conductors
function that is zero in free-space regions and unity in regions where PEC scatterers are present. Hence, the LSF is more versatile in representing multiple
scatterers. It is easy to see that this technique does not require a priori knowledge about the number of scatterers and their approximate locations. However,
when dealing with line-shape PEC scatterers (such as the “L” shape), one has
to use very small squares in order to give a good approximation, which may
significantly increase the computational cost.
PEC scatterers can also be reconstructed by treating them as lossy dielectric
scatterers with high loss [23]. The reconstruction algorithm is, in principle, the
same as that used for reconstructing the conductivity of a penetrable scatterer.
Considering the fact that, for high conductivity, the skin depth of the scatterer
is small, the only meaningful information produced by the reconstruction algorithm is the boundary of the scatterer.
The level set method, devised by Osher and Sethian [24], was first applied to
solve ISPs in [25]. The key idea is to define the boundary of the scatterer as the
level set of a function of higher dimension. A level set of a real-valued function is a set where the function takes a given constant value. For example, for
a two-variable function, a level set is a curve, called a level curve, isoline, or
contour line. For a three-variable function, a level set is called a level surface or
isosurface. The advantage of the level set model is that one can perform numerical computations involving curves and surfaces without having to parameterize
these objects. Also, the level set method makes it easy to follow shapes that
change topology, such as splitting, merging, and developing holes. Although
the level set method is versatile in representing scatterers of arbitrary number
and shape [26], inverse-problem solvers that are based on this representation
require the solution of the corresponding forward problem at each iteration
step, which is often time consuming.
In the following, we present a model that is able to reconstruct both closedcontour and line-shape PEC objects and does not need to solve the forward
problem at each iteration step. In practice, an alternative method for providing a complete bases for PEC scatterers is to use the line-element representation. The domain of interest is discretized into small square cells and then side
edges of square rather than the square itself are used as the basic elements to
represent scatterers, which is a significant difference from the aforementioned
LSF method. It is obvious that the line-element representation method is able
to reconstruct both closed-contour and line-shape objects, in the absence of
a priori information about the number and the approximate locations of the
scatterers. Based on the line-element representation, an inversion model is proposed to solve two-dimensional ISP under both TM and TE cases, where the
forward problem is not solved at each iteration step.
7.3.1
Transverse-Magnetic Case
Suppose that the unknown PEC scatterers are located in a given DOI D ⊂ R2 .
There are Ni incident waves Eip (r) = ẑ Epi illuminating D, p = 1, 2, ..., Ni . For each
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Electromagnetic Inverse Scattering
incidence, the scattered electric field is measured by an array of Ns antennas,
which are located at rsq , q = 1, 2, ..., Ns . The domain of interest is discretized into
small square cells and the side edges of the square are used as the elements to
represent PEC scatterers. Assume that there are M line elements in total in the
domain D and the center of each line-element is located at rm , m = 1, 2, ..., M.
After such a discretization, the method of moments (MoM), employing the
pulse basis function and the delta test function, can be used to solve the forward
problem (see Section 2.9.2).
Define the following three M-dimensional vectors: the vector of induced
current density J = [J(r1 ), J(r2 ), ..., J(rM )]T , the vector of incident field
i
t
E = [Ei (r1 ), ..., Ei (rM )]T , and the vector of total field E = [Et (r1 ), ..., Et (rM )]T ,
where the superscript T denotes the transpose operator. The relationship
between them is given in a compact form,
t
i
E = E + GD ⋅ J,
where the M × M matrix
(7.5)
k 𝜂𝑤
GD has the entries GD (m, m′ ) = − 04 H0(1) (k0 |rm −
{
[ (
]}
k 𝜂𝑤
𝛾k 𝑤 )
1 + i 𝜋2 ln 40 − 1
GD (m, m′ ) = − 04
when
rm′ |) when m ≠ m′ , and
m = m′ . Here k0 is the free space wavenumber, 𝜂 is the impedance of the free
space, 𝑤 is the length of the line element, and 𝛾 ≈ 1.781. The physical meaning
of (7.5) is that the total field is equal to the incident wave coming directly from
transmitters plus the radiation of induced current.
The scattered field received by the antennas is given by
s
E = GS ⋅ J,
where E
s
(7.6)
= [E (rs1 ), Es (rs2 ), ..., Es (rsN )]T
s
k 𝜂𝑤
= − 04 H0(1) (k0 |rsq − rm |).
s
and the Ns × M matrix GS is given by
GS (q, m)
For the forward problem, the requirement that the total electric field at the
boundary of a PEC scatterer vanishes enables us to solve for the induced current J from (7.5). For an inverse problem, however, the PEC boundaries are
unknown yet so that we cannot simply let (7.5) to be zero. Consequently, it
is difficult to build up the objective function. To overcome this difficulty, we
define an M-dimensional vector P, which consists of only 1 or 0, as an indicator of whether an edge belongs to the PEC boundary. In other words, the
dimension of the vector P is equal to the total number of line elements in D,
where a “1” element represents the PEC element and a “0” element represents
the background (air) element. Because of the property of PEC, the total electric
t
field E should vanish in the PEC scatterer, while the induced current J should
only exist on the elements that belong to the PEC boundary. Noticing such a
physical fact, we are able to define the relative residual,
Δsta =
||(I − P) ⋅ J||2
+
||J ||2
t
+
||P ⋅ E ||2
+
||E ||2
(7.7)
Reconstructing Perfect Electric Conductors
where || ⋅ || is the Euclidean length of a vector, P is the diagonal matrix with
P in the diagonal, and I is an M-dimensional identity matrix. For a physical J,
it can be easily checked that both terms in (7.7) are equal to zero no matter
the element of P is equal to “1” (PEC boundary) or “0” (air). Since the fact that
the electric field is zero at the PEC and the current exists on the PEC boundary describes a physical state, (7.7) quantifies a violation of the physical state
and therefore it is referred to as the relative residual in the state equation. The
+
+
denominators J and E , which will be introduced later, have the same unit as
t
J and E , respectively, so that the two terms in (7.7) are both dimensionless.
Following the subspace-based optimization method (SOM) that is proposed
in Chapter 6, we decompose the current J into two complementary and orthog+
−
onal parts, J = J + J . The key step is the singular value decomposition (SVD)
H
of GS , which reads GS = U ⋅ Σ ⋅ V , where U is of size Ns × Ns and is composed of orthonormal left singular vectors uq , V is of size M × M and is composed of orthonormal right singular vectors 𝑣m , Σ is of size Ns × M with the
diagonal terms being singular values 𝜎m that are placed in a nonincreasing order
+
𝜎1 ≥ 𝜎2 ≥ · · ·, and the superscript H denotes the Hermitian. Explicitly, J = V ⋅
−
+
−
𝛼 + + V ⋅ 𝛼 − , where V and V are composed of the first L and the remaining
M − L columns of V , respectively, and L is the number of the total singular values that are above a predefined noise-dependent threshold. The superscripts +
+
and − denote major and minor parts, respectively. The major part of current J
+
is straightforwardly determined from V ⋅ 𝛼 + where
𝛼j+
=
uH
⋅E
j
𝜎j
s
,
j≤L
(7.8)
+
+
The corresponding radiation field in the DOI D is calculated as E = GD ⋅ J .
+
+
We recall that J and E are used as denominators in (7.7). The relative residual
due to the mismatch of the scattering data can be expressed as
−
dat
+
s
||Gs ⋅ V ⋅ 𝛼 − + Gs ⋅ J − E ||2
(7.9)
s
||E ||2
which is referred to as the relative residual in the data equation. The total relative residual is defined to be
Δ
=
Δtot = Δdat + Δsta .
(7.10)
i
For each of the incidence Ep , the total relative residual can be calculated as Δtot
p ,
p = 1, 2, ..., Ni .
Similar to the case of Gs-SOM for dielectric scatterers that has been
presented in Section 6.4.1, there are two approaches to build up the objective
function. The first approach treats 𝛼 − as an intermediate parameter, and it can
be expressed as a function of P by solving a least-squares problem. Thus, the
189
190
Electromagnetic Inverse Scattering
total relative residual depends solely on the binary variable P. This approach is
adopted in [27], where the discrete descent optimization method is chosen to
minimize the objective function. However, numerical simulations show that
this approach exhibits two weaknesses: (1) the computational cost is large since
there is matrix inversion in each step of iteration; (2) the method is not robust
in the sense that the performance of the algorithm depends, in a noncontinuous manner, on the number L of leading singular values. The second approach
treats 𝛼 − as an independent parameter, which is the practice in [28]. Another
important ingredient is to replace the binary indicator function P by a smooth
function of another unknown x that is continuous. Mathematically speaking,
the Heaviside step function (or the unit step function) can be approximated by
several smooth functions, for example,
P=
1
1 + e−ax
(7.11)
where a is a large positive number, such that for each element xm ,
m = 1, 2, ..., M, if xm > 0, Pm ≈ 1 and if xm < 0, Pm ≈ 0. The replacement
of P by a function of x enables us to apply various continuous-parameter
optimization methods to minimize the objective function. The objective
function is given by
f (𝛼 −1 , 𝛼 −2 , … , 𝛼 −Ni , x) =
−
+
s
Ni
∑
||Gs ⋅ V ⋅ 𝛼 −p + Gs ⋅ J p − Ep ||2
s
p=1
||Ep ||2
+
||(I − P) ⋅ J p ||2
+
||J p ||2
t
+
||P ⋅ Ep ||2
+
||Ep ||2
(7.12)
Reference [28] adopts the optimization method used in Gs-SOM (see
Section 6.4.1) to minimize (7.12); that is, alternatively updating 𝛼 −p and x. The
implementation steps are as follows:
+
Step 1: Calculate GS , GD , and the SVD of GS . Obtain J p from (7.8),
p = 1, 2, ..., Ni .
Step 2: Initialization: iteration number n = 0, set x0 = 0, 𝛼 −p,0 = 0 (alternatively,
it is obtained from (7) of [27]); Initialize the search direction 𝜌p,0 = 0.
Step 3: n = n + 1.
Step 3.1: Update 𝛼 −p,n : Calculate gradient (Frechet derivative)
g p,n = ∇𝛼 −p f evaluated at 𝛼 −p,n−1 and xn−1 ; Determine the
Polak–Ribière conjugate gradient search directions 𝜌p,n =
Re[(g p,n −g p,n−1 )H ⋅g p,n ]
g p,n +
𝜌p,n−1 . Plug 𝛼 −p,n = 𝛼 −p,n−1 + dp,n 𝜌p,n
||g p,n−1 ||2
into the objective function, which is quadratic in terms of
the parameter dp,n , and obtain the value of dp,n by solving
a least-squares problem (see Appendix B). Then update
𝛼 −p,n = 𝛼 −p,n−1 + dp,n 𝜌p,n .
Reconstructing Perfect Electric Conductors
Step 3.2: Update xn : for the mth line element, m = 1, 2, ..., M,
−
+
update induced current (J p,n )m = (J p )m + (V ⋅ 𝛼 −p,n )m .
Then update the total field in the mth line element,
t
i
(Ep,n )m = (Ep )m + (GD ⋅ J p,n )m . The objective function has an
analytical derivative with respect to (xn )m and the solution
(xn )m is given by
]
[ N
t
/ Ni
∑i |(Ep,n )m |2
∑ |(J p,n )m |2
1
(xn )m = − ln
+ 2
+ 2
a
p=1 ||E p ||
p=1 ||J p ||
(7.13)
Step 4: Stop iteration if there is no obvious change in the objective function for
continuous three iterations. Otherwise, go to step 3.
Step 5. Determine the binary result P from x: for each element m = 1, 2, ..., M,
if xm > 0, then set Pm = 1; otherwise, Pm = 0.
The following numerical simulation is presented to demonstrate the performance of the proposed algorithm. The DOI D is 2𝜆 × 2𝜆, which is discretized
into 20 × 20 cells when solving the inverse problem. The scatterers are a combination of a closed-contour scatterer and a line-shape scatterer. A square and
an L-shaped PEC scatterer are located in the domain as shown in Fig. 7.1(a),
where the units for the coordinates in the figure are 𝜆. The line elements are
represented by thin line elements. The contour of the PEC objects is represented by thick line elements. A total number of Ni = 10 plane waves are incident from directions evenly distributed on [0, 2𝜋). Ns = 30 receivers are equally
distributed along a circle with radius 5𝜆. The forward scattering problem is
solved by the MoM, with a 40 × 40-cell discretization, denser than is used in
the inverse problem to avoid the inverse crime.
Figure 7.1 (b) shows the reconstructed result by using noise-free data and
choosing L = 12, while Fig. 7.1 (c) applies to the case with 10% white Gaussian
noise and L = 10. The reconstructed pattern is represented by the line elements
marked with triangles at their ends. The numbers of iteration steps are 384 and
410, respectively. From the reconstructed results we clearly see that there is an
“L” shaped scatterer and a square scatterer in the domain and their sizes and
positions approximately match the exact ones.
In some other numerical simulations, occasionally some internal edges
of a closed-contour PEC scatterer are “incorrectly” identified as air. In fact,
since the PEC scatterer is impenetrable, it does not matter whether internal
edges are detected as PEC or air as long as the boundary is correctly detected
as PEC.
From a practical point of view, many properties of Gs-SOM that are presented
in Section 6.4.1 for dielectric scatterers are applicable to PEC scatterers as well.
For example, there are good guidelines in choosing the integer L; the overhead
computational cost can be reduced by implementing a truncated SVD.
191
192
Electromagnetic Inverse Scattering
1
0.5
0
–0.5
–1
–1
0
(a)
1
1
0.5
0.5
0
0
–0.5
–0.5
–1
–1
0
(b)
1
–1
–1
1
0
(c)
1
Figure 7.1 Reconstruction of a combination of closed-contour and line-shape PEC
scatterers. (a) Exact contour, (b) reconstructed contour with noise-free data, and
(c) reconstructed contour under 10% white Gaussian noise. Source: Ye 2011, Inverse
Problems, 27, 055011. [28] Reproduced with permission of IOP Publishing.
7.3.2
Transverse-Electric Case
This section intends to reconstruct PEC scatterers with TE illumination,
without requiring a priori information about the location and quantity of
scatterers. The PEC scatterer can be a closed-contour or line-shape scatterer.
We still use the line-element model to represent scatterers. It is worth stressing
that the TE case ISP is more demanding than the TM case. The reasons are
threefold. (1) The induced electric current for the TM case is in the longitudinal
direction, which is in fact a scalar source for a two-dimensional problem. In
comparison, the TE case involves a vector source since the induced current
flows in the transverse plane. Thus, the TE case is more complex in terms of
both mathematics and physics. (2) The vector property of the induced current
requires careful numerical implementation, especially in the following two
Reconstructing Perfect Electric Conductors
aspects. The first is that the basis function for the current cannot be chosen
as the simple pulse function as in the TM case, because the discontinuous
distribution current will cause fictitious charge accumulation at the junctions
of adjacent elements. The triangular basis function is usually used in solving
the integral equation in the TE case. The second is that the strong effect of
current orientation on the radiation field requires a more accurate model
than the staircase that consists of only horizontal and vertical line elements.
To better represent current that flows in oblique directions, 45-degree and
135-degree oblique line elements have to be added, which will be shown in
later numerical simulations. (3) Although the MFIE is very convenient in
solving the forward problem since only a single component of the magnetic
field H is present, it brings difficulties in imposing boundary conditions in
inverse problems. We do not know which side of a line element is the inner side
of a closed-contour PEC scatterer since the PEC scatterer is as yet unknown.
In addition, the MFIE cannot be applied to a line-shape scatterer, even in the
forward problem [29] (section 1.7). The above factors make it necessary to
use the more complex EFIE, which is, however, convenient in dealing with the
boundary condition in inverse problems.
We use the same notations as in the TM case. A minor difference is that a
total number of M line elements consists of not only horizontal and vertical
line elements, but also 45-degree and 135-degree oblique line elements. The
total electric field on each line element is denoted as Etp and is composed of
the incident field from the sources and the scattered field due to the induced
currents on the PEC scatterers. The relationship between the total field Etp , the
incident field Eip and the induced current on the elements Jp can be written in
an alternative way to (7.4),
Etp (rn ) = Eip (rn )
[
M
∑
k0 𝜂0 (1)
H0 (k0 |rn − r′m |)Jp (r′m )
4
m=1
element m
]
𝜂0
(1)
′
′
′
+
∇H0 (k0 |rn − rm |)∇ ⋅ Jp (rm ) dr′m
4k0
−
∫
(7.14)
where Jp (r′m ) = Jp (r′m )t̂m and t̂m denotes the tangential direction of the mth line
element, m = 1, 2, … , M.
Similarly, the scattered field received by the receiving antennas Esp (rsq ) is given
by
[
M
∑
k0 𝜂0 (1)
H0 (k0 |rsq − r′m |)Jp (r′m )
Esp (rsq ) = −
∫
4
m=1
element m
]
𝜂0
(1)
s
′
′
′
+
∇H0 (k0 |rq − rm |)∇ ⋅ Jp (rm ) dr′m
(7.15)
4k0
193
194
Electromagnetic Inverse Scattering
To avoid fictitious charge accumulation at the junctions of adjacent line
elements that is caused by discontinuous current distribution, we choose
the triangle basis function to represent the current [29] (section 2.4). After
simple algebraic manipulations, (7.14) and (7.15) can be written in compact
forms:
t
i
E = E + GD ⋅ J,
(7.16)
s
E = GS ⋅ J,
t
(7.17)
i
s
where E and E are 2M-dimensional vectors, E is 2Ns -dimensional vector, J
is an M-dimensional vector, and GD and GS are of size 2M × M and 2Ns × M,
respectively.
The objective function can be written in a way similar to the TM case,
f (𝛼 −1 , 𝛼 −2 , … , 𝛼 −Ni , P)
⎧
−
s 2
+
Ni
−
∑
⎪ ||Gs ⋅ V ⋅ 𝛼 p + Gs ⋅ J p − Ep ||
=
⎨
s
||Ep ||2
p=1 ⎪
⎩
1
+
||(I − P) 2 ⋅ J p ||2
+
||J p ||2
1
⎫
t
||P 2 ⋅ (n̂ × Ep )||2 ⎪
+
⎬
+
||Ep ||2
⎪
⎭
(7.18)
where n̂ = ẑ × t̂ is the normal direction of the line element. There are two
important differences from the TM objective function (7.12). The first is
t
that the transverse tangential component of total electric field n̂ × Ep , rather
than the z component, vanishes at the PEC line element. The second is that
1
1
(I − P) 2 and P 2 , rather than (I − P) and P, are used in the relative residual in
the state equation. It is easy to see that the derivative of (7.18) with respect
+
to Pm , m = 1, 2, … , M, equals to −Am + Bm , where Am = |J p,m |2 ∕||J p ||2 and
t
+
Bm = |(n̂ × Ep,m )|2 ∕||Ep ||2 , which is a constant. Then the objective function
(7.18) is linear with respect to Pm in the region [0, 1] so that the minimum of
(7.18) occurs at either Pm = 0 or Pm = 1. A simple criterion of choosing the
value of Pm is
{
0, if Am ≤ Bm
Pm =
(7.19)
1, if Am > Bm
Thus, the auxiliary function xm as defined in the TM case is no longer needed.
The implementation steps of minimizing (7.18) are almost the same as the
TM case; that is, alternatively updating 𝛼 −p and P. Some numerical simulations
are performed to test the proposed algorithm. The DOI D is a 2𝜆 × 2𝜆 square,
where 𝜆 is the wavelength. A group of 15 transmitting antennas and 30 receiving
1
1
0.5
0.5
y (λ)
y (λ)
Reconstructing Perfect Electric Conductors
0
–0.5
–1
–1
0
–0.5
–0.5
0
x (λ)
(a)
0.5
1
–1
–1
–0.5
0
x (λ)
0.5
1
(b)
Figure 7.2 Reconstruction results of rotated line-shape PEC scatterer. (a) 24∘ and (b) 45∘ .
Source: Shen 2013, IEEE Trans. Antennas Propag., 61, 4713–4721. [50] Reproduced with
permission of IEEE.
antennas are evenly distributed on [0, 2𝜋], with a distance of 5𝜆 from the center
of D. Both the x and y components of scattered field are measured. In addition,
10% white Gaussian noise is added.
The first numerical example deals with a line-shape scatterer. The domain D
is discretized to 20 × 20 horizontal line elements, 20 × 20 vertical line elements,
2 × 20 × 20 line elements in the 45∘ direction and 2 × 20 × 20 line elements in
the 135∘ direction. In solving inverse problems, unknown scatterers are not
necessarily aligned with the mesh lines. To challenge the proposed method,
we rotate the line-shape scatterer by 24∘ counterclockwise, as shown by the
thick line in Fig. 7.2(a). In the reconstruction result, it is noticed that although
the real object does not parallel with any mesh line, the algorithm still obtains
the correct location, length, and angle of inclination of the scatterer, though
there are some artifacts. The mesh strategy, including four kinds of line elements, favors the reconstruction of unaligned scatterers. Next, for a line with
45∘ counterclockwise rotation that is aligned with the mesh, the reconstruction result shown as the thick line in Fig. 7.2(b) clearly presents fewer artifacts
compared with the unaligned case.
In the second numerical example, in order to challenge the propped method
in solving multiple scatterers, two circular PEC cylinders are investigated.
Shown as the thick lines depicted in Fig. 7.3, one circle has a 0.3𝜆 radius and is
centered at (−0.45𝜆, 0) and the other one with 0.1𝜆 radius centered at (0.25𝜆, 0).
The reconstruction result by adopting a TSOM scheme (Section 6.4.2) is shown
in Fig. 7.3. Though the problem is very challenging, with different radii and
sub-wavelength gap, the reconstruction result shown in Fig. 7.3 is satisfactory.
195
Electromagnetic Inverse Scattering
1
Figure 7.3 Reconstruction results
of multiple circular PEC scatterers.
Source: Shen 2013, IEEE Trans.
Antennas Propag., 61, 4713–4721.
[50] Reproduced with permission
of IEEE.
0.5
y (λ)
196
0
–0.5
–1
–1
–0.5
0
x (λ)
0.5
1
7.4 Mixture of PEC and Dielectric Scatterers
This section considers a mixed boundary condition inverse scattering problem
where PEC and dielectric scatterers are simultaneously present inside the DOI.
The number of scatterers and their approximate positions not known a priori. Regarding the uniqueness theorem of such a mixed boundary condition
ISP, [30] provides an explicit and constructive proof under certain conditions.
The mixed boundary condition ISP is obviously more challenging than the single boundary condition case; that is, to reconstruct PEC scatterers alone or
dielectric scatterers alone. Existing inversion methods to solve mixed boundary
condition ISP can be categorized as qualitative and quantitative.
Qualitative reconstruction methods, such as the linear sampling method [17]
and factorization method [19], are, in general, independent of the boundary
conditions. This salient feature makes them good candidates to solve the mixed
boundary ISP. Nevertheless, qualitative methods inherently present two weaknesses. The first is that although they are able to retrieve the shapes and locations of scatterers, they cannot distinguish the types of boundary conditions.
The second is that they do not provide the values of relative permittivity of
dielectric scatterers.
Quantitative reconstruction methods not only identify the shapes of PEC
scatterers but also at the same time provide the spatial distribution of the
relative permittivity of dielectric scatterers. A natural idea to represent both
dielectric and PEC scatterers is to use complex permittivity considering the
fact that for high conductivity the skin depth of the scatterer is small so that
the only meaningful information produced by the algorithm is the boundary of
the scatterer [31, 32]. Then both the forward and inverse scattering problems
can be modeled by the standard EFIE, involving a spatial distribution of
complex permittivity. The PEC is distinguished from dielectric scatterers by
Reconstructing Perfect Electric Conductors
a higher value of the imaginary part of complex permittivity. However, this
classification criterion may fail in the case when both high-loss dielectric
scatterers and PEC are simultaneously present, especially when measured
scattered fields are contaminated with considerable noise. Another approach
to tackle the mixed boundary condition problem is the T-matrix method. The
T-matrix method [33] is an important computational technique for solving
wave scattering problems. For any given scatterer, no matter whether it is PEC
or dielectric, the scattered and incident fields are firstly expanded as functions
of multipoles and then multipole coefficients of scattered field are related to
the multipole coefficients of incident field by the so-called T-matrix. Scatterers
with different materials, shapes, and sizes have different T-matrix values. Thus,
the T-matrix method provides a possibility of modeling the mixed boundary
problem for both forward and inverse scattering problems.
We consider a 2D TM mixed boundary condition inverse scattering problem.
The DOI is discretized into a total number of M square meshes and each square
is small enough so that it can be well approximated by a circle of the same area.
We have k0 R ≪ 1, where R is the radius of approximation circle. The T-matrix
model for the scattering problem consists of two steps, that is, to first study the
scattering property of a single cell that is referred to as the subscatterer and then
to study the scattering property of all subscatterers as a whole. In the following,
we present the two steps in sequence.
First, consider a single subscatterer, the center of which is located at
C0 = (r0 , 𝜃0 ) in the cylindrical coordinate system. It is illuminated by an
incident field, which can be represented by the multipole expansion as
i
E (r) =
P
∑
T
Rg[Ψ(k0 , r′ )]p ⋅ [e0 ]p = RgΨ (k0 , r′ ) ⋅ e0 ,
(7.20)
p=−P
and the scattered field is represented by
s
E (r) =
P
∑
T
[Ψ(k0 , r′ )]p ⋅ [a0 ]p = Ψ (k0 , r′ ) ⋅ a0
(7.21)
p=−P
where r′ = r − C0 = (𝜌0 , 𝜙0 ) is under the local coordinate of the subscatterer at
C0 . Here [Ψ(k0 , r′ )]p = Hp(1) (k0 𝜌0 )eip𝜙0 , p = −P, … , P, where P is the truncation
number of multipoles. The Rg indicates the regular part of Hankel function;
that is, Rg[Ψ(k0 , r′ )]p = Jp (k0 𝜌0 )eip𝜙0 . The vectors e0 and a0 are the multiple
expansion coefficients of the incident and scattered field, respectively. Ψ, e0 ,
and a0 are all (2P + 1)-dimensional vectors. The superscript T denotes the
transpose. The scattering ability of the subscatterer is represented by Tp ,
p = −P, … , P, which is defined as the ratio of the multipole-expansion coefficients of the scattered field to those of its incident field; that is, [a0 ]p = Tp ⋅ [e0 ]p .
The values of Tp depend on the material, size, and shape of the subscatterer.
The T-matrix of the subscatterer is defined as T = diag(Tp ); that is, a diagonal
197
198
Electromagnetic Inverse Scattering
matrix with Tp placed in order on the diagonal. To conclude, for a subscatterer,
we have
a0 = T ⋅ e0
(7.22)
Next, the scattering of the original scatterer can be considered as the superposition of the scattered field due to all subscatterers. Assume that the center
of each subscatterer is at Cm = (rm , 𝜃m ), m = 1, 2, … , M. The vectors em and
am , m = 1, 2, … , M, denote the multiple expansion coefficients of the incident
and scattered field, respectively, for the mth subscatterer. The scattered field is
given by
s
E (r) =
M
∑
T
Ψ (k0 , r′m ) ⋅ am
(7.23)
m=1
where r′m = r − Cm = (𝜌m , 𝜙m ), m = 1, 2, … , M, is the local coordinate with
respect to the mth subscatterer. For the mth subscatterer, the total incidence
field upon it consists of the incidence wave that directly comes from the transmitter (em ) and the scattered wave off other subscatterers (Es (rm )). The latter
needs some manipulations to shift the expansion center to Cm before the
definition of T m for the mth subscatterer can be used. This is achieved by the
translational addition theorem,
T
T
Ψ (k0 , r′m′ ) = RgΨ (k0 , r′m ) ⋅ 𝛼 mm′ ,
m ≠ m′
(7.24)
where 𝛼 mm′ , of size (2P + 1) × (2P + 1), is the translational matrix, which can
be derived from Appendix D of [34]. Thus, using the definition of the T-matrix
for the mth subscatterer and using (7.23) and (7.24), we obtain
]
[
M
∑
(7.25)
am = T m ⋅ e m +
𝛼 mm′ ⋅ am′
m′ =1, m′ ≠m
Note that (7.25) is applied to all M subscatterers, which eventually generates a
linear system,
a = O ⋅ [e + AD ⋅ a]
(7.26)
where O has a dimension M(2P + 1) × M(2P + 1), and it is a block-wise diagonal matrix with block entries [O]mm = T m . AD has a dimension M(2P + 1)
× M(2P + 1), with block element [AD ]mm′ = 𝛼 mm′ for m ≠ m′ and zero otherwise, and a and e are both M(2P + 1)-dimensional vectors, with block element
[a]m = am and [e]m = em , respectively. Note that for a plane wave, em has an
explicit analytical form that can be found in [35].
The scattered field Es (rsq ) at the receiver position rsq , q = 1, 2, ...Ns can be
obtained by (7.23). They can be written into a compact form,
s
E = AS ⋅ a.
(7.27)
where the dimension of AS is Ns × (M(2P + 1)). Equations (7.26) and (7.27) are
referred to as the state equation and the data equation, respectively. It is easy
Reconstructing Perfect Electric Conductors
to see that these two equations have the same mathematical structure as their
counterparts in Chapter 6, that is, a replaces the role of J, O replaces the role of
𝜒, and in addition neither AS nor AD depends on the scatterers’ materials. These
properties motive us to use the inversion models developed in Chapter 6, such
as Gs-SOM, TSOM, and NFFT-SOM, to reconstruct O, and the details will not
be repeated in this section.
It is worth discussing the choice of truncation number P. The number of
unknowns in the EFIE model is the number of subscatterers (M), while the
number of unknowns in the T-matrix model is the number of subscatterers
multiplied by 2P + 1. Under the fine meshing assumption (k0 R ≪ 1), the second and above orders can be dropped off since they are much smaller than
the zeroth and first orders. The small-term asymptotic approximations for the
zeroth and first order of T-matrices for both dielectric and PEC subscatterers
are summarized in Table 7.1. We see that the monopole element is the leading term in T-matrices for either PEC or dielectric scatterers. That is, M = 0
is sufficient to represent the scattering effects of either dielectric or PEC subscatterers. However, the dipole element in the T-matrix for PEC subscatterer
is on the same order of k0 R as the monopole element for a dielectric subscatterer. Therefore, when PEC and dielectric scatterers are simultaneously present,
M = 1 should be chosen as the minimum truncation number of multipoles, so
as to accurately represent the scattering effects of both PEC and dielectric scatterers. It is worth mentioning that because of the property of symmetry, the
two dipole elements [T]1 and [T]−1 are equal to each other. Thus the number
of unknowns in O is reduced by one third.
Next, the classification criterion of differentiating PEC from dielectric subscatterers is discussed. After O is retrieved by the optimization process, the
retrieved T-matrix T m for the mth cell, m = 1, 2, … , M, includes the dipole elements [T]±1 and the monopole element [T]0 . We examine the property of the
monopole element [T]0 to determine whether the subscatterer is made of PEC
or dielectric material. For dielectric scatterers, we assume that the real part of
relative permittivity ℜ(𝜖r ) ≥ 1, which is true for most dielectric materials. From
Table 7.1, we conclude that the imaginary part of [T]0 for dielectric scatterer
is positive. For PEC scatterers, the formula of [T]0 shown in Table 7.1 shows
Table 7.1 The small-term (k0 R) asymptotic expansions of T0
and T1 for both PEC and dielectric small circular scatterers.
Source: Ye 2013, IEEE Trans. Antennas Propag., 61, 3774–3781.
[35] Reproduced with permission of IEEE.
Material
T0
PEC
i𝜋
2 ln(k0 R)
Dielectric
i𝜋(k0 R)2
(𝜖r
4
T1
−
− 1)
i𝜋(k0 R)2
4
i𝜋(k0 R)4
(𝜖r
32
− 1)
199
200
Electromagnetic Inverse Scattering
that the imaginary part of [T]0 is negative since k0 R ≪ 1. Therefore, PEC and
dielectric scatterers can be distinguished by the sign of the imaginary part of
[T]0 . For those subscatterers identified as dielectric scatterers, the relative permittivity can be retrieved due to the fact that it is in a linear relationship with
the retrieved [T]0 under small term assumption,
4i[T]0
(7.28)
𝜖r = 1 −
𝜋(k0 R)2
For PEC scatterers, (7.28) has no physical meaning, but it can serve as a mathematical indictor since the real part of 𝜖r evaluated in (7.28) is much less than
zero. In summary, a convenient way to realize both differentiating PEC from
dielectric scatterers and retrieving the relative permittivities of dielectric scatterers is to calculate 𝜖r from (7.28) and classify subscatterers with negative real
part of 𝜖r as PEC. The advantage of the proposed method is that it works well
no matter the dielectric scatterer is lossy or not.
We present two numerical examples to demonstrate the performance of the
T-matrix inversion model. A total number of 10 plane wave incidences come
from angles evenly distributed in [0, 2𝜋), and 10 receiving antennas are evenly
distributed in a circle with a radius 5𝜆 that shares the center with the DOI D,
then 10% white Gaussian noise is added. The inversion model adopted here is
the standard Gs-SOM, in the framework of the state equation (7.26) and the
data equation (7.27).
In the first numerical example, as shown in Fig. 7.4(a), a small square PEC
scatterer is placed on the upper right corner of D, and the dielectric ring with
𝜖r = 4 is placed in the lower left corner. The domain D is a square of size 𝜆 × 𝜆,
which is discretized into 45 × 45 square cells. The ring has an outer radius
0.25𝜆 and an inner radius 0.15𝜆. To visually distinguish PEC from dielectric
scatterers, we use 𝜖r = 0 to mark the shape and location of PEC scatterers.
The reconstruction result displayed in Fig. 7.4(b) shows that both the dielectric ring and PEC square are correctly identified. In addition, the hole of the
ring is clearly seen.
In the second numerical example, as shown in Fig. 7.5(a) and (b), two circular
scatterers with radius 0.1𝜆 are placed inside the domain D. The PEC circular
4
–0.5
4
–0.5
2
0
0
2
0
0
–2
0.5
–0.5
0
(a)
0.5
–4
–2
0.5
–0.5
0
(b)
0.5
–4
Figure 7.4 A ring dielectric scatterer and a square PEC scatterer: (a) original pattern and
(b) reconstructed pattern. Source: Ye 2013, IEEE Trans. Antennas Propag., 61, 3774–3781. [35]
Reproduced with permission of IEEE.
Reconstructing Perfect Electric Conductors
–0.2
0
0.2
–0.5
0
(a)
0.5
–0.2
0
0.2
–0.5
0
(c)
0.5
6
4
2
0
–2
6
4
2
0
–2
–0.2
0
0.2
–0.5
0
(b)
0.5
–0.2
0
0.2
–0.5
0
(d)
0.5
6
4
2
0
–2
6
4
2
0
–2
Figure 7.5 Two circular scatterers: One PEC scatterer and one lossy dielectric scatterer.
(a) Original pattern of the real part of 𝜖r . (b) Original pattern of the imaginary part of 𝜖r .
(c) Reconstructed pattern of the real part of 𝜖r . (d) Reconstructed pattern of the imaginary
part of 𝜖r . Source: Ye 2013, IEEE Trans. Antennas Propag., 61, 3774–3781. [35] Reproduced
with permission of IEEE.
Figure 7.6 Configuration of the scatterers that are
used in the data set “FoamMetExt,” which is the
experimental data collected by the Institut Fresnel.
Adapted from: Geffrin 2005, Inverse Problems, 21,
S117, IOP Publishing. [36]
y
54.25mm
150mm
scatterer is on the right side and is marked as 𝜖r = 0 for visual purpose. The
dielectric circular scatterer is a lossy one with 𝜖r = 4 + 6i and is placed on the
left-hand side. The original patterns for real and imaginary parts of relative permittivity are depicted in Fig. 7.5(a) and (b), respectively. The DOI D is of size
𝜆 × 0.5𝜆 and is discretized into 60 × 30 square cells. From the reconstructed
patterns shown in Fig. 7.5(c) and (d), we observe that both the lossy dielectric
scatterer and PEC object are clearly identified, with the PEC on the right and
dielectric on the left. The numerical example shows that the T-matrix method
can differentiate lossy dielectric scatterer from PEC scatterer.
In the last example, in order to verify the validity of T-matrix method for
experimental data, we adopt the experimental data collected by the Institut
Fresnel. The details of the experimental configuration can be found in [36]. Here
the data set “FoamMetExt” is tested, where a metal and a foam are simultaneously present in the domain of interest. The configuration is a two-dimensional
x
150mm
201
202
Electromagnetic Inverse Scattering
–0.05
1
0
0
–1
0.05
–0.05
0
(a)
0.05
–0.05
0
–1
0.05
–0.05
0
(d)
0.05
–2
1
0
0
–1
0.05
–2
1
0
–0.05
–0.05
0
(b)
0.05
–0.05
0
–1
0.05
–0.05
0
(e)
0.05
–2
1
0
0
–1
0.05
–2
1
0
–0.05
–0.05
0
(c)
0.05
–0.05
–2
1
0
0
–1
0.05
–0.05
0
(f)
0.05
–2
Figure 7.7 Frequency-hopping reconstruction at 2–12 GHz using the T-matrix Gs-SOM for
the “FoamMetExt” experimental data. Real part of relative permittivity: (a) 2 GHz, (b) 4 GHz,
(c) 6 GHz, (d) 8 GHz, (e) 10 GHz, and (f ) 12 GHz. Source: Ye 2013, IEEE Trans. Antennas Propag.,
61, 3774–3781. [35] Reproduced with permission of IEEE.
TM problem. As depicted in Fig. 7.6, the DOI is of size 150 × 150 mm2 . A foam
cylinder with a diameter 80 mm and 𝜖r = 1.45 ± 0.15 is centered at the origin,
and a copper tube with a diameter 28.5 mm is located in touch with the foam.
The data are collected at multiple frequencies. We solve the inverse problem by using the frequency hopping within the range of 2–12 GHz in steps of
2 GHz. That is, the reconstruction result obtained at a frequency is used as
the initial guess for the inversion at the next higher frequency. The DOI is discretized into a grid of 45 × 45 cells. The reconstruction results are shown in
Fig. 7.7, where the unit of length is meters (m). We see that the result is quite
satisfactory. The PEC and dielectric scatterers are clearly seen in Fig. 7.7(f ),
which is obtained for the highest frequency.
7.5 Discussions
Several experimental test databases are available to test the performance
of inversion algorithms for the shape reconstruction of highly conducting
scatterers. Note that perfect electric conductor is defined as the limiting case
of highly conducting scatterers. The Institut Fresnel database introduced in
Section 7.4 has been used to test several inversion algorithms [37]. In [38],
measured data are collected in a controlled environment in the Second University of Naples and are used to test the performance of an inversion algorithm
that is based on the Kirchhoff approximation. In addition, many real-world
applications need to detect highly conducting scatterers. For example, [39]
Reconstructing Perfect Electric Conductors
presents nondestructive testing of concrete with electromagnetic waves, where
the tendon duct and the reinforcement grid are both modeled with infinite
electrical conductivity embedded in homogeneous cement. Reference [40]
demonstrates the disbond detection in strengthened concrete bridge members
retrofitted with composite laminates, which contain carbon that is highly
conductive at microwave frequencies.
For 2D TM inverse scattering problems involving PEC scatterers, if just a
closed-contour scatterer is present; that is, there is no line-shape scatterer, then
the model that treats PEC as a dielectric scatterer with a high imaginary part
of relative permittivity is preferable to the line-element model. The former has
only about half of the unknowns of the latter. Numerical simulations also show
that the former performs more robust than the latter does in presence of noise.
The problem of reconstructing PEC scatterers is closely related to that of
reconstructing piecewise-constant dielectric scatterers. When the relative permittivity of homogeneous dielectric scatterer is known a priori, the ISP problem reduces to a boundary identification problem, which is analogous to the
PEC reconstruction problem [41–44]. The only difference between these two
boundary identification problems is the boundary condition.
For mixed boundary condition ISPs, compared with the model that treats
PEC as a dielectric scatterer with high imaginary part of relative permittivity,
the T-matrix formula has a better ability to distinguish PEC from dielectric
scatterers, but has the weakness of involving much more unknowns since
one cell correspond to several multipole coefficients. We mention in passing
that the T-matrix formula has been applied to ISPs involving only one type
scatterer, that is, PEC or dielectric scatterer alone: PEC scatterers for the 2D
TM incidence [21], dielectric scatterers for the 2D TE incidence [45], and the
3D case [46].
The ISPs that reconstruct a mixture of PEC and dielectric scatterers address
mixed boundary conditions; that is, the Dirichlet boundary and transmission
boundary. The T-matrix formula presented in Section 7.4 can be generalized,
under some mild assumption, to simultaneously reconstruct scatterers with
different boundary conditions such as Dirichlet, Neumann, Robin, and transmission boundaries without a priori information on their locations, shapes, or
physical properties [47]. For scatterers with a mixture of Dirichlet boundary
and impedance boundary, theories and reconstruction algorithms can be
found in [48, 49].
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8
Inversion for Phaseless Data
This chapter deals with inverse scattering problems where the phase information of electromagnetic field is not available. It is generally known that the
accuracy of phase measurements cannot be guaranteed for operating frequencies approaching the millimeter-wave band and beyond. An accurate knowledge of the phase distribution involves sophisticated measurement equipment,
which is increasingly expensive as the working frequency grows, so phaseless
measurements are often necessary at millimeter-wave band and beyond, and
actually mandatory at optical frequencies. In this case, it is important to develop
inversion algorithms to solve phaseless inverse scattering problems.
The organization of the chapter is as follows. Section 8.1 outlines the
basic properties of phaseless ISPs and the main approaches to solving them.
Section 8.2 reconstructs point-like scatterers by subspace methods, which is
a generalization of the subspace method introduced in Chapter 4. Section 8.3
reconstructs point-like scatterers by compressive sensing, which simultaneously locates scatterers and determines their scattering strengths. Section 8.4
reconstructs extended dielectric scatterers, which is far more difficult than its
phase-available counterpart presented in Chapter 6.
8.1 Introduction
When the working frequency is high, it requires sophisticated and expensive
measurement equipment to obtain the phase information. Instead, solving
inverse scattering problems with intensity-only data should lead to simpler
and cost-effective experimental setups. In other words, the complexity and
cost have been shifted from hardware to algorithms. Thus, it is important
to develop inversion models and algorithms for phaseless inverse scattering
problems.
We first look at the uniqueness of the phaseless ISPs. For plane wave incidences and far field measurements, no results on uniqueness are available since
the modulus of the far-field pattern is invariant under translations [1]. Indeed,
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
208
Electromagnetic Inverse Scattering
for plane wave incidences, a translation of scatterer leads to only a phase shift
in far-zone scattered field. For point source incidences, however, results on
uniqueness have been established in [2, 3] for the inverse scattering problem of
determining a nonnegative real-valued potential of a finite-size scatterer from
the phaseless near-field data corresponding to all incident point sources placed
on a surface enclosing the scatterer for all wave numbers in a finite interval.
Next, regarding stability, it is obvious that the phaseless reconstruction is far
more ill-posed than the phase-available reconstruction, the stability estimate of
which is of a logarithmic type as presented in Chapter 6, since a measured signal contains less information. Reference [4] has made some progress in stability
analysis of phaseless reconstruction.
It is worth mentioning that the actual quantity measured in scattering
problems is the total field, instead of the scattered field, in most applications.
Unlike the usual phase-available measurement, where the scattered field is
easily obtained by subtracting the incident wave from the measured total field,
the phaseless measurement cannot yield a modulus square of the scattered
field by subtracting the modulus square of the incident field from the measured
modulus square of the total field. In only a few special cases can the modulus
of the scattered field be directly measured. For example, when the incident
wave is a highly spatially confined beam and the measurements are made at
an appreciable distance from the incidence beam. Most of existing phaseless
inversion algorithms utilize the modulus (or its square) of scattered field, such
as [5–7], and only a small fraction of papers utilize the modulus (or its square)
of the total field [8–10].
There are several methods for solving ISPs with phaseless data [7, 8, 11–13],
and they are essentially categorized into two conceptual approaches. The first
approach splits the problem into two steps [5, 8]. The first step consists of
estimating the phase of scattered field from the measurement of modulus (or
its square) of electric field, while the second step reconstructs the unknown
dielectric properties from the full-data scattered field. In summary, the first step
allows us to estimate the unknown dielectric properties from the second step,
where the second step is a traditional inverse scattering problem. The second
approach is a single step where the unknown dielectric properties are directly
reconstructed by solving a nonlinear optimization problem where the objective
function is defined as the residual between the measured modulus data and the
computed modulus data [6, 9, 12, 14].
In the first approach, phase retrieval (PR) can be realized by two techniques.
The first is the traditional phase retrieval technique that reconstructs the
complex-valued field in the spatial domain or, equivalently, the phase in the
spatial frequency domain from the knowledge of the absolute value of its spatial
Fourier transform and certain constraints in the spatial domain [15–18]. This
reconstruction problem has a long history and arises in many areas of engineering and applied physics, such as optics, X-ray crystallography, astronomical
Inversion for Phaseless Data
imaging, and transmission electron microscopy. It is also an important question
in mathematical society, where both uniqueness theorems and computational
algorithms are provided. The traditional PR is obviously applicable to only
far-field measurement. The second PR technique does not require far-field
measurement, but instead measures richer information on the absolute value
of the field in the spatial domain and reconstructs the phase based on the fact
that the phase is implicitly constrained by electromagnetic propagation theory.
This technique includes phase shift or time delay interferometry, two-probe
scanning, and measuring the field amplitude across two parallel planes [19].
The performances of the two-step approach is compared with the single-step
approach in [5, 14], and it has been found that the former outperforms the
latter, since the separation of the problem into two steps allows better control of the overall nonlinearity with respect to single-step procedures. On the
other hand, it is important to note that the two-step approach can be actually
applied when only some conditions on the measurement setup are satisfied.
When these conditions do not hold true, the measurement data do not provide
enough independent data to accurately pursue phase retrieval. Consequently,
the error in phase occurring in the first step will propagate to the second step.
In this case, the single-step approach has to be preferable.
This chapter deals with the phaseless ISPs that do not require stringent measurement conditions. For example, measurement can be conducted in both
near or far zones, and measurement does not have to be very dense in the spatial domain to ensure a reliable phase retrieval. Thus, the single-step approach
will be adopted. Section 8.2 reconstructs point-like scatterers from the modulus of the scattered field, Section 8.3 reconstructs point-like scatterers from
the modulus of the total field, and Section 8.4 reconstructs extended scatterers
from the modulus of the total field.
8.2 Reconstructing Point-Like Scatterers by Subspace
Methods
For point-like scatterers, that is, the sizes of scatterers are much smaller than
the wavelength, as presented in Chapter 4, the subspace method can be used
to locate the positions of scatterers and to noniteratively retrieve the polarization tensors of scatterers when both the intensity and phase of the scattered
field are measured. For phaseless data, although the mapping from the induced
sources to the intensities of scattered fields is nonlinear, a new set of bases can
be constructed that are mapped linearly to the intensity-only scattering data.
This linear map makes it possible to apply the subspace methods to solve phaseless ISPs involving point-like scatterers.
The configuration of the scattering problem is first introduced. Suppose there are M three-dimensional objects illuminated by time-harmonic
209
210
Electromagnetic Inverse Scattering
electromagnetic waves radiated by an array of Ni antenna units. The
transmitting antenna units are located at ri1 , ri2 , … , riN , each of which consists
i
of three small electric dipole antennas oriented in the x, y, and z directions.
For each incidence, the intensities of scattered electric field are measured
by an array of Ns antenna units that have the same configuration as the
transmitting antennas and their locations are rs1 , rs2 , … , rsN . The size of each
s
of the M objects is much smaller than the wavelength so that they can be
effectively treated as point-like targets. These scatterers can be of any shape
and material, but we consider only anisotropic spherical objects for ease of
presentation. The centers of the scatterers are located at r1 , r2 , … , rM . As
introduced in Section 4.4.1, the polarization tenor 𝜉 m (𝜁 m ), relates the induced
electric (magnetic) current dipole Il(rm ) (Kl(rm )) inside the object to the total
in
in
incident electric field Ein
t (rm ) (magnetic field Ht (rm )) by Il(rm ) = 𝜉 m ⋅ Et (rm )
(and Kl(rm ) = 𝜁 m ⋅ Hin
t (rm )), m = 1, 2, … , M. The analytical expression of 𝜉 m
(𝜁 m ) for anisotropic spheres can be found in Section 2.8.1.
8.2.1
Converting a Nonlinear Problem to a Linear One
Before deriving the intensity of scattered field, it is important to examine the
relationship between scattered electric fields and induced sources. The dyadic
Green’s functions corresponding to electric and magnetic dipoles are given by
G0 (r, r′ ) = [(G0 (r, r′ ))x , (G0 (r, r′ ))y , (G0 (r, r′ ))z ]
(8.1)
and
−∇ × G0 (r, r′ ) = −[(∇ × G0 (r, r′ ))x , (∇ × G0 (r, r′ ))y , (∇ × G0 (r, r′ ))z ]
(8.2)
′
respectively. The closed-form elements of the 3 × 3 matrix G0 (r, r ) is provided
in (2.40), which is repeated here,
)
(
)]
[(
Ru R𝑣
3i
i
1
3
′
′
Gu𝑣 (r, r ) = g(r, r ) 1 +
𝛿u,𝑣 − 2
1+
−
−
k0 R k02 R2
R
k0 R k02 R2
(8.3)
ik0 |r−r′ |
where u, 𝑣 = 1, 2, 3, R = r − r , R = |R|, and g(r, r ) = e
∕(4𝜋|r − r |) is
the scalar Green’s function. The closed-form elements of the 3 × 3 matrix ∇ ×
G0 (r, r′ ) is provided in (2.42), which is repeated here,
(
)
0
−(z − z′ ) (y − y′ ) ⎤
⎡
ik0
1
′
′
⎢
0
−(x − x′ ) ⎥ g(r, r′ )
∇ × G(r, r ) = (z − z )
− 2
⎥
⎢
R
R
′
′
0
⎦
⎣ −(y − y ) (x − x )
(8.4)
′
′
′
For a certain incidence, denote the induced electric and magnetic current dipoles inside the mth scatterer as [Ilx (rm ), Ily (rm ), Ilz (rm )]T and
Inversion for Phaseless Data
[Klx (rm ), Kly (rm ), Klz (rm )]T , respectively, where the superscript T denotes the
transpose. For m = 1, 2, … , M, L = E, M, and l = x, y, z, define
{
if L = E
ik0 (G0 (r, rm ))l
(8.5)
G(m,L,l) (r) =
−(∇ × G0 (r, rm ))l if L = M,
{
𝜂0 Ill (rm ) if L = E
Q(m, L, l) =
(8.6)
if L = M,
Kll (rm )
where k0 is the wavenumber and 𝜂0 is the impedance of the background
medium. Define the index s of the dipole source (m, L, l), which runs from 1 to
6M, so that it sorts all possible combinations of m, L, and l. The dipole source
with a smaller value of m has a lower index. For a given m, the electric dipole
source (L = E) has a lower index than the magnetic source (L = M). For a given
m and L, the x-oriented source (l = x) has the lowest index and the z-oriented
source (l = z) has the highest.
It is easy to find that the scattered electric field is given by
s
E (r) =
6M
∑
(8.7)
Gs (r)Q(s).
s=1
By dropping off the (r), we define a 3Ns × 1 vector Gs = [Gs (rs1 )T , Gs (rs2 )T , … ,
s
Gs (rsN )T ]T . The 3Ns × 1 vector E is defined similarly. The definition of vector
s
G(r, L, l) is similar to that of Gs except that the position of the source is located
at r.
Now we are in a position to define the vector of intensity of scattered field,
s
s
I = (E )∗ ∘E , where ∘ denotes the Hadamard product (or Schur product) that is
an element-wise product. The superscript * denotes a complex conjugate. The
intensity vector I is expanded as
I=
6M
∑
∑
Q(s1 , s2 , 1)U
(s1 ,s2 )
+
s1 =1 s2 ≥s1
6M
∑
∑
U
V
(s1 ,s2 )
=
=
∗
[G(m1 ,L1 ,l1 ) ∘
∗
[G(m1 ,L1 ,l1 ) ∘
(s1 ,s2 )
,
(8.8)
s1 =1 s2 >s1
where the 3Ns × 1 propagator vectors U
(s1 ,s2 )
Q(s1 , s2 , 2)V
G(m2 ,L2 ,l2 ) ]
(s1 ,s2 )
and V
(s1 ,s2 )
are defined by

(8.9)

(8.10)
G(m2 ,L2 ,l2 ) ] ,
where (⋅) and (⋅) denotes the real part and imaginary part operators, respectively. The scalar Q(s1 , s2 , j) is defined as
{
)
(
j=1
2Q (s1 , s2 ) 1 − 12 𝛿s1 ,s2
Q(s1 , s2 , j) =
(8.11)

−2Q (s1 , s2 )
j = 2,
where 𝛿s1 ,s2 = 1 when s1 = s2 and is 0 otherwise, and
Q(s1 , s2 ) = Q∗ (s1 )Q(s2 ).
(8.12)
211
212
Electromagnetic Inverse Scattering
Using the subscript p to denote the pth incidence, we obtain the 3Ns × 3Ni
MSR matrix K = [I 1 , I 2 , … , I 3Ni ]. It can be seen from Eq. (8.8) that the range of
(s ,s )
(s ,s )
K is the span of a set of bases U 1 2 and V 1 2 . It is important to note that Eq.
(8.8) provides a linear mapping under these bases.
8.2.2
Rank of the Multistatic Response Matrix
When applying the subspace method, it is important to determine the rank of
the MSR matrix. Equation (8.8) shows that the rank of the MSR matrix K is
at most (6M)2 . However, due to the polarization of electromagnetic fields, the
rank of the MSR matrix is less than (6M)2 . The first case happens when more
than one magnetic dipoles are induced within a scatterer. From the expression,
0
⎤
| 1 dg |2 ⎡
⎥,
| ⎢
0
(∇ × G0 )∗x ∘(∇ × G0 )y = ||
|
| R dR | ⎢⎣ −(x − x′ )(y − y′ ) ⎥⎦
(8.13)
we conclude that (∇ × G0 )∗x ∘(∇ × G0 )y is purely real, so as (∇ × G0 )∗x ∘(∇ × G0 )z
and (∇ × G0 )∗y ∘(∇ × G0 )z . The second case happens when three electric dipoles
and three magnetic dipoles are induced within a scatterer. The expression
(G0 )∗x ∘(∇ × G0 )x =
where 𝜗(R) =
1 d2 g ∗
(
R3 dR2
−
0
⎤
⎡
i
𝜗(R) ⎢ (x − x′ )(y − y′ )(z − z′ ) ⎥ ,
⎢
k0
′
′
′ ⎥
⎣ −(x − x )(y − y )(z − z ) ⎦
1 dg ∗ dg
) ,
R dR dR
(8.14)
indicates that
(G0 )∗x ∘(∇ × G0 )x + (G0 )∗y ∘(∇ × G0 )y + (G0 )∗z ∘(∇ × G0 )z = 0
(8.15)
Thus, (G0 )∗x ∘(∇ × G0 )x , (G0 )∗y ∘(∇ × G0 )y , and (G0 )∗z ∘(∇ × G0 )z are linearly dependent. Considering the above two cases, we obtain the rank of K to be
rK = 𝛼 2 − 𝛽 − 3𝛾 − 5𝜏,
(8.16)
where 𝛼 is the total number of induced dipole components, 𝛽 is the number of
scatterers in which two components of magnetic dipoles are induced, 𝛾 denotes
the number of scatterers in which three components of magnetic dipoles are
induced but less than three components of electric dipoles are induced, and 𝜏
represents the number of scatterers in which three electric dipoles and three
magnetic dipoles are induced.
The rank given by (8.16) has been tested by many numerical simulations and
all of simulation results agree with Eq. (8.16). Here, two numerical examples
are shown to support the validity of it. The arrays of transmitting and receiving
antennas are chosen to be coincident, and they are located on a spherical
surface centered at the origin, with radius of one wavelength (𝜆). In a spherical coordinate system, the angles of antennas are given by 𝜃 = (𝜋∕2)j∕N𝜃 ,
j = 1, 2, … , N𝜃 and 𝜙 = 2𝜋k∕N𝜙 , k = 1, 2, … , N𝜙 . In the following two
numerical simulations, N𝜃 and N𝜙 are chosen to be 5 and 10, respectively.
5
5
0
0
−5
#73
−10
−15
0
50
100
Singular value number, j
(a)
log10(σj)
log10(σj)
Inversion for Phaseless Data
−5
#77
−10
150
−15
0
50
100
Singular value number, j
(b)
150
Figure 8.1 Singular values of the MSR matrix in the first (a) and the second (b) example.
Source: Chen 2008, J. Opt. Soc. Am. A, 25, 2018–2024. [21] Reproduced with permission of
The Optical Society.
Two spheres with radius 𝜆∕30 are located at [0.2𝜆, 0.1𝜆, 0.1𝜆]T and
[−0.3𝜆, −0.1𝜆, −0.2𝜆]T , respectively. In the first example, the permittivity
and permeability tensors are given by 𝜖 1 = 5𝜖0 I 3 , 𝜇1 = 5𝜇0 I 3 , 𝜖 2 = 𝜖0 I 3 ,
𝜇2 = 5𝜇0 I 3 , where I 3 is an identity matrix of dimension three. For these scatterers, 𝛼 = 9, 𝛽 = 0, 𝛾 = 1, and 𝜏 = 1. Equation (8.16) indicates that the rank
rk is equal to 73. The numerical results of singular values of the MSR matrix
K are shown Fig. 8.1(a). The figure shows that there are 73 dominant singular
values and this agrees with Eq. (8.16). In the second example, 𝜖 1 = 5𝜖0 I 3 ,
𝜇1 = diag{5𝜇0 , 5𝜇0 , 𝜇0 }, 𝜖 2 = diag{𝜖0 , 𝜖0 , 5𝜖0 }, 𝜇2 = 5𝜇0 I 3 . For these scatterers,
𝛼 = 9, 𝛽 = 1, 𝛾 = 1, and 𝜏 = 0. The rank rk is expected to be 77. The numerical
results of singular values of the MSR matrix K are shown Fig. 8.1(b), where we
see 77 dominant singular values, once again consistent with Eq. (8.16).
8.2.3
MUSIC Localization and Noniterative Retrieval
(s ,s )
(s ,s )
Since the range of K is the span of a set of bases U 1 2 and V 1 2 , we can apply
the Multiple Signal Classification (MUSIC) method to locate the positions,
provided that the number of rows 3Ns and the number of columns 3Ni are
both larger than the rank of K. However, the high rank of the MSR makes it
difficult to distinguish the signal subspace from the noise subspace when the
measurement is contaminated with noise. Thus, the MUSIC method is hardly
practical in locating point-like scatterers for phaseless 3D electromagnetic
ISPs. From the two numerical examples presented in Section 8.2.2, we see
that the rank of the MSR matrix is too high already for just two scatterers. As
additional examples, consider the following two most frequently encountered
cases: two small isotropic nonmagnetic dielectric spherical scatterers lead to
a rank rK = 36 (𝛼 = 6, 𝛽 = 𝛾 = 𝜏 = 0), and two small PEC spherical scatterers
lead to a rank rK = 134 (𝛼 = 12, 𝛽 = 𝛾 = 0, 𝜏 = 2). All these data show that
213
214
Electromagnetic Inverse Scattering
the rank of the MSR matrix for phaseless 3D electromagnetic ISPs is too high.
In comparison, for phaseless 3D scalar-wave ISPs, MUSIC is a good candidate
for locating a small number of point-like scatterers since the rank of the MSR
matrix is the square of the number of scatterers. The first paper along this
direction is given by [20], where the theoretical analysis, algorithms, and
numerical simulations are provided.
After obtaining the locations of point-like scatterers, the polarization tensors
of scatterers can be obtained by a noniterative retrieval algorithm, though
the problem is nonlinear when the multiple scattering effect is taken into
account. The noniterative retrieval algorithm contains three steps. The first
and third steps are similar to the two-step least-squares method presented in
Sections 4.4.3 for the case of full data (both phase and intensity) of scattered
field, whereas the second step, that is, determining the phase of induced
current, is unique to intensity-only data retrieval. The details are a little bit
complex and can be found in [21].
8.3 Reconstructing Point-Like Scatterers
by Compressive Sensing
To apply the MUSIC method to locate the positions of point-like scatterers, the
numbers of both incidences and receivers must be larger than the maximum
possible number of independent bases; that is, the rank of MSR matrix. Thus,
MUSIC is not applicable when the number of incidences is few, as encountered
in many real-world applications. In these cases, compressed sensing is a good
candidate for tackling ISPs involving point-like scatterers.
8.3.1
Introduction to Compressive Sensing
The compressive sensing or compressed sensing (CS) is a signal processing
technique that allows one to recover certain signals from far fewer samples
than required by the Shannon–Nyquist sampling theorem. There are two conditions that the CS relies on. The first one is sparsity that pertains to the signal of
interest, and the second one is incoherence that pertains to the sensing modality. While sparsity requires the signal of interest to be sparse in a proper basis,
incoherence requires that the sampling/sensing waveforms have an extremely
dense representation in the same basis [22].
This section briefly introduces the basic principle of CS. For mathematical
details, readers are referred to the monograph [23]. The applications of CS in
electromagnetics are reviewed in [24].
Consider a signal f ∈ ℝn and a linear measurement system records the data
y = S ⋅ f ∈ ℝm , where S denotes the m × n sensing matrix. Let the signal f be
Inversion for Phaseless Data
sparse with respect to the basis Φ ∈ ℝn×n , so that it can be expressed as f = Φx.
Then the abstract problem of recovering f ∈ ℝn from data
y=A⋅x+𝜖
(8.17)
is generally underdetermined when m ≪ n, where A = S ⋅ Φ ∈ ℝm×n is
the sensing matrix with respect to the basis Φ and 𝜖 ∈ ℝm denotes the
measurement noise.
If we know that the solution to (8.17) is sparse in the sense that nonzero
entries occupy only a small fraction, then under certain conditions the solution to the underdetermined linear system (8.17) can be obtained by solving
the following constrained optimization problem
min||x||𝓁0 subject to ||y − A ⋅ x||𝓁2 ≤ 𝜖
(8.18)
where the 𝓁0 norm of a vector is the number of nonzero entries. Equation (8.18)
is, however, an NP-hard (non-deterministic polynomial-time hard) problem,
and consequently its solution is computationally intractable. Nevertheless, [25]
and [26] have proven that under certain conditions on A, for example, satisfying
the restricted isometry property (RIP), together with the sparsity of x, problem
(8.18) is equivalent to the following problem,
min||x||𝓁1 subject to ||y − A ⋅ x||𝓁2 ≤ 𝜖,
(8.19)
where 𝓁1 norm of a vector is the sum of the absolute values of its entries.
Equation (8.19) is a convex problem, known as the second-order cone
programming, for which efficient solution methods already exist [22].
A result that establishes CS as a practical and reliable sensing mechanism
asserts that the solution to (8.19) is robust in presence of noise. The bound of the
reconstruction error is given in [22] as a linear function of the noise level 𝜖. Then
the original ill-conditioned linear problem is cast into a well-posed problem,
provided that the two conditions, that is, sparsity and incoherence, are satisfied.
The CS formulas (8.18) and (8.19) fall into the deterministic strategies and
there is another important category of CS recovery algorithm, the Bayesian
approach. The theory of Bayesian CS is presented in [27] and its application
in solving ISPs can be found in [28].
8.3.2
Solving Phase-Available Inverse Problems by CS
When both the magnitude and phase of scattered field are available, many
publications in both engineering and mathematical communities have applied
CS to solve inverse scattering problems involving point-like scatterers. The first
example is a single-incidence case where the multiple scattering effect is taken
into account. In fact, a single-incidence inverse scattering problem is equivalent
to an inverse source problem. The DOI can be discretized into Q points that
are sufficiently dense such that all M point-like scatterers are located at those
215
216
Electromagnetic Inverse Scattering
points. We use 𝜆 ∈ ℂQ , the vector of complex Q-tuples, to denote the amplitude
of source at Q points. Since the amplitude of the source at points unoccupied by
scatterers is zero, the vector 𝜆 is sparse due to M ≪ Q. The radiated field is measured at discrete positions r′1 , r′2 , … , r′N , where N ≪ Q. The single-incidence
inverse scattering problem of determining the locations and amplitudes of
induced sources consists of solving the following linear equation system,
E(r′n ) =
Q
∑
G(r′n , rq )𝜆q ,
n = 1, 2, … , N
(8.20)
q=1
where G(r′n , rq ) denotes the Green’s function and 𝜆q is the amplitude of induced
source at the qth point. It is worth mentioning that multiple scattering effect
has been taken into account here. Obviously (8.20) is a linear equation with a
sparsity constraint. The second example is the multiple-incidence case under
the Born approximation; that is, the effect of multiple scattering is neglected.
In this case, a linear equation with a sparsity constraint can be established,
where the unknowns are the scattering strengths of point-like scatterers.
Before applying CS to solve these two underdetermined linear equations, we
should first check the two conditions that CS relies on. While the sparsity of
unknown vector is obvious for point-like scatterers, the incoherence of measurement, or equivalently the RIP condition of the sensing matrix A in (8.17) is,
in practice, not straightforward. Reference [29] has provided an extensive analysis on this topic for inverse scattering problems. At the same time, it should be
noted that publications of related problems largely take the compressed sensing
theory for granted and assume, either explicitly or implicitly, either the incoherence or the RIP property without proof. For these two problems, [30] has
concluded that in the absence of noise, CS theory recovers exactly the target of
sparsity up to the dimension of the data. Stability with respect to noisy data is
proved for weak or widely separated scatterers.
Unlike the subspace methods presented in Section 8.2 where multiple
incidences are mandatory and the solution of induced current is available only
after the positions of scatterers have to be first determined by the MUSIC
method, the compressed sensing approach requires only a single incidence and
the positions of point-like scatterers are automatically identified as entries with
nonzero amplitude. After introducing the application of CS to phase-available
ISPs, we are in a position to discuss the application of CS to phaseless ISPs.
8.3.3
Solving Phaseless Inverse Problems by CS
We consider a two-dimensional scattering problem (̂z is the longitudinal direction) under time-harmonic transverse magnetic (TM) electromagnetic wave
illumination. The background medium is free space. Since it is a scalar wave
problem, we will henceforth conveniently suppress the unit vector ẑ , wherever
we refer to the electric field and the induced current.
Inversion for Phaseless Data
A total number of M point-like scatterers are embedded in a DOI D, which is
discretized into a total number of Q cells, with the centers of the cells located
at rn (n = 1, 2, · · · , Q), and the area and relative permittivity of the nth cell are
denoted by An and 𝜖r (rn ), respectively. We assume each point-like scatterer just
occupies a cell with good approximation. The contrast, denoted by 𝜒(rn ) (n =
1, 2, · · · , Q), is defined as
{
𝜖r (rn ) − 1, if the cell is a point-like scatterer,
𝜒(rn ) =
0,
otherwise.
In our problem, we assume the point-like scatterers are made of lossless
dielectric material, and consequently the value of contrast is a real number.
Outside the DOI D, a total number of Ni transmitters (TX) are located at rip
(p = 1, 2, · · · , Ni ), and a total number of Ns receivers (RX) are located at rsq
(q = 1, 2, · · · , Ns ).
The received field Erec (rsq , rip ) consists of two portions,
Erec (rsq , rip ) = Ei (rsq , rip ) + Es (rsq , rip ),
(8.21)
where the first portion is the incident field directly coming from the TX,
Ei (rsq , rip ), and the second portion represents the field scattered off the
point-like scatterers, which is expressed by
Es (rsq , rip ) =
Q
∑
k02 ⋅ An ⋅ g(rsq , rn ) ⋅ Et (rn , rip ) ⋅ 𝜒(rn ).
(8.22)
n=1
where the total incident electric field onto the nth cell at rn satisfies the
Foldy–Lax equation,
Et (rn , rip ) = Ei (rn , rip ) +
Q
∑
k02 ⋅ An′ ⋅ g(rn , rn′ ) ⋅ Et (rn′ , rip ) ⋅ 𝜒(rn′ ),
(8.23)
n′ =1
n′ ≠n
Under the Born approximation, which is valid when point scatterers have weak
contrasts or are sufficiently separated, (8.22) can be approximated by
Es (rsq , rip ) ≈
Q
∑
k02 ⋅ An ⋅ g(rsq , rn ) ⋅ Ei (rn , rip ) ⋅ 𝜒(rn )
(8.24)
n=1
qp
For convenience, we introduce yn = k02 ⋅ An ⋅ g(rsq , rn ) ⋅ Ei (rn , rip ) to simplify
the expression.
In phaseless problem, the square of the modulus of the total received electric
field F rec (rsq , rtp ) can be expressed by
I(rsq , rip ) = Erec (rsq , rip )Erec (rsq , rip )∗
= Ei (rsq , rip )Ei (rsq , rip )∗ + Es (rsq , rip )Es (rsq , rip )∗
+ 2Re{E
i
(rsq , rip )Es (rsq , rip )∗ }
(8.25)
217
218
Electromagnetic Inverse Scattering
where the notation Re(⋅) stands for the real-part operator. Since
Ei (rsq , rip )Ei (rsq , rip )∗ is known, we treat I(rsq , rip ) − Ei (rsq , rip )Ei (rsq , rip )∗ as the
measured data, denoted as y. Substitution with Eq. (8.24) yields
y(rsq , rip ) =
Q
∑
qp∗
2Re{Ei (rsq , rip )yn } ⋅ 𝜒(rn )
n=1
+
Q Q
∑
∑
qp
qp∗
yn′ ⋅ yn ⋅ 𝜒(rn′ ) ⋅ 𝜒(rn )
(8.26)
n′ =1 n=1
For convenience of description, we simplify y(rsq , rip ) and 𝜒(rn ) to succinct notations yq,p and 𝜒n . The data y for all TX and RX pairs can be written as a column
vector of length Ns Ni ,
y = [y1,1 , y2,1 , · · · , yNs ,1 , y1,2 , y2,2 , · · · , yNs ,2 , · · · , y1,Ni , y2,Ni , · · · , yNs ,Ni ]T
The contrast information can be recast as a column vector of length Q in the
form
𝜒 = [𝜒1 , 𝜒2 , · · · , 𝜒Q ]T .
We also build a vector of length Q(Q + 1)∕2 to record the information of the
products of contrast, which is written in a column vector as follows [31].
𝜆 = [𝜒1 𝜒1 , 𝜒1 𝜒2 , 𝜒1 𝜒3 , 𝜒1 𝜒4 · · · , 𝜒1 𝜒Q ,
𝜒2 𝜒2 , 𝜒2 𝜒3 , 𝜒2 𝜒4 , · · · , 𝜒2 𝜒Q ,
𝜒3 𝜒3 , 𝜒3 𝜒4 , · · · , 𝜒3 𝜒Q ,
𝜒4 𝜒4 , · · · , 𝜒4 𝜒Q ,
· · · , · · · , · · ·,
𝜒Q 𝜒Q ]T
Combining 𝜒 and 𝜆, we define a vector x of length Q(Q + 1)∕2 + Q as follows:
T
x = [𝜒 T , 𝜆 ]T .
Now Eq. (8.26) can be expressed in a compact form:
y=A⋅x
(8.27)
where A is a [Ns Ni ] × [Q(Q + 1)∕2 + Q] matrix, whose elements can be determined by the coefficients in Eq. (8.26). Now the phaseless imaging problem is
reduced to finding the contrast vector 𝜒 given y.
8.3.4
Applicability of CS
We know from (8.27) and the definition of vector x that the problem of recovering 𝜒 from y is intrinsically nonlinear and therefore the framework of compressive sensing does not seem applicable to this problem. However, if we simply
ignore the nonlinear relationship between 𝜒 and 𝜆 and treat each entry of x as
Inversion for Phaseless Data
an independent variable, then (8.27) can be regarded now as a linear equation.
This idea provides the possibility of modeling the phaseless ISPs in the framework of compressive sensing.
In order to evaluate the applicability of compressive sensing, we also examine
the two principles on which the compressive sensing relies. These two principles are sparsity and incoherence. In this phaseless ISP, the unknown is represented by vector x. Since M of the Q cells are occupied by the point-like
scatterers (M << Q), then the number of nonzero elements in vector x is M +
M(M + 1)∕2, thus x can be regarded as a rather sparse signal, in view of the
fact that the length of x is Q + Q(Q + 1)∕2. Regarding the incoherence condition, a closely relevant paper [32], where the received phaseless data is the
square of the modulus of scattered field rather than total field, has proven that
the sensing matrix satisfies the RIP condition under certain conditions; that is,
the CS recovers the locations and the contrasts of the scatterers exactly. Numerical simulations show that the CS reconstruction result is robust for noisy data
provided that multiple illuminations are used, although the theoretical justification is still an ongoing effort [32]. Here, we assume, without proof, that the
conclusions of [32] can be extended to the problem (8.27) and consequently we
directly apply CS to solve it.
8.3.5
Numerical Examples
One numerical example is given in this section. As depicted in Fig. 8.2(a),
five point-like scatterers are placed at (0.85𝜆, −0.55𝜆), (−0.6𝜆, 0.55𝜆),
(0.15𝜆, −0.1𝜆), (0.7𝜆, 0.7𝜆), and (−0.7𝜆, −0.45𝜆), with their relative permittivities being 1.8, 2.0, 1.6, 1.7, and 1.9, respectively, in the DOI D, which is a square
of 2.0𝜆 on each side and has been discretized into a grid of 12 × 12 cells. The
data acquisition scheme has the following properties: (1) a single TX antenna
(Ni = 1) is placed at a position randomly chosen outside D; (2) a total number
of Ns = 50 RX antennas are placed at random positions outside D; and (3)
overlap between these antennas should be avoided.
In numerical simulations, we have chosen an open-source solver, the
“𝓁1-MAGIC” solver [33], which is based on standard interior-point methods
and is suitable for large-scale problems. In this open-source solver, we have
chosen a CS model, referred to as the Dantzig selector, which is made slightly
T
different from (8.19) by replacing the constraint by ||A ⋅ (y − A ⋅ x)||∞ ≤ 𝜖.
Figure 8.2(b) shows the retrieved image in the absence of noise, where we
see a perfect match with the exact pattern Fig. 8.2(a). Figure 8.2(c) exhibits
the retrieved image in presence of additive white Gaussian noise with SN R =
20 dB, where locations of the five small scatterers are exactly recovered and
their permittivities only have slight errors. For a noise that is too high, as shown
in Fig. 8.2(d) where SN R = 10 dB, our method cannot robustly reconstruct the
exact image.
219
Electromagnetic Inverse Scattering
2
1
1.5
–0.5
y(λ)
y(λ)
0.5
0
–1
–1 –0.5
1.5
0
–0.5
0
0.5
1
1
–1
–1 –0.5
x(λ)
(a)
0
0.5
1
1
x(λ)
(b)
2
1
2
1
0.5
1.5
0
y(λ)
0.5
1.5
0
–0.5
–0.5
–1
–1 –0.5
2
1
0.5
y(λ)
220
0
x(λ)
(c)
0.5
1
1
–1
–1 –0.5
0
0.5
1
1
x(λ)
(d)
Figure 8.2 The pattern consisting of five point-like objects. (a) Exact pattern of relative
permittivity. (b) Reconstructed pattern with no noise added into measurement.
(c) Reconstructed pattern with Gaussian white noise added into measurement (SNR =
20 dB). (d) Reconstructed pattern with Gaussian white noise added into measurement (SNR
= 10 dB). Source: Pan 2012, IEEE Trans. Antennas Propag., 60, 5472–5475. [31] Reproduced
with permission of IEEE.
At the end of the section, it is worth mentioning that [32] has solved phaseless
ISPs in which data are the square of scattered field. Different from the approach
used this section, that is, to define a long vector so that the original nonlinear
problem is reformulated into a linear vector equation, [32] directly solves for
a matrix. The sparsity condition requires that the matrix has a minimum rank
(the number of its nonzero singular values). It has been proven that under certain conditions on the array imaging configuration and on the scatterers, the
rank minimization problem is equivalent to the minimization of the nuclear
norm (the sum of its singular values) of the decision matrix, which makes the
problem convex and solvable in polynomial time.
8.4 Reconstructing Extended Dielectric Scatterers
This section deals with reconstructing extended dielectric scatterers from the
knowledge of measured magnitude of total electric field.
We consider a two-dimensional configuration under TE incidence, where
the system is invariant in the z direction. Dielectric cylindrical scatterers are
Inversion for Phaseless Data
located in the DOI D ⊂ ℝ2 in the x-y plane. There are Ni magnetic-current line
sources and Ns receivers that measure both the x or y components of the total
field. The total number of scattering data is 2Ni Ns . The DOI D is discretized
into a total number of M cells.
For the pth incidence, p = 1, 2, … , Ni , we have the following two equations
to solve the forward scattering problem,
i
J p = 𝜉 ⋅ (Ep + GD ⋅ J p ).
s
Ep
(8.28)
= GS ⋅ J p .
(8.29)
where J p is 2M-dimensional column vector denoting the induced contrast displacement current due to the pth incidence. The 2M × 2M diagonal matrix 𝜉
consists of scattering strength, which depends on the relative permittivity 𝜖 r .
As shown in (2.81), for a circular cylinder, 𝜉 = (−i2𝜋k0 a2 ∕𝜂0 )(𝜖r − 𝜖0 )∕(𝜖r + 𝜖0 ).
The phaseless data may conveniently be taken to be the square of the intensity of the total received field. For the pth transmitting antenna, the measured
phaseless data can be written as a column vector of size 2Ns .
i
s
i
s
F p = (Ep + Ep )∗ ∘(Ep + Ep ),
(8.30)
where ∘ denotes the Hadamard product (or Schur product), which is an
element-wise product, and the superscript ∗ denotes the complex conjugate.
Equations (8.28) and (8.30) are referred to as the state equation and the data
equation, respectively. It is easy to see that the state equation remains the same
for both phase-available and phaseless problems, whereas the data equations
are different. In solving phaseless IPSs, a straightforward way is to define an
objective function that quantifies the mismatch between measured and calculated phaseless data, where the unknown parameter 𝜉 is obtained by optimization process. However, the performance of this approach is not stable in
presence of measurement noise and depends very much on the initial guess. In
addition, a direct optimization process is like a black box, which does not provide us with much insight. Instead, the subspace-based optimization method
(SOM) presented in Chapter 6 provides far better and more stable reconstruction results.
H
The SVD of GS can be expressed as GS = U ⋅ Σ ⋅ V . The induced currents
(J p ) on scatterers due to the pth incidence are partitioned into two orthogonally
+
−
complementary portions (namely, the major part J p and the minor part J p ):
+
−
+
−
+
−
J p = J p + J p = V ⋅ 𝛼 +p + V ⋅ 𝛼 −p , where V and V comprise the first L and
the remaining 2M − L columns of the V matrix. Note that L is the total number
of singular values in the Σ matrix that are above a predefined noise-dependent
threshold.
Since phaseless data do not linearly depend on the induced current, the
straightforward truncated-SVD inversion of the data equation cannot be
221
222
Electromagnetic Inverse Scattering
applied to obtain the major part of induced current. On the other hand, since
the scattered electric field is mainly due to the major part of induced current,
it naturally follows that the dominant contribution to the intensity of the total
i
field is due to the Ep and the major part of induced current. This inference
+
paves the way to estimating the major part of induced current J p by solving a
nonlinear optimization problem. On basis of this idea, an objective function is
proposed as follows,
+
i
+
i
𝛼 +p = arg min ||F p − (Ep + GS ⋅ V ⋅ 𝛼 +p )∗ ∘(Ep + GS ⋅ V ⋅ 𝛼 +p )||2 , (8.31)
𝛼 +p
where “arg min” means the value of the argument for which the function
reaches its minimum. The objective function is a quartic polynomial and
the gradient of the function can be computed in a straightforward manner. The Levenberg–Marquardt (LM) algorithm, which is a mixture of the
Gauss–Newton algorithm and the method of gradient descent, is employed
to minimize the objective function. Numerical simulations show that the
convergence is fast, which will be shown in later numerical examples.
After the major part of induced current is determined, the 𝜉 of the scatterers
can be obtained by minimizing the sum of relative mismatches in both the state
equation and the data equation. The relative residual in the state equation is
defined to be
Δsta =
Ni
∑
||A ⋅ 𝛼 −p − Bp ||2
+
p=1
||J p ||2
−
,
(8.32)
−
i
+
+
where A = V − 𝜉 ⋅ GD ⋅ V , and Bp = 𝜉 ⋅ (Ep + GD ⋅ J p ) − J p . Due to the truncation of the singular values, the relative residual in the data equation can be
defined as
∗
dat
Δ
=
Ni
∑
||F p − C p ∘C p ||2
p=1
i
||F p ||2
+
,
(8.33)
−
where C p = Ep + GS ⋅ J p + GS ⋅ V ⋅ 𝛼 −p . Here we treat 𝜉 as the only independent parameter. Consequently, 𝛼 −p is an intermediate parameter that can be
obtained as the least-squares solution (see Appendix B) of (8.32); that is, 𝛼 −p =
H
H
(A ⋅ A)−1 ⋅ (A ⋅ Bp ).
The scattering strength 𝜉, and consequently the relative permittivity 𝜖r , is
obtained by minimizing the following objective function.
𝜉 = arg min (Δsta + Δdat ).
𝜉
(8.34)
We apply the Levenberg–Marquardt algorithm again to minimize this objective function.
0.8
2
0.8
2
0.4
1.5
0.4
1.5
1
0
y(λ)
y(λ)
Inversion for Phaseless Data
1
0
–0.4
0.5
–0.4
0.5
–0.8
–0.8 –0.4 0 0.4 0.8
x(λ)
0
–0.8
–0.8 –0.4 0 0.4 0.8
x(λ)
0
(a)
(b)
Figure 8.3 The pattern consisting of a circle and an annulus. (a) Exact relative permittivity.
(b) Reconstructed relative permittivity with 31.6% Gaussian white noise. Source: Pan 2011,
IEEE Trans. Geosci. Remote Sens., 49, 981–87. [34] Reproduced with permission of IEEE.
The performance of the proposed method is tested in the following
numerical example. A total number Ni = 14 transmitting antennas are
evenly distributed on a circle of radius 5𝜆, with their locations given by
, 5𝜆 ⋅ sin 2𝜋p
) p = 1, 2, ..., Ni . A total number of Ns = 30 receivers
(5𝜆 ⋅ cos 2𝜋p
Ni
Ni
are evenly distributed on a circle of radius 5.5𝜆, with their locations given by
, 5.5𝜆 ⋅ sin 2𝜋q
) q = 1, 2, ..., Ns . As depicted in Fig. 8.3(a), a circle
(5.5𝜆 ⋅ cos 2𝜋q
Ns
Ns
(with relative permittivity of 2 and radius of 0.15𝜆) is placed in the central hole
of a concentric annulus (with relative permittivity of 1.6 and inner and outer
radii of 0.4𝜆 and 0.6𝜆, respectively). For the DOI, we have chosen a square
region (with width of 1.6𝜆), which is discretized into a grid of 30 × 30 cells.
The measured intensity is contaminated with 31.6% additive white Gaussian
noise. It takes the Levenberg–Marquardt optimization algorithm 20 and 10
iterations to converge for finding 𝛼 +p and 𝜉, respectively. The reconstructed
relative permittivity is reproduced in Fig. 8.3(b), where we see that all the main
features of the scatterer’s structure (including the gap with a width of 0.25𝜆
between the central circle and the concentric annulus) can be identified clearly.
Phaseless ISPs are difficult to solve due to the missing phase information,
and many publications have cautioned that choosing a proper starting point
is crucial for their algorithms to eventually yield satisfactory reconstructed
results. Numerical simulations presented in [34] show that the extraction of
the first L major part of induced current significantly reduces the dependence
on initial guesses.
8.5 Discussions
For point-like scatterers, the two inversion methods for phase-available data,
that is, the subspace method and the compressive sensing, can be modified to
solve phaseless problems. Although the phaseless data depend nonlinearly on
223
224
Electromagnetic Inverse Scattering
the unknowns, new bases are constructed so that the phaseless ISP becomes
a linear one. The new bases inherit the property of the original bases of the
phase-available problem. For the subspace method, the singularity of the
Green’s function is the key, and the new bases exhibit the same spatial position
for which singularity occurs as the original bases do. For CS, the sparsity of the
unknown is the key, and the new bases make it possible that the corresponding
new unknown is still sparse.
For extended scatterers, the subspace-based optimization method for
phase-available ISPs can be adopted, after some modifications, to solve phaseless problems, as discussed in Section 8.4, therefore it is helpful to acquire in
advance a good understanding of the methods for phase-available ISPs.
Finally, we mention in passing that, in this chapter, the background medium
is homogeneous so that wave propagation is fully coherent. The reason why
phase information is unavailable is the high frequency of the electromagnetic
wave. This type of imaging is therefore different from diffuse optical tomography where wave propagation is incoherent (i.e., the phase information cannot
be measured) because of strong multiple scattering [35].
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9
Inversion with an Inhomogeneous Background Medium
Previous chapters dealt with inverse scattering problems with a homogeneous
background medium. This chapter deals with reconstructing dielectric scatterers embedded in an inhomogeneous background. Such inverse scattering
problems have important practical applications in nondestructive evaluation,
biomedical imaging, through wall imaging, geophysical inversion, and so on.
The organization of the chapter is as follows. Section 9.1 outlines the
main methods for solving inhomogeneous background ISPs. When a
(semi-)closed-form expression of the inhomogeneous background Green’s
function is not available, there are mainly three approaches, which are
discussed in Sections 9.2–9.4 for two-dimensional problems. Section 9.2 introduces an approach that numerically obtains an inhomogeneous background
Green’s function by solving differential equations (DE) and then applies
integral-equational based inversion algorithms to solve ISPs. Section 9.3
presents an approach that directly solves differential equations without calculating a numerical Green’s function. Section 9.4 introduces an approach that
treats the known inhomogeneous background medium as a known scatterer
rather than part of the background. Section 9.5 presents the models and
reconstruction results for two practical imaging modalities. Section 9.6 briefly
discusses several topics on inhomogeneous background ISPs.
9.1 Introduction
If an inhomogeneous background medium is so simple that a (semi-)
closed-form expression of the corresponding Green’s function is available, then
the integral-equational (IE) based inversion algorithms introduced in Chapter
6 can be used directly to solve the ISPs. For example, a half-space background
medium ISP is solved in [1–3], a through-wall-imaging (TWI) problem is
solved in [4, 5], and an ISP in presence of a conducting circular cylinder is solved
in [6]. The first two examples are typical multiple-planary-layer ISPs [7, 8].
IE-based inversion algorithms are preferred in these cases since Green’s
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
228
Electromagnetic Inverse Scattering
function provides long-range interactions between two spatial points and in
addition the inducted contrast current, as an intermediate parameter, exhibits
important physical meanings and provides much mathematical flexibility to
manipulate.
When a (semi-)closed-form expression of the inhomogeneous background Green’s function is not available, there are three main approaches
to reconstruct dielectric scatterers. The first approach is to numerically
obtain an inhomogeneous background Green’s function by solving differential
equations (DE), and then apply IE-based inversion algorithms introduced
in Chapter 6 to solve ISPs. For example, [9] applies the finite difference
(FD) method to obtain the Green’s function and then adopts the contrast
source inversion (CSI) method for inversion; [10] and [11] apply the finite
element method (FEM) to obtain the Green’s function and then, respectively,
adopt the subspace-based optimization method (SOM) and the CSI for
the inversion. The second approach directly solves differential equations
without calculating a numerical Green’s function. For example, the FD is
used in [12, 13] and the FEM is used in [14, 15]. The third approach treats
the known inhomogeneous background medium as a known scatterer rather
than part of the background. Thus, the background is simply homogeneous
and then the original inhomogeneous-background ISP can be solved by
the IE-based inversion algorithms introduced in Chapter 6. For example,
[16] adopts the SOM for the inversion. We mention in passing that these
three approaches do not assume dielectric scatterers to be homogenous. For
homogenous scatterers both the forward and inverse models will be different
from their inhomogeneous-scatterer counterparts. For example, [17] presents
an inversion method for subsurface sensing of homogeneous dielectric objects
embedded in a dispersive lossy ground with an unknown rough surface. The
topic of homogenous scatterer is beyond the scope of this chapter.
This chapter will discuss all three approaches. The first and second involve
truncating the FEM or FD computational domain, and it is very important to
choose the truncation boundary so that the computation domain is as small
as possible during each iteration step of the optimization process. There are
three kinds of boundaries in inhomogeneous background ISPs, which are
shown in Fig. 9.1. The boundary B1 encloses all scatterers, that is, any contrast
with respect to the inhomogeneous background medium, the boundary B2
encloses all inhomogeneities, that is, the region exterior to B2 is homogenous,
and the boundary B3 requires reflectionlessness for all outgoing waves. It
is obvious that B1 ⊆ B2 ⊆ B3, as far as the domains that they enclose are
concerned. If the domain of scatterers is not known a priori, then we have to
choose B1 = B2. For B3, the reflectionlessness can be realized by adding the
perfectly matching layer (PML) or applying mathematical exact or approximate
radiation boundary condition (RBC). In order to ensure high accuracy and at
the same time keep computational size at a minimum, boundary integral (BI)
Inversion: Inhomogeneous Background Medium
Figure 9.1 The illustration of three kinds of
boundaries for an inhomogeneous
background ISP where some scatterers are
located inside a square annulus
background. The boundary B1 encloses all
scatterers, i.e., anything that is different
from the inhomogeneous background
medium, the B2 encloses all
inhomogeneities, i.e., the region exterior to
B2 is homogenous, and the B3 requires
reflectionlessness for all outgoing waves.
B3
B2
B1
equation is a good candidate. In this case, the boundary B3 can be chosen to
be conformal to the B2, and very close to it in practice. In some references, the
BI equation method is referred to as the boundary element method (BEM).
An important result for inhomogeneous background ISPs is that when B1
is known a priori, the BI equation can be formulated on B1 rather than B3,
which has the advantage of limiting the computational domain within B1 at
each iteration during the optimization process. The details will be given in
Section 9.5.
We mention in passing that this chapter deals with full-wave inversion, and
thus inversion algorithms that are based on Born-type approximations will
not be discussed, such as the distorted wave Born approximation method [18]
and the FDFD-based (finite-difference frequency-domain) Born inversion
method [19].
9.2 Integral Equation Approach via Numerical Green’s
Function
Consider a two-dimensional (2D) acoustic or electromagnetic transverse magnetic (TM) scattering problem, in which the field satisfies the scalar Helmholtz
equation. Some known objects are distributed in an otherwise homogeneous
background medium (the wavenumber of which is k0 ). The known objects
and the otherwise homogeneous background are together referred to as the
known inhomogeneous background, and the distribution of the wavenumber
is denoted as kb (r), where r = (x, y) represents the spatial coordinate. Some
unknown scatterers are embedded in the inhomogeneous background. The
distribution of wavenumber of the scattering system, including the unknown
scatterers and the known inhomogeneous background, is denoted k(r). To
229
230
Electromagnetic Inverse Scattering
reconstruct the unknown scatterers, the test domain is successively illuminated by a number of incident waves and the scattered waves are measured
at an array of receivers. A total number of Ni transmitters (also known as the
primary source) are located at rip , p = 1, 2, ..., Ni , and a total number of Ns
receivers are located at rsq , q = 1, 2, ..., Ns .
The problem considered here does not know a priori within which part of
B2 the scatterers locate, then we have to choose B1 = B2. The boundary B3 is
chosen to be conformal to the B2 and very close to it.
Let’s denote the region interior to B3 to be the domain of interest (DOI) D,
and for convenience the B3 is conventionally denoted the boundary 𝜕D of D,
̂ Transmitters and receivers
the outward normal direction of which is denoted n.
are located outside D, and the medium outside D is homogeneous. The relative refractive index is introduced as n(r) = k(r)∕k0 . In electromagnetic problems, the refractive n(r) is equal to the square root of the relative permittivity
√
𝜖r (r). The relative refractive index of the known inhomogeneous background
medium is denoted as nb (r). We first let one of the transmitters to illuminate
the DOI. In D, the background field ub (r) and the total field u(r) satisfy the
following Helmholtz equations, respectively:
[∇2 + k02 n2 (r)]u(r) = 0,
(9.1)
[∇2 + k02 n2b (r)]ub (r) = 0.
(9.2)
The combination of (9.2) and (9.1) yields
[∇2 + k02 n2b (r)](u(r) − ub (r)) = −k02 [n2 (r) − n2b (r)] u(r).
(9.3)
The right-hand side is written in shorthand as −J con (r), which is interpreted as
the contrast source (also known as the secondary source). The contrast field
u(r) − ub (r) is denoted as uc (r). It is easy to see that the contrast field can be
understood as the reradiated field by the contrast source in the inhomogeneous
background. (9.3) can be converted to the “weak” form by multiplying both
sides with a testing function T(r) and performing an integration over D,
∫ ∫D
(∇T(r) ⋅ ∇uc (r) − k02 n2b (r)T(r)uc (r))ds
=
∫ ∫D
T(r)J con (r)ds +
∮𝜕D
T(r)
𝜕uc (r(t))
dt
𝜕n(t)
(9.4)
𝜕
means the
where t is a parametric variable along the boundary 𝜕D and 𝜕n
derivative along the normal direction of 𝜕D. In the exterior of 𝜕D, the contrast
field can be formulated by the Green’s theorem,
[
]
𝜕u (r′ (t ′ ))
𝜕G(r, r′ (t ′ ))
− c ′ G(r, r′ (t ′ )) + uc (r′ (t ′ ))
dt ′ , (9.5)
uc (r) =
∮𝜕D
𝜕n(t )
𝜕n(t ′ )
where G denotes the Green’s function for the homogeneous medium, the
wavenumber of which is k0 , as discussed at the beginning of this section.
Inversion: Inhomogeneous Background Medium
When (9.5) is evaluated at the boundary 𝜕D (approaching the boundary from
outside), the boundary condition yields
[
]
′ ′
𝜕uc (r′ (t ′ ))
′ ′
′ ′ 𝜕G(r(t), r (t ))
uc (r(t)) +
G(r(t), r (t )) − uc (r (t ))
dt ′ =0.
∮𝜕D
𝜕n(t ′ )
𝜕n(t ′ )
(9.6)
Once the contrast field on the boundary and its normal derivative are obtained
from Eqs. (9.4) and (9.6), the scattered field is subsequently obtained from (9.5)
by choosing r to be the position of receiver.
In this section, the two-dimensional inhomogeneous background scattering
problem is solved by the FEM method (see chapter 3 of [20] and section 10.2 of
[21]). The domain D is discretized into a total number of Ncell rectangular-cell
meshes. The number of interior nodes is denoted as Nint and the the number
of nodes located on the boundary 𝜕D is denoted as Nbou . The mesh size is
so small that the relative refractive index is treated as a constant nl within
the lth cell, l = 1, 2, ..., Ncell . The interior contrast field is expanded in basis
functions
uc (r) ≅
Nint
∑
∑
Nint +Nbou
uint
n Bn (r) +
n=1
ubou
n Bn (r),
(9.7)
n=Nint +1
bou
where Bn (r) denotes the basis function centered at node n, and uint
n and un
con
represent the value of contrast field at that node. The contrast source J (r)
is expanded on the same basis. Here we adopt the bilinear basis function for
rectangular cells (see section 9.3 of [20]). The contrast field at the boundary 𝜕D
is automatically given by
∑
Nint +Nbou
uc (r(t)) ≅
ubou
n Bn (t),
(9.8)
n=Nint +1
where Bn (t) is the projection of Bn (r) onto the boundary. Note that the approximate sign in (9.7) and (9.8) is due to numerical discretization, and the same
applies to (9.9).
The notation J bou (t) is introduced as a shorthand for 𝜕uc (r(t))∕𝜕n(t) and it is
interpreted as the equivalent surface source. The J bou (t) can also be expanded
in basis functions as
J bou (t) ≅
Nbou
∑
Jnbou B̃ n (t),
(9.9)
n=1
where B̃ n (t) denotes the basis function for the equivalent boundary source and
is chosen as the piecewise-constant function. For a boundary point with parameter t that is located in the n-th boundary line element, the value of J bou (t) is
given by Jnbou .
231
232
Electromagnetic Inverse Scattering
If Eqs. (9.7) and (9.9) are substituted into to (9.4), a total number of Nint + Nbou
linearly independent equations can be generated, where the testing functions
are chosen as the basis functions,
n = 1, 2, ..., Nint + Nbou .
T(r) = Bn (r),
(9.10)
This method is the well known Galerkin FEM. The resulting matrix equation
has the form
] [
[
]
int
cros, A
X
X
uint
c
⋅
cros, B
bou
ubou
c
X
X
]
[
[
]
int
cros, A
0
Z
Z
con
⋅J +
=
(9.11)
bou
cros, B
bou
Y ⋅J
Z
Z
int
con
where J is an (Nint + Nbou )-dimensional vector, X , X
have the common form
X mn =
int
cros, A
∫ ∫D
∇Bm ⋅ ∇Bn − k02 n2b Bm Bn ds,
cros, B
cros, A
,X
cros, B
, and X
bou
(9.12)
bou
Z ,Z
,Z
, and Z have the common form Zmn = ∫ ∫D Bm Bn ds, and
the entries of Y are given by Y mn = ∮𝜕D Bm B̃ n dt.
If Eqs (9.8) and (9.9) are substituted into (9.6) and in the meanwhile testing
functions are chosen as the Dirac delta function located in the center of each
line element around the boundary 𝜕D, a total number of Nbou linearly independent equations are generated, which are expressed in the matrix form as
L ⋅ ubou
+M⋅J
c
bou
= 0,
(9.13)
where the matrices L and M are calculated as integrals over the line element of
the boundary 𝜕D, the detail of which can be found in [20] (section 3.10). The
substitution of (9.13) into (9.11) yields
] [
[
] [
int
cros, A ]
int
cros, A
X
X
uint
Z
Z
con
c
⋅J ,
⋅
=
cros, B
bou
−1
cros, B
bou
bou
u
c
X
X +Y ⋅M ⋅L
Z
Z
(9.14)
For convenience, the matrices on the left- and right-hand sides are denoted as
A and B, respectively. From (9.14), the contrast field is obtained by
uc = GD ⋅ J
−1
con
,
(9.15)
where GD = A ⋅ B has a size of (Nint + Nbou ) × (Nint + Nbou ). The ordering
of the nodes presented in this paper yields a dense matrix only at the bottom right-hand corner of the matrix A. Considering the fact that Nbou is
usually much smaller than Nint , the inversion of (9.14) can be implemented
Inversion: Inhomogeneous Background Medium
by the matrix-partitioning method. The overall computational complexity is O ((Nint + Nbou )1.5 ), noting that the number of nodes on one side of
square-shape 𝜕D is roughly (Nint + Nbou )0.5 . The M is a square matrix of size
−1
Nbou . Since the size of M is much smaller than the size of A, the value of M ⋅ L
can be obtained by solving a linear equation using the LU decomposition and
the computational cost is much smaller compared with the one used in solving
−1
A ⋅ B.
It is noted that the operator GD obtained by the FEM defines a mapping from
the source at the nodes of cells to the field evaluated also at the nodes of cells. It
is different from its counterpart in earlier versions of SOM with homogeneous
background, where the mapping is from the source at the centers of cells to
the field evaluated also at the centers of cells. Furthermore, define the matrix
P, which picks up the boundary nodes out of all nodes, as [0Nbou ×Nint , I Nbou ×Nbou ],
where I denotes an identity matrix. Thus, the contrast field at the boundary is
obtained from (9.15),
ubou
= P ⋅ GD ⋅ J
c
con
.
(9.16)
The scattered fields at all receivers can be obtained from (9.5) and can be
written in an Ns -dimensional vector,
bou
usca
− Ms ⋅ J
c = −Ls ⋅ uc
bou
,
(9.17)
where Ls and Ms are similar to L and M, respectively, with only minor differences. Upon the substitution of Eqs (9.13) and (9.16), the scattered fields can
be written in a compact form as,
usca
c = (−Ls + M s ⋅ M
−1
⋅ L) ⋅ P ⋅ GD ⋅ J
con
,
(9.18)
con
which is written in shorthand as usca
, where the dimension of GS is
c = GS ⋅ J
Ns × (Nint + Nbou ). Once again, the operator GS obtained by the FEM defines a
mapping from the source at the nodes of cells to the scattered field measured at
receivers. It is different from its counterpart with an homogeneous background,
where the mapping is from the source at the centers of cells to the scattered field
measured at receivers.
The definitions of J con (r) and uc (r) imply the following equation,
J con (r) = k02 [n2 (r) − n2b (r)](ub (r) + uc (r)).
(9.19)
When this equation is evaluated at the centers of the cells, it can be written in
a compact form,
C⋅J
con
con
= k02 𝜉 ⋅ (ub + C ⋅ GD ⋅ J
con
),
(9.20)
where J , evaluated at the nodes of cells, has been defined earlier, C is a matrix
of dimension Ncell × (Nint + Nbou ) that assigns the value to the center of a cell
233
Electromagnetic Inverse Scattering
by averaging the values at its four nodes, and 𝜉, which is referred to as the
contrast matrix, is a diagonal matrix of dimension Ncell × Ncell with diagonal
elements being [n2 (r) − n2b (r)]. It is noted that ub is an Ncell -dimensional vector evaluated directly at the centers of cells, which is readily available for the
given inhomogeneous background. Equations (9.18) and (9.20) are referred to
as the data equation and the state equation, respectively. They are the inhomogeneous background counterpart of the data equation and the state equation
that are presented in Chapter 6, and thus the IE-based inversion algorithms can
be adopted to solve the inhomogeneous background ISPs.
In the following numerical example, the subspace-based optimization
method (SOM) is adopted to solve the inhomogeneous background ISP. The
DOI D is a square of side length 2𝜆, where 𝜆 is the wavelength in air, and D
is discretized into 44 × 44 square cells in the inverse problem. The number of
transmitters is 30 and the number of receivers is 40, and they are uniformly
distributed on a circle of radius 3𝜆. The FEM is employed as the forward
problem solver. To avoid the inverse crime, the forward problem uses a 60 × 60
discretization, which is finer than the one used in the inverse problem. Then
10% additive white Gaussian noise is added to the exact data, and the resultant
noisy data is treated as the measured scattering data. The inhomogeneous
background medium, as shown in Fig. 9.2, is a square wall, with an outer√side
length 1.6𝜆 and inner side length 1.1𝜆, and its relative refractive index is 1.5.
There are three scatterers, as illustrated in Fig. 9.3(a). The first scatterer is an
annulus, with inner√and outer radii 0.25𝜆 and 0.4𝜆, respectively, and its relative
refractive index is 2. The second scatterer is a square inhomogeneity located
inside the wall. Its side length
is 0.2𝜆 and is centered at (0.65𝜆, −0.3𝜆). Its rel√
ative refractive index is 2.5. The third scatterer is an extra rectangle attached
to the wall, and it has the same relative refractive index as that of the wall.
The rectangle is located at (−0.2𝜆, 0.875𝜆) and its size is 0.4𝜆 × 0.15𝜆. After
1
2.5
0.5
2
y (λ)
234
0
1.5
–0.5
–1
–1
–0.5
0
x (λ)
0.5
1
1
Figure 9.2 The configuration of the
inhomogeneous background
medium. It is a square wall with an
outer side length 1.6𝜆 and inner side
length
1.1𝜆, and its refractive index is
√
1.5. The shaded bar represents the
value of the square of refractive
index, i.e., n2 . Source: Chen 2010,
Inverse Problems, 26, 074007. [10]
Reproduced with permission of IOP
Publishing.
Inversion: Inhomogeneous Background Medium
1
2
0
1.5
–0.5
–1
–1
0
1
1
2.5
0.5
y (λ)
0.5
y (λ)
1
2.5
2
0
1.5
–0.5
–1
–1
0
x (λ)
x (λ)
(a)
(b)
1
1
Figure 9.3 The configuration of the scatterer. (a) Exact profile, where three scatterers are
present, i.e., an annulus, a square inside the wall, and a rectangle attached to the wall. (b)
Reconstructed profile at the 50th iteration. Source: Chen 2010, Inverse Problems, 26, 074007.
[10] Reproduced with permission of IOP Publishing.
50 iterations, the FEM-based SOM obtains the reconstructed result shown
in Fig. 9.3(b). The annulus and the square inside the wall are obviously well
reconstructed. The reconstruction of the extra rectangle attached to the wall is
a little poor. The reason for this is that the extra rectangle has a weaker contrast
with the background medium. Nevertheless, Fig. 9.3(b) clearly shows that
there is a weak scatterer present in the correct position. The reconstruction
can also be quantitatively evaluated by the relative error, which is defined as
∑Ncell |n2ret (ri ) − n2 (ri )|
i=1
n2 (ri )
,
𝛿r =
Ncell
(9.21)
where nret (ri ) denotes the “retrieved” refractive index of the ith cell. The
relative error of this numerical simulation is equal to 3.7%.
9.3 Differential Equation Approach
This section directly solves differential equations without calculating numerical
Green’s function. The presented inversion model, based on the finite difference,
does not need to solve the forward problem in each iteration of the optimization
process.
We consider a two-dimensional (2D) acoustic or TM electromagnetic inverse
scattering scattering problem. The configuration of the ISP is the same as that
in Section 9.2, except that the boundary B3 (that is also 𝜕D) is chosen to be
the smallest rectangle enclosing the B2 for the purpose of using the FD. Once
again, we do not know a priori within which part of B2 the scatterers locate and
consequently we set B1 = B2.
235
236
Electromagnetic Inverse Scattering
We use the FD method to discretize the DE. We first discretize the rectangular D into Mx × My small cells. The contrast is spatially uniform within each cell.
We assume that contrast fields are linearly changing from cell to cell, and we
denote the contrast fields at the centers of cells as (uc )m,n with 1 ≤ m ≤ Mx and
1 ≤ n ≤ My , which is different from the method in Section 9.2 where the contrast fields at the corners of cells are sampled. Since there is a Laplace operator
in (9.3), we apply the finite difference scheme and define an operator as,
′
Pm,n
≡ (∇2 uc )m,n =
(uc )m+1,n + (uc )m−1,n − 2(uc )m,n
+
Δx2
(uc )m,n+1 + (uc )m,n−1 − 2(uc )m,n
Δy2
(9.22)
for m ∈ [1, Mx ] and n ∈ [1, My ], where Δx and Δy are the dimensions of the
small cells along the x and y directions, respectively. It is obvious that the finite
difference scheme requires to extend the computational domain to the −x, +x,
−y, and +y directions with one additional cells. The contrast fields at the centers
of these extended cells are denoted as (uc )m,n , where m = 0 or Mx + 1, n =
1, … , My are for vertical cells and m = 1, … , Mx , n = 0 or My + 1 are for
horizontal cells, as shown in Fig. 9.4.
However, the extension introduces 2(Mx + My ) additional unknowns at the
centers of extended cells. On the other hand, these 2(Mx + My ) additional
unknowns are related to the field inside the domain D by the Huygens’
surface equivalence principle. The integral path in (9.5) is chosen as 𝜕D that
is shown in Fig. 9.4. For the contrast fields that appear in the integrand of
(9.5), we use the fields on both sides of the boundary to approximate them,
such as [(uc )0,n + (uc )1,n ]∕2 with n ∈ [1, My ] on one of the boundaries. For
the normal derivative of the field, we apply the finite difference rule, such as
∆y
∂D
∆x
Figure 9.4 The FD scheme requires
extension of the computational
domain with one additional cell
outward. The cells that are exterior to
the boundary 𝜕D are the extended
cells. Their centers are marked with
crosses. The cells marked with solid
dots are referred to as the boundary
cells and the cells marked with circles
are referred to as interior cells.
Inversion: Inhomogeneous Background Medium
[(uc )0,n − (uc )1,n ]∕Δx with n ∈ [1, My ] on one of the boundaries. If we place the
observation point of (9.5) at the centers of the extended cells, meaning that the
left-hand side of the equation are the fields at the centers of those extended
cells, then we obtain a total number of 2(Mx + My ) linear equations, which can
be written as
bou
uext
+ B ⋅ uext
c = A ⋅ uc
c
(9.23)
is a [2(Mx + My ) − 4]-dimensional vector, representing the contrast
where ubou
c
field at the center of boundary cells (marked with solid dots in Fig. 9.4), uext
c is a
[2(Mx + My )]-dimensional vector, representing the contrast field at the center
of extended cells, and A and B are matrices involving both the Green’s function and its normal derivative. With the help of (9.23), the contrast fields at the
extended cells can be expressed as a linear superposition of the contrast fields
at the boundary cells,
−1
uext
⋅ A ⋅ ubou
c = (I − B)
c
(9.24)
which consequently means that the unknown field variables now consist of only those in the domain D. The contrast field in D, denoted as
an Mx My -dimensional vector uc , consists of two parts: a total number of
(Mx − 2)(My − 2) fields at the centers of interior cells (marked with circles
in Fig. 9.4) and a total number of [2(Mx + My ) − 4] fields at the centers of
boundary cells. The latter has been denoted as ubou
and the former is now
c
int T
bou T T
defined as uint
c . That is, uc = [(uc ) , (uc ) ] , where the superscript T denotes
the transpose.
The Helmholtz equation (9.3) can be written into the FD format,
[P + k02 n2b ] ⋅ uc = −k02 𝜉 ⋅ (ub + uc ).
(9.25)
where 𝜉, which is referred to as the contrast matrix, is a diagonal matrix
with the dimensions Mx My × Mx My with the diagonal elements being
[n2 (r) − n2b (r)], and the matrix n2b of the same size is defined similarly. The
Mx My × Mx My -dimensional matrix P is defined with the help of the definition
of FD Laplace operator P′ in (9.22) and the relationship between the fields at
the centers of extended and boundary cells (9.24).
From (9.5) and (9.23), we write the scattered fields measured at receivers as
−1
usca
⋅ A] ⋅ ubou
c = [As + Bs ⋅ (I − B)
c ,
(9.26)
where As and Bs are defined in a way similar to A and B, respectively, with the
only minor difference that the observation positions are shifted. Equation (9.26)
can be written in shorthand as
bou
usca
c = GS ⋅ uc
(9.27)
where the dimension of GS is Ns × [2(Mx + My ) − 4], which maps from the contrast electric field at the center of boundary cells to the scattered field measured
237
238
Electromagnetic Inverse Scattering
at receivers. Equation (9.25) and (9.27) are the state equation and data equation,
respectively.
In solving the inverse problem, motivated by the modified gradient method
[22], we treat the contrast and the contrast field as independent unknowns,
which has the merit of avoiding solving the forward problem. The objective
function consists of two parts, that is, the relative residual in the state equation
Δsta (uc , 𝜉) = ‖(P + k02 n2b ) ⋅ uc + k02 𝜉 ⋅ (ub + uc )‖2 ∕‖k02 ub ‖2
(9.28)
and the relative residual in the data equation,
bou 2
sca 2
Δdat (uc ) = ‖usca
c − GS ⋅ uc ‖ ∕‖uc ‖
(9.29)
Then the objective function accounts for the total relative residuals for all
incidences,
f (uc1 , uc2 , … , ucNi , 𝜉) =
Ni
∑
dat
Δsta
p + Δp
(9.30)
p=1
It is easy to see that the objective function is a quadratic in both uc and 𝜉 individually, and consequently it is a quartic function of the unknowns uc and 𝜉
simultaneously.
The objective function (9.30) can be minimized by updating the contrast and
the contrast field alternatively during the optimization process. Nevertheless,
the performance can be improved by incorporating the subspace technique. As
practiced in Gs-SOM in Chapter 6, we are able to first determine the major part
at boundary cells by using the singular value decomof the contrast fields ubou
c
position (SVD) of GS , and then obtain the minor part of the contrast fields by
∑
,
solving an optimization problem. Denoting the SVD formula as GS = j uj 𝜎j 𝑣H
j
where the superscript H denotes the Hermitian, the major part and the minor
part of the contrast field at boundary cells can be, respectively, expressed as
)
(
⋅ usca
∑ uH
∑
c
j
bou+
bou−
H
uc
=
𝑣j
, uc
= I 2(Mx +My )−4 −
𝑣j 𝑣j ⋅ 𝛼, (9.31)
𝜎j
𝜎 ≥𝜎
𝜎 ≥𝜎
j
L
j
L
where 𝜎L is a chosen singular value that is above a certain noise level and the
method of choosing the cutoff integer L can be found in Chapter 6. I 2(Mx +My )−4
is a 2(Mx + My ) − 4 dimensional identity matrix, and 𝛼 is a 2(Mx + My ) − 4
dimensional unknown vector, which is to be reconstructed via optimization.
The objective function (9.30) now depends on uint
cp , 𝛼 p , p = 1, 2, … , Ni , and 𝜉.
In implementing the alternative update scheme in each iteration of the
inversion, when updating the contrast fields (uint
cp and 𝛼 p ), the contrast 𝜉 is
treated to be known, and likewise for updating the contrast. The update of the
scattered fields uses the Polak–Ribière conjugate gradient (CG) method (see
Section 6.4.1). Since the DE is used to describe the field behaviors, the scope of
such a description is a local one; that is, there is no long-distance interaction
Inversion: Inhomogeneous Background Medium
that has been displayed by the Green’s function in the integral equation
method. Therefore, when updating the scattered fields, we will increase the
number of iterations of the CG algorithm in every inversion iteration, such
that the fields can be updated in a large domain instead of the local updating.
The number of the CG iterations should be at the level of the largest number
of cells in one direction of the DOI.
The contrast 𝜉 is updated by solving a least squares problem in every iteration
of the optimization, which is
∑Ni
2
2 2
∗
p=1 [−k0 (ub(m) + uc(m) )]p [[P ⋅ uc ](m) + [k0 nb ⋅ uc ](m) ]p
(9.32)
𝜉(m) =
∑Ni
2
2
p=1 |−k0 (ub(m) + uc(m) )|p
where the superscript ∗ denotes the complex conjugate and the subscript m
denotes the mth cell, m = 1, 2, … , Mx My .
To summarize, in each iteration of the inversion, there are a number of iterations of the CG method for the update of the contrast fields, followed by one
update for the contrast function.
The performance of the proposed DE-based inversion algorithm is tested
for the well-known Austria profile, with relative permittivity 𝜖r = 2.0, that is,
n2 = 2.0. Other parameters in the system setup can be found in Section 6.2.5.
The background medium is chosen to be air; that is, n2b = 1.0. The measured
scattered field is contaminated with 10% additive white Gaussian noise. In solving the inverse problem, the domain is discretized into 60 × 60 meshes, the
parameter L in the Gs-SOM is chosen as 15. The number of CG iterations is
chosen to be 50 in every iteration of the inversion to update the contrast fields.
The inversion results after 100 iterations are shown in Fig. 9.5, where we see
that the DE inversion method successfully reconstructs the scatterer profile.
0.5
0.5
0
0
–0.5
–0.5
–0.5
1
0
1.5
0.5
2
–0.5
–0.1
0
0
0.5
0.1
Figure 9.5 Reconstruction results by the finite-difference inversion method: Left and right
panels show the real and imaginary parts of relative permittivity, respectively. Source: Zhong
2014, Differential equation based inversion method for solving inverse scattering problems,
Inverse Problems – from Theory to Applications (IPTA 2014), 90–94. [35] Reproduced with
permission of IOP Publishing.
239
240
Electromagnetic Inverse Scattering
9.4 Homogeneous Background Approach
This section presents an approach that treats the known inhomogeneous background medium as a known scatterer rather than part of the background. Thus,
the background is simply homogeneous and then the ISPs can be solved by
IE-based inversion algorithms. The advantages of this approach are that numerical evaluation of Green’s function is avoided, the homogenous background
Green’s function provides long-range interactions, and the induced contrast
current, as an intermediate parameter, exhibits important physical meanings
and provides much mathematical flexibility to manipulate.
The known objects and the otherwise homogeneous background (with
the permittivity 𝜖h ) are together referred to as the known inhomogeneous
background, and its permittivity distribution is denoted as 𝜖b (r). When
some unknown scatterers are embedded in the inhomogeneous background
medium, the final distribution of the permittivity is denoted as 𝜖(r). In
terms of the contrast with the homogeneous background medium, we have
𝜒(r) = 𝜒s (r) + 𝜒b (r), where 𝜒s (r) = [𝜖(r) − 𝜖b (r)]∕𝜖h denotes the normalized
contrast of unknown scatterer and 𝜒b (r) = [𝜖b (r) − 𝜖h ]∕𝜖h is the normalized
contrast of known background scatterer. Inversion algorithms that treat the
known inhomogeneous background medium as a known scatterer can be
categorized into two types. The first type deals with the case that unknown
scatterers do not overlap with the known inhomogeneous background scatterer, that is, for any spatial point r, if 𝜒s (r) is nonzero, then 𝜒b (r) must be
zero. Simply speaking, the unknown scatterers are separable from the known
background scatterers that are, for convenience, referred to as the obstacle.
For convenience, this type of problem is denoted an SOP; a separable obstacle
problem. Many real-world problems fall into this type, such as inspecting
the contents of closed parcels in standard shipment packages. The second
type allows unknown scatterers to be overlapped with the obstacle, which is
denoted as OP; an obstacle problem. Since an homogeneous background is
used, the inversion algorithms that solve these two kinds of ISPs are referred
to as the SOP- and OP-homo scheme, respectively.
An SOP is solved in [16] where the IE-based inversion method, the SOM,
is adopted to reconstruct 𝜒s (r) after a minor modification that restricts the
domain of updating the contrast to only those spatial points where 𝜒b (r) = 0.
Numerical simulations show that although there are various advantages for
the SOP-homo scheme, such as avoiding the computational time and numerical inaccuracies in evaluating inhomogeneous background Green’s functions, a
trade-off is also involved. In the SOP-homo scheme, although we have excluded
the obstacle from the process of updating unknowns, the induced contrast currents exist in both the known obstacle and the unknown scatterers. That is,
the contrast current occurs wheresoever 𝜒s (r) ≠ 0 or 𝜒b (r) ≠ 0. The multiple
scattering between unknown scatterers and the known obstacle increases the
nonlinearity of the problem formulated as the SOP-homo scheme. In com-
Inversion: Inhomogeneous Background Medium
parison, in OP-inhomo or SOP-inhomo schemes, inhomogeneous background
Green’s functions are numerically evaluated and consequently the contrast currents occur only at positions with 𝜒s (r) ≠ 0. This is because the multiple scattering effect between the scatterers and the obstacles has already been accounted
for in the inhomogeneous background Green’s function. As a consequence,
the SOP-homo has a higher nonlinearity compared to OP/SOP-inhomo and
thus may reduce the quality of reconstruction; for example, in cases where the
known obstacle has a higher permittivity than the unknown scatterer.
The OP/SOP-homo schemes are tested by experimental data and
are compared with the OP/SOP-inhomo schemes. A two-dimensional
through-wall-imaging (TWI) experiment has been conducted at Zhejiang
University [23]. The experimental setup is of the transverse magnetic (TM)
type. Figure 9.6(a) and (b) show the schematic diagram and the actual experimental setup. The system is designed to work at 2.4 GHz. Twenty-four regular
patch antennas linearly polarized along the vertical direction are evenly
distributed on a circle with a diameter of 113 cm. Figure 9.6(c) and (d) show
the configuration of the scatterer and the inhomogeneous background. Four
Teflon boards with the same length of 100 cm and a relative permittivity of
wall (target inside)
imaging
domain
receiver
microwave
switches
emitter
target
emitter
receiver
trun table
(a)
(b)
3.5 cm
foam
cylinder
5 cm
21 cm
wall
1 cm
(c)
(d)
Figure 9.6 Experimental setup of the through-wall-imaging problem: (a) 2D schematic
diagram of the experimental setup; (b) photograph of the experimental setup; (c) cylindrical
scatterer is at the center; and (d) cylindrical scatterer is off center. Source: Meng 2016,
Electronics Letters, 52, 1933–1935. [23] Reproduced with permission of Institution of
Engineering and Technology (IET).
241
242
Electromagnetic Inverse Scattering
–λ
3
–λ
3
0
2
0
2
1
λ
–λ
λ
–λ
0
(a)
λ
0
(b)
λ
1
–λ
3
–λ
3
0
2
0
2
1
λ
–λ
λ
–λ
0
(c)
λ
0
(d)
λ
1
Figure 9.7 Comparison of inversion results for the off-center cylinder. Inversion result by (a)
OP-homo; (b) SOP-homo; (c) OP-inhomo; and (d) SOP-inhomo. Source: Meng 2016,
Electronics Letters, 52, 1933–1935. [23] Reproduced with permission of Institution of
Engineering and Technology (IET).
around 2.1 are used to construct the wall, which is known a priori and treated
as the background medium. A 100-cm-long organic glass cylinder is used as
the scatterer, whose relative permittivity is around 3.0. The cylinder is placed
at the center of the wall in Fig. 9.6(c) and it is moved to an off-center place
3.5 cm from one side of the wall in Fig. 9.6(d). For the given background
wall, by subtracting the incident fields from the total fields, we obtain a
12 × 23 matrix recording the scattered fields. These scattering data are further
calibrated using the “incidence calibration” method [23]. The inversion results
for the centered-cylinder (Fig. 9.6(c)) are presented [23]. The inversion
results for the more challenging off-center case (Fig. 9.6(d)), obtained by the
OP/SOP-homo and OP/SOP-inhomo schemes, are shown in Fig. 9.7. Note
that the standard twofold subspace-based optimization method (TSOM)
described in Section 6.4.2 is used to solve the OP-homo scheme. It can be
Inversion: Inhomogeneous Background Medium
seen that the inversions based on the inhomogeneous-background Green’s
function (Fig. 9.7(c) and (d)) produce better reconstruction results than their
homogenous-background counterparts (Fig. 9.7(a) and (b)). The gap between
the cylinder and the wall is more sharply distinguished.
When we directly apply the SOM or TSOM to solve inhomogeneous
background ISPs under the OP/SOP-homo schemes, usually we use the
background’s information 𝜒b (r) as the initial guess. At the first sight, it
seems hard to incorporate other information about the known background
obstacle under the OP/SOP-homo schemes. Recently, [24] provided a detailed
analysis of the induced contrast current to improve the performance of
OP/SOP-homo schemes. It explicitly used the knowledge of the field Eb (r)
generated when only the known background obstacle exists. The total
field is written as E(r) = Eb (r) + Es (r), where Es (r) is the scattered field
due to the presence of unknown scatterer. The contrast current is then
J(r) = [𝜒s (r) + 𝜒b (r)] ⋅ [Eb (r) + Es (r)], which radiates in the homogeneous
background. The details of the method can be found in [24].
It is worth noting that these OP/SOP-homo schemes are exact models; that
is, there is no approximation made. When certain approximations are made,
alternative OP/SOP-homo schemes can be proposed. For example, in solving
an SOP, [25] assumes that the scattered field Es (r) is much smaller than the
background field Eb (r) in the domain of the scatterer, which can be considered as a Born approximation with respect to an inhomogeneous background
medium. In [25], the “normal medium” is a horizontally layered medium. The
so-called “normal medium”, a terminology from [25], is a more general concept
than the homogenous background medium since it covers all cases where an
analytical Green’s function is available.
9.5 Examples of Three-Dimensional Problems
In many real-world inhomogeneous background ISPs, it is known that the scatterers are located within a certain region; that is, the B1 in Fig. 9.1 is known a
priori. In this case, the BI equation can be formulated on the B1, instead of
conventionally on the B3, which has the tremendous advantage of limiting the
computational domain within B1 at each iteration during the optimization process. This section focuses on three-dimensional inhomogeneous background
ISPs where B1 is interior to B2.
In the region of scatterer, that is, B1 and its interior region, we apply the
three-dimensional FEM-BI method (see section 10.5 of [21]). We label the interior region as V and its boundary B1 as S. The outward normal direction is
̂ The field in V satisfies the vector wave equation
denoted as n.
∇ × ∇ × E(r) − k02 𝜖r (r)E(r) = 0
(9.33)
243
244
Electromagnetic Inverse Scattering
which is in fact (2.114) for nonmagnetic materials. Using a test function T to
dot multiply (9.33), integrating it over the domain V , and using the divergence
theorem, we obtain the weak-form equation,
∫ ∫ ∫V
(∇ × T) ⋅ (∇ × E) − k02 𝜖r T ⋅ EdV + i𝜔𝜇0
∯
T ⋅ (n̂ × H)dS = 0
S
(9.34)
where E is the electric field in the domain V and H is the magnetic field on the
boundary S.
In the exterior region, that is, the region outside of B1, the electric field is
given by the Huygens’ principle introduced in Section 2.12,
E(r) = Ei (r) +
∯S
̂ ′ ) × H(r′ )]
i𝜔𝜇0 GE (r, r′ ) ⋅ [n(r
̂ ′ )]dS(r′ )
+ GEM (r, r′ ) ⋅ [E(r′ ) × n(r
(9.35)
̂ (9.35) leads to
For an r that approaches the boundary S, n×
n̂ × E = n̂ × Ei + n̂ ×
∯S
̂
i𝜔𝜇0 GE ⋅ [n̂ × H] + GEM ⋅ [E × n]dS
(9.36)
If we discretize the rectangular cuboid domain V into rectangular brick elements, the boundary S is consequently discretized into rectangular patch elements. We choose the rectangular brick edge-based vector elements Nei , which
is curl-conforming, as the expansion bases for the electric field Ee in the eth
brick element, e = 1, 2, … , M, where M is the total number of brick elements.
∑12
We have Ee = i=1 𝛼ie Nei , where 12 is the total number of expansion bases in the
rectangular brick element. We choose the rectangle edge element Ssj = n̂ × Nsj
to expand the boundary field n̂ × Hs in the sth rectangular patch element, s =
1, 2, … , Ms , where Ms denotes the total number of rectangular patch elements.
∑4
We have n̂ × Hs = j=1 𝛽js Ssj , where 4 is the total number of expansion bases in
the rectangular patch elements. Since a rectangular patch element is one of the
six facets of an outermost brick element, the expansion of n̂ × E on the boundary S automatically follows from the expansion of E in the brick elements. We
apply Galerkin’s technique to (9.34), that is, the test function T is chosen as the
basis function Nei , use a global numbering of all edges, and eventually yield a
linear equation system,
K ⋅𝛼+B⋅𝛽 =0
(9.37)
We apply the test function T = Nsj (using a global numbering of all edges) to
(9.36), integrate it over the surface S, and obtain
P⋅𝛼+Q⋅𝛽 =b
(9.38)
Inversion: Inhomogeneous Background Medium
In the forward problem, the vector b on the right-hand side of (9.38) is given
and the unknown coefficients 𝛼 and 𝛽 are solved from (9.37) and (9.38). Then
the scattered field can be calculated by the integral that appears in (9.35),
s
s
s
E = P ⋅ 𝛼b + Q ⋅ 𝛽
where 𝛼 b
s
(9.39)
denotes extracting the values of 𝛼 for the edges located on the boundary S. P is slightly different from P, with the observation points shifted to the
s
positions of receivers. The same applies to Q and Q.
When we apply the above procedure to solve the forward or inverse problem,
we first need to numerically obtain the dyadic Green’s functions GE (r, r′ ) and
GEM (r, r′ ) since their analytical formulas usually do not exist. In such computations, we place on the boundary S the electric dipole and magnetic dipole that
are tangent to the surface. Electric fields are evaluated at both the boundary
S and at the receivers. Such computations can be implemented in commercial
software. It is obvious that the computation of dyadic Green’s functions can be
implemented in parallel.
For the inverse problem, the objective function can be constructed as a standard one that quantifies the mismatch between measured and calculated scattered fields. The unknown parameters, which are the relative permittivities of
brick elements inside the boundary B1, are updated during the optimization
process. It is worth mentioning that at each iteration of optimization, the matris
s
ces P , Q , P, Q, and B are unchanged since they are independent of scatterers.
Thus, the numerical evaluation of dyadic Green’s functions is implemented only
once and the data are stored in a library for later use. It is stressed that the
computational domain is always restricted to the boundary B1 and its interior
region during the optimization process.
There are two important issues to note when dealing with numerical dyadic
Green’s functions. The first is the choice of medium in the region V , which is
interior to B1. As discussed in Section 2.12 when applying the Huygens’ principle to express the scattered field outside of the boundary B1, we have the freedom to choose the medium that is interior to B1. In principle, the medium interior to B1 can be chosen to be an arbitrary isotropic medium, but care should be
taken of the region in the near neighborhood of B1. For the sake of simplifying
the evaluation of the integrals appearing in P and Q, it is desirable to choose the
medium at the interior side of B1 to be the same as the medium on the other
side. That is, we expect that the electric/magnetic dipoles placed at the center
of rectangular patch elements on B1 are surrounded by a homogenous medium
within a finite region. The second issue concerns with the evaluation of the integrals appearing in P and Q. When the source cell (with coordinate r′ ) is different
from the field cell (with coordinate r), the integral can be evaluated numerically
245
246
Electromagnetic Inverse Scattering
using the mid-point integration or more accurate three-point Gauss–Legendre
quadrature. When the source cell coincides with the field cell, care should to
taken to deal with the singularity. We decompose the Green’s function into the
homogeneous Green’s function and the perturbation dyadic Green’s function.
The homogeneous Green’s function has an analytical form and the corresponding integral can be calculated by the standard method in FEM (see section 10.3
of [21]). The perturbation dyadic Green’s function is a smooth function and its
integral can be numerically evaluated using two sets of integration points that
do not coincide.
This procedure has been applied to two real-world applications, which will
be introduced in the next two subsections.
9.5.1
Confocal Laser Scanning Microscope
Confocal laser scanning microscopy (CLSM) is a valuable tool for obtaining
high resolution images and has been applied in many scientific disciplines,
ranging from biology to semiconductor industry. Figure 9.8 illustrates the
schematic of a confocal optical microscope. Its imaging model consists of
three subsystems: focusing of incident light, interaction of focal field with the
sample, and detection of the scattered light [26] (section 4.3). For a scanning
system, the sample is scanned relative to the optical system and for each
scanning point the total intensity of the light passing through a finite-sized
detector pinhole is collected at the detector.
The laser light with a wavelength of 405 nm is focused by the objective lens
and the focal field illuminates object structures attached to the substrate, which
Huygens’ surface
Laser Beam
(405 nm)
PMT
PH
TL
BS
OL
Structure
on the substrate
Figure 9.8 Schematic diagram of a confocal optical microscopy system. The modeling
consists of three subsystems: focusing of incident light, interaction of focal field with the
object structures, and the detection of the scattered light. OL, Objective lens; TL, Tube lens;
BS, Beam splitter; PH, pinhole; PMT, Photomultiplier tube. The inset shows the focal spot and
computational domain (the line box with the arrow) used in the subsystem II. The laser light
of 405 nm wavelength is focused by 150× NA0.9 objective lens and the intensity is recorded
by a PMT with a pinhole of 20 μm diameter. The sample is expanded and shown on the
left-hand side. Source: Chen 2016, Optica, 3, 1339–1347. [28] Reproduced with permission of
The Optical Society.
Inversion: Inhomogeneous Background Medium
is referred to as the sample. The interaction of the focal field with the sample is
treated as an electromagnetic scattering problem in an inhomogeneous
back√
ground, where the refractive index nr (r), which is equal to 𝜖r (r), of both the
background medium and the sample are spatially varying. The scattered light
from the sample is refracted by the objective lens and then focused by the tube
lens, and eventually the intensity of light is collected by the detector pinhole.
For each scanning point, that is, each relative position of the microscopy with
respect to the sample, we obtain one intensity signal I. After the scanning process is finished, we obtain a vector I by stacking all measurements.
The numerical method to solve the forward problem is as follows. The vector
diffraction theory is used to analyze the first and third subsystems. The second
subsystem (interaction of the focal field with the sample) is dealt with by the
FE-BI method. As shown in the inset of Fig. 9.8, the closed boundary S of the
domain V divides the problem into an interior and an exterior region. The BI is
applied to the S and consequently electric/magnetic dipoles should be placed
on S to numerically evaluate the Green’s function. For a given object with the
refractive index nr (r), we denote the simulated intensity distribution vector as
I sim (nr ), where nr denotes the discretized version of nr (r).
The inverse reconstruction is to estimate the unknown refractive index distribution so as to minimize the discrepancy between the computed image and
experimental image. The cost function with a regularization term is defined as
‖ I (n ) − I (n ) ‖2
‖
mea r ‖
(9.40)
f (nr ) = ‖ sim r
‖ + 𝛾Δ(nr )
‖
‖
I
(n
mea r )
‖
‖
where I mea is the calibrated image data from the experimental CLSM image data
I exp . In the cost function, the first term is a fidelity term measuring the discrepancy between the estimated and the measured data and the second term is a
regularization term, which enforces a priori knowledge to stabilize the ill-posed
problems. The regularization parameter 𝛾 is chosen empirically to balance the
accuracy and stability. Here the total variation regularization term is chosen to
enforce the sparsity on the gradient of refractive index distribution [27]
√
|∇nr |2 + 𝜁 2 d𝑣
(9.41)
Δ(nr ) =
∫V
The main advantages of the total variation regularization are to preserve the
edges in the image and to smooth homogeneous areas. Thus, such a choice is
suitable here since the sample is often piecewise constant. A small constant
𝜁 > 0 in (9.41) makes the objective function differentiable at nr = 0 and
𝜁 = 1 × 10−10 is taken in the computation. For the initial guess of the optimization, we choose a homogeneous refractive index distribution that is simply the
value of the substrate. The image reconstruction (unconstrained optimization
problem) is solved by the conjugate gradient method.
To demonstrate the capability of the proposed approach, the four-square
and four-disk patterns are fabricated and then imaged by the CLSM system at
247
248
Electromagnetic Inverse Scattering
160 nm
(a)
160 nm
(b)
(c)
(e)
(f)
40 nm
40 nm
(d)
(g)
(h)
Figure 9.9 Inverse reconstruction using four-square and four-disk patterns with 160 nm
center-to-center and 40 nm edge-to-edge distances: (a) and (b) SEM images of four-square
and four-disk patterns; (c) and (d) The simulated images of four-square and four-disk
patterns using the proposed optical model; (e) and (f ) The calibrated images from
experimental images of four-square and four-disk patterns using the CLSM setup; (g) and (h)
Inverse reconstruction images based on images in (e) and (f ) using the proposed inversion
approach with 56 iterations. Scale bars in (a) and (b) are 80 nm. Source: Chen 2016, Optica, 3,
1339–1347. [28] Reproduced with permission of The Optical Society.
the National University of Singapore [28]. The scanning electron microscopy
(SEM) images of them are shown in the Fig. 9.9(a) and (b), where the fabricated
patterns show 160 nm center-to-center and 40 nm edge-to-edge distances. It
is obvious from Fig. 9.9(e) and (f ) that the calibrated images from experimental
images that are obtained by the CLSM cannot resolve the four squares or
disks. In comparison, the four squares or disks can be easily identified in
Fig. 9.9(g) and (h), which are reconstructed images from the images in (e)
and (f ), respectively. The reconstructed image (Fig. 9.9(g)) for four-square
pattern provides four separated spots with artifacts. This is because the CLSM
image in Fig. 9.9(e) has some distortions due to the contamination of noises
compared to the simulated images in Fig. 9.9(c). In contrast, the simulated and
CLSM images (Fig. 9.9(d) and Fig. 9.1(f )) for the four-disk pattern are much
similar and thus the reconstructed image of the four-disk pattern is much
better, as shown in Fig. 9.9(h).
Inversion: Inhomogeneous Background Medium
9.5.2
Near-Field Scanning Microwave Impedance Microscopy
Near-field scanning microwave impedance microscopy (MIM) has attracted
intense scientific and industrial interests in the past decade due to its considerable abilities to determine the composition and physics of nanoscale materials
and devices. A schematic of the MIM is shown in Fig. 9.10. In an MIM
measurement, GHz voltage modulation is delivered to the tip of a metallic
probe, usually of pyramid or cone shape with a base length of several μm and
an apex diameter of the order of tens of nm. When the tip is brought close to
and scanned across the surface of a sample, variations of tip-sample-ground
admittance are recorded, the real and imaginary parts of which are denoted as
MIM-Re and MIM-Im signals, respectively. Thus, the MIM-Re and MIM-Im
signals depend on the property of the sample, and the problem of inferring the
property of the sample from the MIM signals is a typical inverse problem. We
mention in passing that the MIM technique is closely related to the modulated
scatterer technique [29], the application of which to microwave diffraction
tomography can be found in [30].
In the near-field region, the sizes of tip and sample, as well as their gap,
are much smaller than the wavelength and, consequently, the probe-sample
interaction is actually in the electro-quasistatic regime. In this regime, the electric surface charges located on the surface of metallic probe have short action
distances due to the absence of spatial phase oscillation. Numerical simulations have shown that only a limited region beneath the probe, referred to as
the effective window, contributes to the measured signal [31]. We decompose
the whole 3D simulation domain into two regions: the effective window and the
Detector
Reflected Signal
Laser
Coupler
Source
Mixer and
Amplifiers
ΔY
MIM-Re
MIM-Im
Figure 9.10 A schematic of the MIM. GHz voltage modulation is delivered to the tip of a
metallic probe. When the tip of the probe is brought close to and scanned across the surface
of a sample, variations of tip-sample-ground admittance are recorded, the real and
imaginary parts of which are denoted as MIM-Re and MIM-Im signals, respectively. Source:
Wei 2016, IEEE Trans. Microw. Theory Techn., 64, 1402–1408. [31] Reproduced with permission
of IEEE.
249
Electromagnetic Inverse Scattering
region outside of it. The FEM is used in the effective window and the BI is
applied to the closed boundary of the effective window. To apply the BI, we
numerically evaluate and then store the inhomogeneous background Green’s
function, which accounts for the tip structure and the material property outside
of the boundary. Thus, the FEM-BI model applies to arbitrary tip structures.
The detailed information on the forward problem solver can be found in [31].
The MIM image obtained by scanning over the sample records the contrast
capacitance, but does not directly show the permittivity of sample. In addition,
when the MIM tip scans a point of sample, the received signal comes from
not only the single point that is directly beneath the tip, but instead all other
points inside the effective window. Thus, the measured MIM signal is in fact
a complex function of the permittivity of sample. Therefore, we need to solve
an inverse problem to reconstruct the permittivity of sample from the measured data. Here, we formulate the inverse problem as an optimization problem,
where the difference between calculated and measured contrast capacitance
16
Ws
0.1
14
0.2
Ws1
ε1
Ws2
Ws
hp
hs1
hp1
y (um)
Ls2
εb
0.3
12
0.4
10
0.5
0.8
0.1
0.2
5
0.2
4
0.4
3.5
0.5
3
0.6
2.5
0.8
(c)
0.6
0.8
0.6
0.8
4
16
14
0.3
12
0.4
10
0.5
8
0.6
6
1.5
0.7
4
1
0.8
2
0.7
y (um)
4.5
0.3
0.4
x (um)
(b)
5.5
0.4
x (um)
0.2
F× 10–12
0.1
0.2
6
0.7
hs
(a)
8
0.6
εb
y (um)
250
0.2
0.4
x (um)
0.6
0.8
2
(d)
Figure 9.11 (a) A three-dimensional sample with a “51” shape perturbation presented. The
substrate provides a background relative permittivity 𝜖b = 3.9 and the perturbation has
𝜖1 = 16; (b) top view of exact distribution of relative permittivity in (a); (c) the simulated MIM
signal, where 5% white Gaussian noise is added; and (d) reconstruction of relative
permittivity from the signal in (c).
Inversion: Inhomogeneous Background Medium
is minimized. The conjugate gradient (CG) algorithm is used to minimize the
objective function. In solving the inverse problem for the near-field scanning
MIM, the fact that only a limited region beneath the probe contributes to the
measured signal is of great importance since it saves much computational time
during the raster-scanning process.
Figure 9.11(a) presents a three-dimensional “51” shape perturbation sample [36]. The total sample size is Ws × Ws × hs with Ws = 6 μm and hs = 1 μm.
As illustrated in Fig. 9.11(a), a “51” shape perturbation is distributed in a top
layer of the sample with the thickness hp = 0.4 μm, width Ws1 = 100 nm, Ws2 =
250 nm, and length Ls1 = 600 nm, Ls2 = 150 nm. Figure 9.11(b) presents the
top view of exact distribution of relative permittivity for a “51” shape perturbation sample. Figure 9.11(c) presents the simulated MIM signal with 5% Gaussian noise. Note that only one channel of the MIM signals is present since the
other channel is zero for lossless samples. The reconstructed permittivity profile from the received noisy signal is shown in Fig. 9.11(d), which shows that
the proposed method is able to reconstruct the properties of samples and at
the same time provide improved resolution than the measured MIM signal.
What is more interesting is that, in addition to providing an improved lateral
resolution, near-field scanning MIM is also able to provide depth information.
Figure 9.12 shows a two-layer structure. The substrate provides a background
relative permittivity 𝜖b = 3.9. Two vertical bars with 𝜖2 = 6 are located on the
top layer and one horizontal bar with 𝜖1 = 16 is located in the lower layer.
Ls
Ws
(a)
0.2
0.1
0.2
12
0.2
10
0.3
8
0.4
0.5
F/m × 10–12
5
0.1
4.5
4
0.2
3.5
3
0.3
2.5
2
0.4
1.5
1
0.5
0.3
0.4
0.5
0.1 0.2 0.3 0.4 0.5
x (um)
(d)
y (um)
14
0.1
0.2 0.3
x (um)
0.4
0.5
5.5
5
0.3
4.5
0.4
6
0.5
4
0.1
(b)
0.1
0.2
0.3 0.4
x (um)
(e)
0.2 0.3
x (um)
0.4
0.5
4
(c)
7.5
7
6.5
6
5.5
5
4.5
4
3.5
y (um)
y (um)
0.1
0.1
0.5
y (um)
ε2
ε1
y (um)
Ls
ε2
6
16
Si02
Ws
0.1
7
0.2
6
0.3
5
0.4
4
0.5
0.1
0.2 0.3 0.4
x (um)
0.5
(f)
Figure 9.12 (a) Three-dimensional view of a two-layer medium with Ws = 100 nm,
Ls = 400 nm, 𝜖1 = 16, and 𝜖2 = 6. Top view of the exact distribution of relative permittivity
for the (b) bottom layer and (c) top layer. (d) The simulated received MIM with 1% additive
white Gaussian noise. Reconstructed distribution of relative permittivity for the (e) bottom
layer and (f ) top layer.
3
251
252
Electromagnetic Inverse Scattering
The simulated received MIM data with 1% additive white Gaussian noise is
shown in Fig. 9.12(d), where a single 2D image cannot distinguish the two-layer
structure. By solving a 3D inverse problem, we obtain the distribution of reconstructed relative permittivity. Figure 9.12(e) and (f ) shows the reconstructed
sectional image for the bottom and top layers, respectively. Although the reconstructed relative permittivity of the lower bar is much lower than the exact
value, the position, size, and shape of the bar are satisfactorily reconstructed.
It is worth commenting that it is the nature of the inverse problem that the
lower-layer structure is difficult to reconstruct since its interaction with the
probe is much weaker than that of the top layer. In other words, a lack of information cannot be remedied by mathematical calculations.
9.6 Discussions
Three methods for solving inhomogeneous-background ISPs are presented in
this chapter. The first two methods, introduced in Sections 9.2 and 9.3, require
use of the truncation boundary B3. In order to ensure high accuracy and at
the same time keep computational domain to a minimum, the BI equation
approach is a good candidate. In this case, the boundary B3 can be chosen to be
conformal to and very close to the B2. Section 9.2 introduces an approach that
numerically obtains an inhomogeneous background Green’s function for every
point in the DOI and then applies IE-based inversion algorithms to solve ISPs.
The advantage of this approach is that it inherits the desirable properties of
IE-based inversion algorithms, such as long-range interaction and good physical and mathematical properties of the induced current. The weakness of this
approach is the high computational and storage overhead in numerical evaluation of the Green’s function GD and GS . Nevertheless, such computations are
conducted only once during the whole inversion process since GD and GS are
independent of the property of scatterers. Section 9.3 presents an approach that
directly solves differential equations without calculating the numerical Green’s
function. Compared with the first approach, the method has much lower computational and storage overheads and but it needs more iterations to converge
due to the local-range interaction of FD methods. The third approach, introduced in Section 9.4, treats the known inhomogeneous background as a known
scatterer rather than part of the background. The advantages of this approach
are mainly little or no computational and storage overheads and the availability of existing fast forward problem solver, such as the conjugate gradient fast
Fourier transform (CG-FFT) method. Nevertheless, the approach has a higher
degree of nonlinearity and thus might perform poorly when the known background obstacle has a higher permittivity than the unknown scatterer.
Inversion: Inhomogeneous Background Medium
If it is known that the scatterers are located within a certain region, that is, the
B1 in Fig. 9.1 is known a priori, then the BI equation can be formulated on the
B1, instead of conventionally on the B3, which has the tremendous advantage
of limiting the computational domain within B1 at each iteration during the
optimization process. Section 9.5 uses this approach to solve 3D ISPs. Numerical evaluation of Green’s functions is needed, but the test electric/magnetic
dipoles are placed only on the boundary B1, instead of in the whole DOI, which
should be distinguished from the method presented in Section 9.2.
In the FEM implementations presented in this chapter, the finite element is
chosen to be a square element in 2D or a rectangular brick element in 3D, for
the sake of ease of mesh-generation and visualization. Note that for ISPs involving dielectric scatterers, such elements are greatly desirable for the following
two reasons. First, although other elements, such as triangular or tetrahedral
element, are more flexible and accurate in representing the geometry of scatterers, they are advantageous in the forward problem, but not necessarily in the
inverse problem. Since scatterers are unknown in ISPs, the triangle or tetrahedral element has to be applied to the whole domain of interest, which inevitably
increases the number of unknowns, resulting in not only an increased difficulty for the inversion, but also a heavy computational cost at each iteration of
the optimization process. What is more significant is that the fine features of
the scatterer, such as sharp corners, cannot be reconstructed anyway in solving IPSs. In practice, sharp corners are usually reconstructed to be smooth
round corners, unless special regularization methods that incorporate a priori
information about sharp corners are used. Second, the induced current inside
dielectric scatterers is volumetric, which does not have a strict requirement on
the directionality, which is quite different from the scattering of perfect electric
conductors (PECs). Considering these two reasons, a square element in 2D and
a rectangular brick element in 3D are good candidates for elements in solving
ISPs involving dielectric scatterers.
In many real-world inhomogeneous background ISPs, the experimental
environment is controllable and the background Green’s function can be
experimentally obtained. For example, [32] experimentally determines the
point spread function (PSF), which is a simple function of the Green’s function,
via a calibration measurement of a known electrically very small (point) scatterer, which is placed in the known environment formed by the background
medium and the acquisition system.
As far as the application of differential equation solver to ISPs is concerned,
a promising direction in solving large-scale 3D electromagnetic ISPs involving
penetrable scatterers is to apply the domain decomposition methods. Recently,
[33] adopts the finite-element tearing and interconnecting full-dual-primal
(FETI-FDP2) method as the forward problem solver in solving 3D electromagnetic ISPs. The idea is to split the computational domain into smaller nonover-
253
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Electromagnetic Inverse Scattering
lapping subdomains in order to simultaneously solve local subproblems. The
FETI-FDP2 method is efficiently coupled with the inversion algorithm, keeping
the memory requirement and the computational time as low as possible.
Finally, we briefly mention that when the scatterers are known to be small
scatterers, much smaller than the wavelength, that are imbedded in the
known inhomogeneous background medium, the multiple signal classification
(MUSIC) that has been introduced in Chapter 4 can be used to locate small
scatterers [34]. The applicability of MUSIC is supported by the fact that the
singularity property of the homogeneous-background Green’s function that
is used in Section 4.2 is also applicable to an inhomogeneous-background
Green’s function when the observation position approaches the source point.
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10
Resolution of Computational Imaging
The purpose of this chapter is to discuss the resolution of an image that is
obtained by solving inverse scattering problems, rather than to provide a
comprehensive review of super-resolution imaging theories and schemes. An
important conclusion is that the classical half-wavelength resolution widely
used in traditional optical microscopy is not applicable to inverse scattering
problems. The reason is that instrumental imaging and computational imaging
are two different imaging strategies.
The organization of the chapter is as follows. Section 10.1 discusses the
resolution of a traditional optical microscopy, which is a kind of instrumental
imaging. Section 10.2 introduces computational imaging, where images are
generated by numerical reconstruction. Both the inverse source problem
and inverse scattering problem are discussed. Section 10.3 introduces the
Cramér–Rao bound (CRB), which quantifies a lower bound on the variance of
any unbiased estimator. The accuracy of computational imaging is quantified
by the CRB. Section 10.4 presents the resolution of image obtained by the
Born Approximation (BA) that is applicable to weak scatterers. The analytical
tool for the BA-based imaging provides a deep insight into the resolution
of computational imaging. Section 10.5 discusses several other topics on
resolution.
10.1 Diffraction-Limited Imaging System
It is well known that the resolution of traditional optical microscopy is limited
by diffraction due to the wave nature of light. A point source will be imaged into
a finite-sized spot. Thus, when two identical point sources are in close proximity, the microscopy cannot resolve them as two distinct points. The minimum
resolvable distance is defined as the spatial resolution. The theory of resolution
was formulated by Abbe and Rayleigh in the nineteenth century. Since then
there have been several criteria defining spatial resolution, such as Rayleigh’s
criterion, Abbe’s criterion, and Sparrow’s criterion [1].
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
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Rayleigh’s criterion states that two incoherent point sources are barely
resolved when the center of the intensity pattern generated by one point
source falls exactly on the first zero of the intensity pattern generated by the
second [2] (section 6.5.2). The spatial resolution under Rayleigh’s criterion is
given by
𝜆
,
(10.1)
n sin 𝜃
where 𝜆 is the wavelength of light in air, n is the refractive index of the object
space, and 𝜃 is the angular semi-aperture, as depicted in Fig. 10.1(a). The resolutions defined under other criteria are slightly different from (10.1), but they are
all close to Δ = 0.5𝜆∕(n sin 𝜃). When the angular semi-aperture is large enough
so that sin 𝜃 ≈ 1, the spatial resolution is expressed as
ΔR = 0.61
Δ = 0.5𝜆0 ,
(10.2)
where 𝜆0 = 𝜆∕n is the wavelength in the object space. Equation (10.2) shows the
well-known loosely adopted statement that the spatial resolution of the traditional optical microscopy is limited by the half wavelength due to diffraction.
The resolution discussed in this chapter is mainly for coherent waves. The
theory of resolution lies in how a single point source, as shown in Fig. 10.1(a),
is mapped to a spreading spot in the image plane, as shown in Fig. 10.1(b). The
intensity distribution in the image plane is referred to as the point spread function. For a traditional optical microscopy as shown in Fig. 10.1(a), we analyze
how the point spread function is generated. Note that the single lens depicted
inside the dashed box in Fig. 10.1(a) is only a schematic diagram of the lens system. In fact, all imaging elements are lumped into a single “black box,” which
might be composed of several lenses with various distances between them. The
passage of light inside the black box is adequately described by geometrical
optics, and the imaging system is aberration-free in the sense that a diverging
spherical wave incident upon the black box in the object space is converted by
the system into a converging spherical wave in the image space. To avoid relying
Image plane
Point spread function
y
O
2θ
z
(a)
(b)
Figure 10.1 A single point source, located at the origin, is mapped to a spreading spot in a
traditional optical microscopy. (a) A schematic diagram of the lens system, where the
dashed box might be composed of several lenses, with various distances between them. (b)
Distribution of intensity in the image plane.
Resolution of Computational Imaging
on the knowledge in optical microscopy, such as the magnification and the sine
condition, we consider a simple case, where the object and image spaces have
the same focal lengths and are both air filled; that is, the black box lens system
that is placed in air presents left-right symmetry.
We place a point source at the focus in the object space, which is defined as the
origin of the coordinate system, as shown in Fig. 10.1(a). The field distribution
in the object space is a well-known Green’s function G(r) = eikr ∕(4𝜋r). From
the Weyl identity, for any point r, the spherical wave can be expressed in terms
of plane waves (section 2.2.2 of [3]):
i
eikx x+iky y+ikz |z|
eikr
dkx dky
=
4𝜋r
8𝜋 2 ∫−∞ ∫−∞
kz
∞
∞
(10.3)
where the wavenumber along the longitudinal direction (i.e., the z direction) is
defined as
⎧ √ 2
k − kx2 − ky2 k 2 − kx2 − ky2 > 0
⎪
kz = ⎨ √
(10.4)
⎪ i kx2 + ky2 − k 2 k 2 − kx2 − ky2 < 0
⎩
This definition assures outgoing waves and a finite energy density at infinity. It
can be seen that for each pair of kx and ky , the complex amplitude of the plane
wave is given by i∕(8𝜋 2 kz ). From the definition of 2D Fourier transform, we find
that the Fourier transform of Green’s function in object space is given by
̃ x , ky ) =
G(k
i
eikz |z|
8𝜋 2 kz
(10.5)
where ∼ denotes the Fourier transform of a function. Since the focal length of
the optical lens is often much larger than the wavelength, evanescent waves
(k 2 − kx2 − ky2 < 0) decay to a negligible level before it reaches the lens system.
Due to the finite size of the lens system, it receives only a fraction of the travelling waves (k 2 − kx2 − ky2 > 0). This portion of travelling waves, after passing
through the lens system, converges to the focal point in the image space. If we
define a local coordinate system in image space and the origin is located at the
focus, then the converging property in image space, in contrast to the diverging property in object space, yields a field distribution that is in some sense
reflected as a flip of signs of kx and ky in (10.3); that is, the (kx , ky , kz ) plane wave
in object space conjugates with the (−kx , −ky , kz ) wave in image space. Consequently, due to the left–right symmetry of the lens system, for point pairs r′
and r with x′ = x, y = y′ , and z′ = −z, we find that the field distribution in the
image space is an inverse Fourier transform of windowed Fourier components
of field distribution in object space.
̃ x , ky )]
P(r′ ) = IFT[LF (kx , ky )G(k
(10.6)
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Electromagnetic Inverse Scattering
where the LF (kx , ky ) is a windowed low-pass filter defined as
{
1, kx2 + ky2 ≤ k 2 sin 𝜃 2
LF (kx , ky ) =
0, Otherwise
(10.7)
We see from (10.6) that the field distribution in object space, which is singular
at the point source, is mapped into a continuous field distribution in image
space, which is an inverse Fourier transform of windowed Fourier components
of the field distribution in object space. This interpretation is referred to as the
field–field mapping. If interpreted from the source-field mapping point of view,
the point source that is a delta-function distribution in object space is mapped
into a spreading continuous field distribution, as shown on the right-hand side
of (10.6) in image space. Consequently, (10.6) is defined as the point spread
function. This spreading is a direct consequence of spatial-frequency filtering;
that is, evanescent waves are lost and only a fraction of travelling waves are
collected by the lens system.
This analysis is for a unit point source at the focus. For a general source distribution S(r) near the focus in object space, the application of the convolution
theorem yields
̃ x , ky )S(k
̃ x , ky )]
FI (r′ ) = (P ∗ S)(r′ ) = IFT[LF (kx , ky )G(k
(10.8)
The spatial resolution of traditional optical microscopy can be understood
as a hardware implementation of the inverse Fourier transform of windowed Fourier components. The image is physically formed on the film or
charge-coupled device (CCD) sensor placed in the focal plane. This imaging
modality is referred to as instrumental imaging. The hardware implementation
of the inverse Fourier transform can be equally successfully replaced by the
software implementation. That is, if the lens system inside the black box in
Fig. 10.1(a) is replaced by sensors that read the windowed Fourier components
and then pass the data to a computer, then a numerical implementation of
the inverse Fourier transform in the computer is able to generate in a display
a digital image that is, in principle, the same as the physical image formed
on the film. This imaging modality is referred to as computational imaging.
Naturally, a question is raised about computational imaging: if the computer,
upon receiving the windowed Fourier components, implements an algorithm
that is different from the inverse Fourier transform, is it possible to achieve
a better spatial resolution? The answer is confirmative, which will be the
topic in Section 10.2. To summarize, computational imaging is different from
instrumental imaging and thus the spatial resolution of traditional optical
microscopy is not applicable to computational imaging [4].
Before moving to the next section, we’d like to discuss two issues. First, the
previous analysis on imaging is for a coherent wave and, in addition, both the
magnitude and the phase of the field in the focal plane is studied. In many cases,
an imaging system might work in different ways. For example, sources emit
Resolution of Computational Imaging
incoherent waves; only the intensity at the focal plane can be measured without
the access to the phase information. The spatial resolution for those imaging
modalities might exhibit different values from the one presented in this section,
but the angular spectrum framework presented here still plays an important
role in other imaging systems. Second, this section presents the result for scalar
waves. For a vectorial wave case, the derivation of point spread function can be
found in [5] and section 4.1 of [6]. In this case, the dyadic Green’s function
is a 3 × 3 matrix, and the resolution depends on the orientation of the dipole
source. Consequently, it is not easy to define a simple and well-accepted criterion of resolution. Thus, we will mainly discuss the resolution of scalar waves
in the remaining sections.
10.2 Computational Imaging
Computational imaging problems are categorized into two types. The first
type is the inverse source problem, where the distribution of source amplitude
is reconstructed from the measured field distribution in image space [7]. The
source can be either an active source that radiates by itself or an induced
source that is due to illumination by active sources. This topic will be detailed
in Section 10.2.1. The second type is the inverse scattering problem, where
the distribution of permittivity of the scatterer placed in object space is
reconstructed from the measured scattered fields [8]. The scatterer has to
be illuminated by an array of active sources. For each incidence, scattered
fields are captured by an array of detectors. This topic will be detailed in
Section 10.2.2. For a single incidence, the inverse scattering problem reduces
to the inverse source problem.
10.2.1
Inverse Source Problem
For a general source distribution S(r) in object space, where the source can be
either an active or induced source, the field distribution in image space is given
by (10.8); that is,
FI (r′ ) = (P ∗ S)(r′ )
(10.9)
The inverse source problem involves recovery of the distribution of source
S(r) from the measured field distribution in image space FI (r′ ). In the spatial
domain, the inverse source problem is about recovering the source distribution
by deconvolving the measured field distribution in image space. If viewed
from the spatial frequency domain, Section 10.1 shows that traditional optical
microscopy actually conducts a hardware implementation of the inverse
Fourier transform of windowed Fourier components. In computational
imaging, algorithms indeed exist that, different from a direct inverse Fourier
transform, are able to achieve a better spatial resolution.
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Electromagnetic Inverse Scattering
A natural idea is to first extrapolate the Fourier components that are outside
of the window from the Fourier components that are inside the window, and
then implement the inverse Fourier transform. Mathematically, it is written as
̃ x , ky ) → S(k
̃ x , ky ) ⇒ S(r) = IFT[S(k
̃ x , ky )]
(10.10)
LF (kx , ky )S(k
For spatially bounded objects, in the absence of noise it is, in principle,
possible to resolve infinitesimally small details of the object. Resolution
beyond the classical diffraction limit is often referred to as super resolution.
The fundamental reasons lie in the following basic mathematical principles
(section 6.6 of [2] and [9]). The two-dimensional Fourier transform of a spatially bounded function is an analytic function and once an analytic function is
known exactly in an arbitrarily small but finite region, then the entire function
can be uniquely found by means of analytic continuation. In other words,
although evanescent waves cannot be directly measured in the far field, they
are mathematically linked to propagation waves and thus can be numerically
recovered from propagation waves. Once the entire object spectrum is exactly
found, then the exact object can be reconstructed with arbitrary precision
by the inverse Fourier transform. However, in practice, only a very limited
range of spectrum extrapolation is achievable due to the effects of noise in the
measured data [10] (section 13.1).
In addition to the aforementioned extrapolation method, one can also
directly solve (10.9) for the S(r) in the spatial domain, taking some a priori
information of the source. For example, [11] presents the mathematical theory
of super-resolution by directly solving (10.9) for discrete point sources. The
method can recover such sources precisely, that is, determining the exact
locations and amplitudes by solving a convex optimization problem, provided
that the distance between sources is above a certain threshold.
There are many other algorithms that are able to achieve super resolutions.
For example, an iterative extrapolation method works by iterating between
the source domain and the spectral domain, making changes in each domain
to reinforce prior knowledge of the source (such as located in a certain finite
domain) or measured data [12]. Many other algorithms can be found in [10]
(section 13.1) and [2] (section 6.6).
One topic worth discussing is the nonradiating source. Since nonradiating
sources contribute zero within the windowed spatial frequency, the entire spectrum will be automatically zero by analytical continuation. The inverse Fourier
transform of the zero function yields a zero source. Thus, when no prior information about the source is available, the inverse source problem does not have
a unique solution due to the presence of nonradiating sources [13].
10.2.2
Inverse Scattering Problem
Consider a distribution of relative permittivity 𝜖r (r) of an object placed in
the domain of interest (DOI) D. The inverse scattering problem consists of
Resolution of Computational Imaging
determining the values of 𝜖r (r) from multiple incidence-scattering pairs (̂ri , r̂ s ),
where r̂ i and r̂ s are the directions of incident and scattered fields, respectively.
Here the incident wave are plane waves and the measurements are carried out
in the far field. The contrast, also known as the scattering potential, is defined as
O(r) = 𝜖r (r) − 1. For each incident-scattering pair (̂ri , r̂ s ), we have the following
two equations. Inside D, the field satisfies the Lippmann–Schwinger equation,
𝜓(r, k̂ri ) = eik̂r ⋅r + k 2
i
∫D
G(r, r′ )O(r′ )𝜓(r′ , k̂ri )d3 r′ , r ∈ D
(10.11)
ik|r−r′ |
e
where the G(r, r′ ) is the free-space Green’s function G(r, r′ ) = 4𝜋|r−r
and
′|
i
i
𝜓(r, k̂r ) is the total field at r for the incidence direction r̂ . The far-field
scattered field reads
𝜓 s (k̂rs , k̂ri ) =
s
eikr
e−ik̂r ⋅r O(r)𝜓(r, k̂ri )d3 r
4𝜋r ∫D
(10.12)
where O(r)𝜓(r, k̂ri ) has a physical meaning of induced source and the phase
s
factor e−ik̂r ⋅r is the same as the one appearing in (2.101) due to the far-field
approximation.
For any incident wave, 𝜓(r, k̂ri ) can be obtained from (10.11) and then is substituted to (10.12). We obtain the relationship,
𝜓 s (k̂rs , k̂ri ) = P[̂rs ,O(r)] (eik̂r ⋅r )
i
(10.13)
where P is an operator that maps the incident field to the scattered field. Obviously, the P depends on the direction of scattered field r̂ s and the contrast O(r).
When solving inverse scattering problems, continuous equations have to be
discretized. We assume that there are Ni incidence directions and scattered
fields are measured in Ns directions. The DOI is divided into a total number
of M cells. The inverse scattering problem consists of reconstructing the value
of O(rm ), m = 1, 2, ..., M from a total number of Ni Ns measured scattered
fields, 𝜓 s (k̂rsq , k̂rip ), p = 1, 2, ..., Ni and q = 1, 2, ..., Ns . For any given candidate
of O(rm ), m = 1, 2, ..., M, the corresponding operator P can be calculated and
consequently we are able to calculate the scattered field from (10.13). Next,
we check whether the calculated scattered fields match measured ones for all
incidence-scattering pairs (̂rip , r̂ sq ), p = 1, 2, ..., Ni and q = 1, 2, ..., Ns . If we are
extremely lucky and the first trial O(rm ) satisfies the match condition, then
this trial O(rm ) is identified as the solution to the inverse scattering problem.
If we are extremely unlucky and have to conduct brute-force trials, that is,
enumerating all possible candidates for the solution and checking whether
each candidate satisfies the match condition, then the computational time will
be exceedingly long. In practice, it is unlikely that we are so lucky to have only
one attempt, and we cannot wait long enough for brute-force checking either.
Instead, a more practical approach is to convert the inverse problem to an
263
264
Electromagnetic Inverse Scattering
optimization problem,
min
O(rm )
Ni Ns
∑
∑
p=1 q=1
|𝜓 s (k̂rsq , k̂rip ) − P[̂rsq ,O(rm )] (eik̂rp ⋅r )|2
i
(10.14)
The process of searching for the solution among all candidates is guided by
optimization algorithms.
If there is only a single incidence, the inverse scattering problem reduces to an
inverse source problem. In this case, the source O(r)𝜓(r, k̂ri ) is the induced contrast current, which is a secondary source. When there are multiple incidences,
the inverse scattering problem, in which the contrast O(r) is to be determined,
is different from the inverse source problem, where the amplitude of source
is to be recovered. The former is a nonlinear problem, whereas the latter is a
linear problem. Consequently, the diffraction-limited resolution, that is, half
wavelength as a rule of thumb, as discussed in Section 10.1, does not apply to
inverse scattering problems. It can be easily seen that, in absence of noise, if
the trial O(r) happens to be chosen as the exact contrast, by either good luck,
brute-force checking, or optimization algorithms, then the objective function
(10.14) is equal to zero. In this case, the resolution of the reconstructed contrast
is in principle unlimited. When measured data are contaminated with noise, we
need to determine the fundamental limits of the accuracy of the reconstructed
contrast. In fact, an inverse scattering problem can be treated as a parameter
estimation problem, where the contrast is estimated, based on measured scattering data that are contaminated with noise. The use of the CRB approach,
which is widely used in estimation theory and statistics, quantifies the best
precision with which parameters of interest can be estimated in the statistical framework of unbiased estimation under given noise models. The CRB will
be discussed in the next section.
10.3 Cramér–Rao Bound
The CRB quantifies a lower bound on the variance of any unbiased estimator.
Consider a model function that is capable of fitting the measurement data
perfectly in the absence of noise. In the presence of noise, fitting is imperfect
causing statistical errors in the parameter estimates. Thus, when a measurement is repeated many times, the resulting model parameters exhibit a spread.
The minimal attainable values of variance corresponding to these spreads are
called the CRBs. The smaller the value of CRB, the better the precision. Different estimators usually have different precision, but the CRB is independent
of estimators (in terms of inverse scattering, the CRB is inversion-algorithm
independent). There exist unbiased estimators, including the maximum
likelihood (ML) estimator, that achieve the CRB asymptotically [14]
(chapter 3).
Resolution of Computational Imaging
The CRB theory is based on the likelihood function. Suppose that the measured complex-valued data X have N components, X1 , X2 , …, XN . In presence
of noise, the measured data are expressed as,
X = X̂ + B
(10.15)
where X̂ = [X̂ 1 , X̂ 2 , ..., X̂ N ]T denotes the noiseless data and B = [B1 , B2 , ..., BN ]T
is noise. In the following, we assume additive independent, identically distributed complex white Gaussian noise, though it is technically feasible to
derive the CRB under other noise models. For complex-valued noise, we
further assume that the real part is independent of the imaginary part, each
having a Gaussian distribution  (0, 𝜎 2 ∕2). The joint probability function of
the measurement data X, also known as the likelihood function, equals the
product of the probability function of all sample components,
)
( N
∑ 1
1
2
exp −
|Xi − X̂ i |
P(X) =
2
𝜋 N ΠNi=1 𝜎i2
i=1 𝜎i
)
( N
∑ 1
1
2
̂
=
exp −
|X − Xi |
𝜎2 i
(𝜋𝜎 2 )N
i=1
)
(
1
1
̂ 2
(10.16)
=
exp
−
|X
−
X|
𝜎2
(𝜋𝜎 2 )N
It is a common practice to use the log-likelihood function,
L(X) = ln[P(X)]
(10.17)
Fisher information essentially describes the amount of information that
data carry about unknown parameters. If the measured data X depends
on the real-valued model parameter vector 𝜃 = (𝜃1 , 𝜃2 , ..., 𝜃M )T , the Fisher
information matrix (FIM) F takes the form of an M × M matrix, with element
]
[
𝜕L(X) 𝜕L(X)
F(𝜃)i,j = E
(10.18)
𝜕𝜃i 𝜕𝜃j
where E stands for expectation value. An algebraic calculation shows that the
FIM can be expressed as
H
2
F(𝜃) = 2 ℜ{D ⋅ D}
(10.19)
𝜎
where ℜ{⋅} denotes the real part operator, the superscript H represents
the conjugate transpose, and D is a matrix of size N × M, with element
Dij = 𝜕 X̂ i ∕𝜕𝜃j . The CRB, which sets the lower bound of variance of the estimated parameters 𝜃, is obtained by inverting the FIM and taking the diagonal
elements,
−1
Var{𝜃i } ≥ CRB(𝜃i ) = [F (𝜃)]ii ,
i = 1, 2, ..., M
(10.20)
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Electromagnetic Inverse Scattering
Inverse scattering problems can be categorized into two types, as far as
how scatterers are represented is concerned. The first type is a parameterized
model, where scatterers are known a priori to be represented by a few parameters describing geometry, location, material, and so on. The second type is a
pixel (or voxel) based model, where the domain of interest is discretized into
pixels and the material property at each pixel will be reconstructed. In the
inverse scattering problem, the CRB is usually calculated for parameterized
models. For example, [15] analyzes the precision of contrast estimators with
the CRB when the target is homogeneous, infinitely-long, and has a circular
cross section; [16] calculates the CRB for ideal point scatterers; [17, 18]
consider the CRB for spherical or circularly cylindrical scatterers of finite
size. Study [19] evaluates the CRB on shape estimation accuracy using the
domain derivative technique from nonlinear inverse scattering theory, where
the true shape is described by the parametric model. In comparison, due to
large number of unknowns and the corresponding large size FIM, it is not
common to calculate the CRB in inverse scattering problems for pixel (or
voxel) based model. In [20], the two-dimensional electromagnetic inverse
problem of imaging an isotropic dielectric scatterer is considered, where an
optimum trade-off between the accuracy of reconstruction and the resolution
(defined as the size of pixel) is defined by virtue of the CRB.
Here we consider a numerical example of calculating the CRB for an inverse
scattering problem. The purposes of the numerical example are twofold. Firstly
we illustrate the numerical procedure of evaluating the CRB, and secondly we
show that the errors in estimating the positions of two point-like scatterers
from far-field data can be much smaller than half wavelength. Consider a 2D
problem where ẑ is the longitudinal direction, and the system is invariant in
this direction. The wave number k in air is equal 2𝜋; that is, the wavelength is
equal to 1. Two nonmagnetic scatterers with circular cross section are placed
at (x1 , y1 ) = (−1∕16, 0) and (x2 , y2 ) = (3∕16, 0), respectively. The radii of both
scatterers are R = 0.02, which is much smaller than the wavelength. The relative
permittivity 𝜖r of two scatterers is equal to 10. In this condition, the two scatterers can be effectively treated as point-like (in the xy plane) scatterers. Under the
TM illumination, the scattering strength of the circular
small√ scatterer
is mainly
√
√
due to the monopole scattering, that is, 𝜉 =
𝜖r J0 (kR)J0′ ( 𝜖r kR)−J0 ( 𝜖r kR)J0′ (kR)
−4
,
√
√
√
(1)′
k𝜂0 H0 (kR)J0 ( 𝜖r kR)− 𝜖r J0′ ( 𝜖r kR)H0(1) (kR)
where 𝜂0 is the intrinsic impedance of the air. Ni transmitting and Ns receiving
antennas are uniformly distributed on a circle with a radius of 20. In the forward scattering problem, the scattered z-component electric field is stored in
the multistatic response (MSR) matrix K with size Ns × Ni [16]. The forward
problem solver is based on the Foldy–Lax equation and can be found in [16].
We assume that Y , the measurement of MSR matrix, deviates from the
model by additive independent, identically distributed complex white Gaussian noise with variance 𝜎 2 ; that is, Y = K + W . The noise level is defined to
Resolution of Computational Imaging
be the signal-to-noise ratio (SNR) in dB, that is 20log10 (||K||F ∕||W ||F ), where
|| ⋅ ||F denotes the Frobenius norm of a matrix. Due to the property of the
noise W , the SNR is found to be just 10log10 (||K||2F ∕(Ns Ni 𝜎 2 )). The eight real
parameters to be estimated are represented by a vector 𝜃, which is defined as
(x1 , x2 , y1 , y2 , 𝜉1R , 𝜉2R , 𝜉1I , 𝜉2I )T . The Fisher information matrix (FIM), F(𝜃)
has the element,
)H
(
⎤
⎡
𝜕K(𝜃) ⎥
2 ⎢ 𝜕K(𝜃)
Fi,j (𝜃) = 2 ℜ
⋅
, i, j = 1, 2, … , 8,
(10.21)
⎢
𝜎
𝜕𝜃i
𝜕𝜃j ⎥
⎦
⎣
where K is the vectorized version of K by stacking columns sequentially. Since
the CRB involves inverting the 8 × 8 FIM, there is obviously no analytical solution and we can only numerically calculate the CRBs of the eight real parameters. For the FIM, we calculate the derivative 𝜕K(𝜃)∕𝜕𝜃i either analytically or
by finite differences.
Two cases are considered. The first one deals with a single transmitter that
is located at angle 0.2𝜋 with respect to the x-axis, whereas the second case
has 10 transmitters uniformly distributed on the circle with a radius of 20. In
both cases, 10 receivers are uniformly distributed on the aforementioned circle.
The CRBs for the positions x1 and x2 , under different noise levels, are plotted in Fig. 10.2. As expected, the CRBs are much smaller for 10-incidence case
than single-incidence case. It is worth mentioning that although the two small
scatterers are separated by a quarter wavelength along the x-axis, the CRBs
of the positions x1 and x2 are in the order of 10−4 wavelength for the case of
10 incidences. Noting that the wavelength is equal to 1, the result means that
super-resolution reconstruction can be achieved, though the measurements
4
× 10–3
x1
x2
3
1.5
× 10–4
x1
x2
CRB
CRB
1
2
0.5
1
0
10
15
20
SNR (dB)
(a)
25
30
0
10
15
20
SNR (dB)
(b)
25
Figure 10.2 The CRBs of x1 and x2 for (a) a single incidence and (b) 10 incidences. The
wavelength is equal to 1.
30
267
268
Electromagnetic Inverse Scattering
are conducted in far-field zone. It is important to emphasize that the number of
scatterers is equal to the rank of the MSR matrix, as discussed in Chapter 4, and
this information can be directly used by inversion algorithms without requiring
to know a priori the number of scatterers.
10.4 Resolution under the Born Approximation
When scatterers are weak ones, that is, when the permittivity of scatterer differs only slightly from that of the background medium, the inverse scattering
problem can be solved by the first-order Born approximation, which is sometimes simply referred to as the Born Approximation (BA) method. In this case,
the inverse scattering problem is linear and there exists a special relationship
between scatterer’s contrast and scattered fields, which provides a rough insight
into the resolution of reconstructed image.
The BA method has been introduced in Section 6.2.1 for general incident waves and measurement positions. This section focuses on plane
wave incidences and far-field measurements. A three-dimensional scalar
wave and vectorial wave are discussed in sequence, followed by the twodimensional case.
First, we consider the 3D scalar wave inverse scattering problem. Under a
plane wave incidence, 𝜓 i (r) = exp(ik̂ri ⋅ r), the far-field scattered field in the
direction r̂ s is
s ′
eikr
e−ik̂r ⋅r O(r′ )𝜓(r′ )d3 r′
(10.22)
𝜓 s (̂rs ) =
4𝜋r ∫D
where 𝜓(r) = exp(ik̂ri ⋅ r) + 𝜓 s (r) is the total field. By dropping off the factor
expikr ∕(4𝜋r), we define the scattering amplitude
f (̂rs , r̂ i ) =
e−ik̂r ⋅r O(r′ )𝜓(r′ )d3 r′
s
∫D
′
(10.23)
which plays an important role in scattering theory.
Consider a scatterer that occupies a finite domain in air and its relative permittivity is only slightly different from unity. Since the scattered field is very
weak compared with the incident field, it is plausible to assume that one will
obtain a good approximation to the total field if 𝜓(r) is replaced by the incident
field exp(ik̂ri ⋅ r). Under these circumstances, the scattering amplitude is found
to be
f (̂rs , r̂ i ) =
e−ik̂r ⋅r O(r′ )eik̂r ⋅r d3 r′
s
∫D
i
′
′
(10.24)
Recalling the definition of Fourier transform,
̃
O(K)
=
′
∫D
O(r′ )e−iK⋅r d3 r′
(10.25)
Resolution of Computational Imaging
we see that (10.24) implies
̃ rs − r̂ i )]
f (̂rs , r̂ i ) = O[k(̂
(10.26)
It implies that within the accuracy of the first-order Born approximation, the
scattering amplitude, where the incident direction is r̂ i and the measurement
direction in the far zone is r̂ s , depends entirely on one and only one Fourier
component of the scattering potential, K = k(̂rs − r̂ i ). Let us examine the
totality of all Fourier components of the scattering potential that can be
deduced from such experiments. The following geometric construction gives
us the answer.
In the first step, we consider the incident wavevector ki = k̂ri = O1 A (as
shown in Fig. 10.3(a)), and the measurement is made in the far field in all
directions r̂ s . It is easy to see that one obtains all those Fourier components
̃
O(K)
of the scattering potential, which are labeled by K vectors whose end
points lie on a sphere 𝜎1 , of radius k, centered at O1 . We refer to 𝜎1 as the
Ewald’s sphere of reflection, as shown in Fig. 10.3(a). Next, let us suppose
that the object is illuminated in a different direction of incidence and that
the scattered field is measured in the far zone in all possible directions. For
each incidence, one obtains those Fourier components, which are labeled by
K-vectors whose end points lie on another Ewald’s sphere of reflection, say 𝜎2 .
ΣL
σ1
σ2
Σ0
B2
B1
A
O1
σ1
B1
ks
O1
K
ki
ki = kri, ks = krs
(a)
O2
A
k
2k
(b)
Figure 10.3 (a) Ewald’s sphere of 𝜎1 , associated with the incident wavevector ki = k̂ri = O1 A.
(b) Ewald’s limiting sphere ΣL . It is the envelope of the spheres 𝜎1 , 𝜎2 , …, associated with all
possible wavevectors O1 A, O2 A, …of the incident fields. Adapted from: Born and Wolf 1999,
figure 13.4, p. 701. [10] Reproduced with permission of Cambridge University Press.
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270
Electromagnetic Inverse Scattering
For all possible directions of incidence, one obtains Fourier components that
are labeled by K-vectors whose end points are the interior of a sphere ΣL of
radius 2k, which is called the Ewald limiting sphere as shown in Fig. 10.3(b).
In the first-order Born approximation, if one were to measure the scattered
field in the far zone for all possible directions of incidence and all possible direc̃
tions of scatterings, one could determine all those Fourier components O(K)
of
the scattering potential labeled by K-vectors of magnitude
|K| ≤ 2k
(10.27)
Thus, the low-pass filtered approximation OLP (r) to the scattering potential O(r)
is found to be
1
iK⋅r 3
̃
d K
(10.28)
OLP (r) =
O(K)e
(2𝜋)3 ∫|K|≤2k
The scattering potential itself contains all the Fourier components, being given
in the inverse of (10.25), namely:
O(r) =
1
iK⋅r 3
̃
O(K)e
d K
(2𝜋)3 ∫
(10.29)
where the integration extends over the whole K-space. Alternatively,
̃ LP (K) can be expressed as
the low-pass filtered Fourier components O
̃ LP (K) = H(K)O(K),
̃
O
where
{
1,
for |K| ≤ 2k
H(K) =
(10.30)
0,
for |K| > 2k
The low-pass filtered approximation OLP (r) to the scattering potential O(r) can
be obtained by the convolution theorem,
OLP (r) =
∫
h(r − r′ )O(r′ )d3 r′
(10.31)
One can prove that the inverse Fourier transform of the filter H(K), which is
referred to as the point spread function, is given by
[
]
4k 3 j (2k|r|)
h(r) = 2 1
(10.32)
𝜋
2k|r|
where j1 (⋅) is the spherical Bessel function of the first order. If we define
the imaging resolution as the distance between the main lobe and the first
zero of the point spread function, then the resolution of the reconstructed
results under the Born approximation is given by 2kΔr ≈ 4.493; that is,
Δr ≈ 0.357𝜆.
Next, we consider 3D electromagnetic wave inverse scattering. Under a plane
̂ where the polarization direction p̂ is perwave incidence, Ei (r) = exp(ik̂ri ⋅ r)p,
pendicular to the incidence direction r̂ i , the far-zone scattered electric field in
Resolution of Computational Imaging
the direction r̂ s is
s ′
−k 2 eikr
Es (̂rs ) =
e−ik̂r ⋅r r̂ s × r̂ s × O(r′ )E(r′ )d3 r′
4𝜋r ∫D
(10.33)
which can be derived from (2.101). For weak scatterers, the Born approximation
applies and it is plausible to replace E(r) by Ei (r),
Es (̂rs ) =
s
i
′
−k 2 eikr
̂ 3 r′
e−ik(̂r −̂r )⋅r O(r′ )̂rs × r̂ s × pd
4𝜋r ∫D
(10.34)
If we denote the polarization direction and the amplitude of Es (̂rs ) as ês (̂rs ) and
̂ rs × r̂ s × p|
̂ and
Es (̂rs ), respectively, then it is clear that ês (̂rs ) = r̂ s × r̂ s × p∕|̂
Es (̂rs ) =
s
i
′
−k 2 eikr
̂ 3 r′
e−ik(̂r −̂r )⋅r O(r′ ) sin 𝜃(̂rs , p)d
4𝜋r ∫D
(10.35)
̂ is the angle between the directions r̂ s and p.
̂ For measured Es (̂rs )
where 𝜃(̂rs , p)
data, we define the normalized amplitude of the scattered field as
4𝜋r
.
(10.36)
Ens (̂rs ) = Es (̂rs ) 2 ikr
̂
−k e sin 𝜃(̂rs , p)
Consequently, Ens (̂rs ) is the same as the scattering amplitude (10.24) that is presented for scalar wave case. For a given incident direction r̂ i of an electromagnetic wave, although we have the freedom of choosing the polarization
direction p̂ so long as it is perpendicular to r̂ i , the diversity of p̂ actually does
not increase the diversity of scattered field, which can be inferred from the
normalized amplitude of the scattered field. Consequently, the resolution of
reconstructed scatterer under the Born approximation for an electromagnetic
wave remains the same as that for a scalar wave.
This analysis in the 3D case applies to the 2D case as well. Under the Born
approximation, if one were to measure the scattered field in the far zone for
all possible directions of incidence and all possible directions of scattering, one
̃
could determine all those Fourier components O(K)
of the scattering potential
labeled by K-vectors of the magnitude K ≤ 2k. The inverse Fourier transform
of the low-pass filter H2D (K) is given by
[
]
2k 2 J1 (2k|r|)
h2D (r) =
,
(10.37)
𝜋
2k|r|
J (x)
which is referred to as the point spread function. Since the first zero of 1 x
occurs at x = 3.832 and its main lobe locates at x = 0, the resolution of the
reconstructed results under the Born approximation is given by 2kΔr ≈ 3.832;
that is, Δr ≈ 0.305𝜆.
In fact, (10.32) and (10.37) can be obtained from the following inverse Fourier
transform of the indicator of an n-dimensional ball of radius a,
an Jn∕2 (a|r|)
1
iK⋅r n
e
d
K
=
,
(10.38)
(2𝜋)n ∫|K|≤a
(2𝜋a|r|)n∕2
271
272
Electromagnetic Inverse Scattering
𝜈
𝜋
where the identity J𝜈 (z) = √ (z∕2) ∫0 cos(z cos 𝜃)sin2𝜈 𝜃d𝜃 is used (9.1.20
𝜋Γ(𝜈+0.5)
of [21]).
In obtaining (10.32) and (10.37), the inverse Fourier transform of a lower-pass
filter is used. If higher spatial resolution is to be obtained, as mentioned in
Section 10.2.1, analytical continuation will be used before implementing the
IFT. However, in practice, only a very limited range of spectrum extrapolation
is achievable due to the effects of noise in measured data.
10.5 Discussions
Four miscellaneous topics on resolution will be briefly discussed in this section.
The first topic is mainly for instrumental imaging of a point source and the other
three are for inverse scattering problems. The purpose of this section is not
to provide tools to solve imaging problems, but instead to comment on some
existing interpretations of super-resolution imaging.
First, in many far-field imaging systems, the observed super-resolution imaging is claimed to be due to the conversion of evanescent wave to propagating
wave. However, in many cases, such a qualitative conclusion alone, without further quantitative evidence, is barely convincing. For example, consider that we
aim to instrumentally obtain an image of a point source. The angular spectrum
(or Fourier spectrum) representation of the point source is formulated by the
Weyl identity, (10.3), which reads,
eikr
i
eikx x+iky y+ikz |z|
dkx dky
=
2
4𝜋r
8𝜋 ∫−∞ ∫−∞
kz
∞
∞
(10.39)
If we analyze resolution in terms of Fourier spectrum, the ultimate criterion
is that, during the image generation process, either instrumentally or numerically, the linear superposition of the Fourier spectrum must keep the original
coefficient ieikz |z| ∕(8𝜋 2 kz ) of the Fourier spectrum radiated by the source. Consequently, if the measured far field data contain a wider Fourier spectrum, it
does not automatically mean a better resolution. To convert evanescent waves
to propagating waves, the point source has to be placed in the vicinity of another
object, such as a sphere [22], a grating [23], the surfaces of special materials
[24], and so on. Regarding the conversion of evanescent wave to propagating
wave, the following two comments are made. (1) Although these objects convert evanescent waves into propagating waves, they destroy the original propagating waves as well. Consequently, in the far field, evanescent waves are still
not measured and at the same time propagating waves are altered. (2) If the
argument of the conversion of evanescent waves to propagating waves is used,
then a convincing quantitative approach is to extract the evanescent wave spectrum from the measured propagating wave spectrum and in addition show that
Resolution of Computational Imaging
the extracted coefficients of the evanescent wave spectrum are the same as the
original ones radiated by the point source. However, due to complex scattering mechanisms, such an extraction is not an easy task and consequently there
has barely been any research into it. To conclude, a qualitative argument that
evanescent waves are converted to propagating waves, without further quantitative evidence, is not convincing enough to explain super-resolution imaging.
Second, in discussing the resolution of reconstructed scatterer by solving
inverse scattering problems, the Fourier spectrum of the contrast distribution
is often analyzed. When the multiple scattering effect cannot be ignored, measured data contain a spectrum that is outside of the Ewald limiting sphere. It is
stressed that containing a spectrum outside the Ewald limiting sphere does not
automatically lead to super resolution, since the scattered field depends on the
spectrum in a nonlinear way. To exactly retrieve the coefficients of spectrum
outside the Ewald limiting sphere, a nonlinear equation should be solved.
Multiplying the Lippmann–Schwinger equation
𝜓(r, k̂ri ) = eik̂r ⋅r + k 2
i
∫D
G(r, r′ )O(r′ )𝜓(r′ , k̂ri )d3 r′
(10.40)
by O(r), we obtain the source-type integral equation,
O(r)𝜓(r, k̂ri ) = O(r)eik̂r ⋅r + k 2
i
∫D
O(r)G(r, r′ )O(r′ )𝜓(r′ , k̂ri )d3 r′
The Fourier transform of Eq. (10.41) yields
∞ ̃
2
O(𝜷 − 𝜶)T(𝜶, k̂ri ) 3
̃ − k̂ri ) − k
T(𝜷, k̂ri ) = O(𝜷
d 𝜶
3
8𝜋 ∫−∞
k 2 − 𝛼 2 + i𝜖
(10.41)
(10.42)
with
T(𝜷, k̂ri ) =
∫D
e−i𝜷⋅r O(r)𝜓(r, k̂ri )d3 r
(10.43)
where the eigen-function expansion of Green’s function G(r, r′ ) has been used
(section 2.2.2 of [3])
∞
exp[i𝜶 ⋅ (r − r′ )] 3
−1
G(r, r′ ) =
(10.44)
d 𝜶
8𝜋 3 ∫−∞ k 2 − 𝛼 2 + i𝜖
The quantity 𝜖 is an infinitely small positive value, representing loss, introduced
to remove the singularity when 𝛼 2 = k 2 . The far-field scattered field, given by
(10.12), now can be written as
eikr
(10.45)
T(k̂rs , k̂ri ),
4𝜋r
̃ Once the whole
which, from (10.42), obviously contains the whole spectrum O.
spectrum is obtained, the application of IFT yields super-resolution imaging.
However, it is a challenging task to retrieve those spectra, since it involves solving a nonlinear equation system (10.42) for several incidences. Thus, the fact
𝜓 s (̂rs , r̂ i ) =
273
274
Electromagnetic Inverse Scattering
that the scattered field contains a spectrum outside the Ewald limiting sphere
does not automatically lead to super resolution.
Third, generally speaking, the higher the contrast of scatterer, the stronger the
multiple scattering effect. In solving inverse scattering problems, for any specific inversion method, the role of multiple scattering does not always guarantee
an improved resolution. The reason is as follows. For a scalar wave scattering
problem, the scattered field is formulated as
s
i
E = GS ⋅ [(I − 𝜉 ⋅ GD )−1 ⋅ 𝜉 ⋅ E ]
(10.46)
The degree of nonlinearity of (10.46) as a function of 𝜉, a matrix with diagonal
term being the contrast, is categorized as
• If ||𝜉 ⋅ GD || ≪ 1, where || ⋅ || denotes the norm of an operator [25] (section
s
i
15.1), (10.46) is linear, E ≈ GS ⋅ [𝜉 ⋅ E ], which is in fact the first-order Born
approximation.
• If ||𝜉 ⋅ GD || < 1, the matrix inversion can be expanded into a Neumann series,
s
∑∞
i
E = GS ⋅ [(I + n=1 [𝜉 ⋅ GD ]n ) ⋅ 𝜉 ⋅ E ]. The truncation of the series yields a
polynomial relationship.
• If ||𝜉 ⋅ GD || > 1, the relationship is nonpolynomial.
We make the following comments on these three conditions.
• For weak scatterers, the multiple scattering effect can be ignored and
scattered fields depend linearly on the contrast. Since the matrix GS is
ill-conditioned, the reconstruction result is unstable in the presence of
noise.
• For moderate scatterers, a multiple scattering effect provides a lower-degree
polynomial dependance. For example, when the dependence is the
second-order polynomial, the scattered field is formulated as
s
i
E ≈ GS ⋅ [(I + 𝜉 ⋅ GD ) ⋅ 𝜉 ⋅ E ]
(10.47)
Converting the equation to an optimization problem, where the mismatch
between the left- and right-hand sides is minimized, we have the following
objective function,
s
i
s
i
min ∶ ||E − GS ⋅ 𝜉 ⋅ E ||2 − 2ℜ{(E )H ⋅ GS ⋅ 𝜉 ⋅ GD ⋅ 𝜉 ⋅ E }
𝜉
(10.48)
where the terms involving the third and fourth order of 𝜉 have been ommitted. The superscript H denotes the Hermitian operator. The minimum occurs
when the derivatives of objective function with respect to the elements of 𝜉
are equal to zero. If the term −2ℜ{⋅} was absent, the problem would reduce
to the case of weak scatterers, which is unstable. Due to the presence of the
term −2ℜ{⋅}, which is a quadratic function of 𝜉, the stability of the solution
to (10.48) is improved in the presence of measurement noise.
Resolution of Computational Imaging
• For strong scatterers, a multiple scattering effect is intensive, which provides
a higher degree of polynomial or even nonpolynomial dependance. The high
degree of nonlinearity makes it challenging to obtain the solution even in the
absence of noise; that is, stability is not the primary concern anymore.
The following numerical example provides validation of the aforementioned
conclusions. We consider a two-dimensional transverse magnetic (TM)
configuration. In order to test the effect of multiple scattering, we choose
the well-known “Austria” profile [27]. As shown in Fig. 10.4, it consists of an
annular and two disks, with the background being air. The discs of radius
0.2 m are centered at (0.3, 0.6) m and (−0.3, 0.6) m. The ring has an exterior
radius of 0.6 m and an inner radius of 0.3 m, and is centered at (0, −0.2)
m. The domain of interest is a 2 × 2 m2 square centered at the origin. The
relative permittivity 𝜖r of the “Austria” profile is chosen to be the following six
values, 1.01, 1.05, 1.1, 1.3, 2, and 3.5. The scatterers are illuminated with 20
plane waves at 400 MHz from different angles that are evenly distributed in
[0, 2𝜋). The scattered waves are collected by an antenna array with 40 antennas
uniformly placed on a circle centered at the origin with the radius 5𝜆 (𝜆 = 0.75
m is the wavelength in the background medium of air). White Gaussian noise
e is added to the exacts scattered field and
the noise level is defined to be the
s
SNR in dB 20log10 (||E ||∕||e||), where E is the exact scattered field. We apply
the subspace-based optimization method that is introduced in Chapter 6 to
reconstruct scatterers. The reconstructed results for 20 dB and 10 dB cases
are shown in Fig. 10.5 and Fig. 10.6, respectively, which demonstrate that (1)
inversion is unstable for weak scatterers, especially when noise is high; (2) for
strong scatterers, the optimization method is unable to find a solution that is
close to the exact one, regardless of the level of noise; and (3) for moderate
1
Figure 10.4 Inverse experiment of
the Austria profile: exact profile.
y(m)
0.5
0
–0.5
–1
–1
–0.5
0
x(m)
0.5
1
275
276
Electromagnetic Inverse Scattering
1.01
0.5
1
0
–0.5
0.99
1.03
1.02
0.5
0
1.01
0
–0.5
1
–0.5 0 0.5
(b)
–0.5 0 0.5
(a)
1.3
0.5
1.2
0.5
0
1.1
0
–0.5
1
1.1
0.5
–0.5
–0.5
0.99
1.8
1.6
1.4
1.2
1
0.8
1
–0.5 0 0.5
(c)
2
0.5
1.5
0
1
–0.5
–0.5 0 0.5
(e)
–0.5 0 0.5
(d)
1.05
0.5
–0.5 0 0.5
(f)
Figure 10.5 Reconstruction results of an Austria profile for 𝜖r = (a) 1.01, (b) 1.05, (c) 1.1, (d)
1.3, (e) 2, and (f ) 3.5 with SNR = 20 dB, respectively.
0.5
0
1.02
1.04
1.01 0.5
1.02
1
1
0
0.98
0.99 –0.5
–0.5
0.96
1.3
0.5
1.2
0
1.1
–0.5
1
–0.5 0 0.5
(d)
0.5
0
–0.5
–0.5 0 0.5
(e)
1.05
0
1
–0.5
–0.5 0 0.5
(b)
–0.5 0 0.5
(a)
1.1
0.5
0.95
–0.5 0 0.5
(c)
1.8
1.6
1.4
1.2
1
0.8
0.6
2
0.5
1.5
0
1
–0.5
0.5
–0.5 0 0.5
(f)
Figure 10.6 Reconstruction results of an Austria profile for 𝜖r = (a) 1.01, (b) 1.05, (c) 1.1, (d)
1.3, (e) 2, and (f ) 3.5 with SNR = 10 dB, respectively.
scatterers, the optimization method is able to find a solution that is close to
the exact one and the inversion is relatively stable in the presence of noise.
Fourth, we mention in passing two items regarding the effect of multiple scattering on solving ISPs:
• The multiple scattering effect may be either constructive or destructive. A
closed-form analytical design rule is proposed in [18] to construct a scattering system in which the multiple scattering model is definitely destructive.
Even if the interaction is constructive, in which multiple scattering is able to
Resolution of Computational Imaging
transfer more information of scatterers into the far field, the inversion process is also more complex.
• In some literature, comparisons are made between a full-wave nonlinear
inversion method that includes the multiple scattering effect and the BA
inversion that excludes the multiple scattering effect, given an experimental
setup in which multiple scattering does exist in nature and cannot be
neglected [26]. Thus the full-wave nonlinear inversion model is an exact
inversion model. However, the BA inversion model neglects multiple
scattering and this approximation inevitably introduces deterministic
errors, thus degrading the quality of the reconstructed results. Thus, [26]
perfectly illustrates that an exact inversion model performs better than
an approximate inversion model. However, it is not proper to cite this
reference as an experimental proof that the multiple scattering effect helps
in enhancing the reconstruction result.
10.6 Summary
The definition and reporting of spatial resolution for coherent imaging methods vary widely in the imaging community. There is an abundance of literature on the mathematics and physics of imaging principles. This chapter has
focused on discussing the resolution of imaging that is obtained by solving
inverse scattering problems, rather than providing a comprehensive review of
super-resolution imaging theories and schemes. Various topics have been discussed in this chapter and the main conclusions are summarized as follows.
It is important to note that instrumental imaging and computational imaging
are two different imaging strategies. For any instrumental imaging system, its
imaging mechanism can be formulated as a certain imaging algorithm, and
instrumental imaging is in fact just a hardware implementation of such an
algorithm. In comparison, computational imaging is implemented in computers, where various imaging algorithms, including the one corresponding to the
instrumental imaging, can be implemented. Thus, the classical half-wavelength
resolution widely used in traditional optical microscopy is not applicable to
inverse scattering problems.
Computational imaging problems are categorized into inverse source and
inverse scattering problems. The former deals with active objects that radiate
by themselves, whereas the latter deals with passive objects that are illuminated
by other primary sources. The accuracy of computational imaging depends on
the noise level of measured data. In a noise-free case, it might be possible to
achieve perfect reconstruction, that is, unlimited resolution, for certain computational imaging problems, such as for point-like sources/scatterers. Computational imaging problems can be treated as parameter identification problems,
and the accuracy of computational imaging is quantified by the Cramér–Rao
277
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Electromagnetic Inverse Scattering
bound (CRB), which quantifies a lower bound on the variance of any unbiased
estimator.
The Born Approximation (BA) is an important concept in wave-imaging
theory, which is applicable to weak scatterers. For plane wave incidences and
far-field measurements, BA reconstruction can be understood as a low-pass
filter. The analytical tool for the BA-based imaging provides a deep insight
into the resolution of computational imaging. The imaging resolution are
analytically derived for both two- and three-dimensional cases.
When we analyze the resolution in terms of the Fourier spectrum, the ultimate criterion is that during the image generation process, either instrumentally or numerically, the linear superposition of Fourier spectrum must keep the
original coefficient of Fourier spectrum radiated by the source. Consequently,
unless quantitative evidences are provided, qualitative arguments (such as the
improved resolution being due to the conversion of evanescent waves to propagating waves or the fact that the measured data contain a spectrum that is outside of the Ewald limiting sphere) alone are not convincing enough to explain
the super-resolution imaging results that are observed in many experiments.
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A
Ill-Posed Problems and Regularization
This appendix introduces some of the basic definitions and theorems of
ill-posed problems and regularization. The materials presented are accessible
to engineering researchers, with a focus on practical problem solving. Readers
do not have to be familiar with functional analysis. For a rigorous treatment of
ill-posed problems and regularization, literature in mathematical communities
is abundant, such as [1–8]. Regularization methods used in solving inverse
scattering problems can be referred to [9, 10] and references therein.
Four questions are discussed in this appendix. The first is, what are the properties of ill-posed problems? Out of the three properties, instability is of most
practical concern. The second is, what is the regularization theory? In order
to construct a stable approximate solution of an ill-posed problem, a process
called regularization is needed. The third is, what are commonly used regularization schemes? Three schemes will be introduced. The fourth is, how do we
select the regularization parameter that is used in regularization schemes? Four
selection methods will be introduced.
A.1 Ill-Posed Problems
The concept of a well-posed problem stems from a definition given by
Hadamard [11]. Let K be a linear or nonlinear mapping from the space X to Y .
The equation
Kx = y
(A.1)
is said to be well-posed (or properly posed) if the following three conditions are
satisfied
1) Existence: For each y ∈ Y there is x ∈ X such that Kx = y.
2) Uniqueness: For each y ∈ Y there is at most one x ∈ X such that Kx = y.
3) Stability: The solution x depends continuously on y, that is, if Kx∗ = y∗ and
Kx = y, then x → x∗ whenever y → y∗ .
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
282
Electromagnetic Inverse Scattering
Equations for which one or more of these conditions do not hold are said to
be ill-posed. We note that the above three conditions are, in general, not independent of each other. For example, when K is a bounded linear operator, if the
existence and uniqueness conditions are satisfied, then the stability condition
is automatically satisfied due to theorems in functional analysis.
For many physical and engineering problems, the existence and uniqueness
have been well studied. Thus, the instability is often the primary concern in the
study of ill-posed problems. For example, in linear algebra, let x ∈ ℂn , y ∈ ℂm ,
and K ∈ ℂm×n be a full-rank ill-conditioned matrix, that is, its condition number, which is defined as the ratio of the largest to smallest singular values, is
much larger than one. When solving an overdetermined linear equation system (A.1), where m>n, the solution is practically unstable since a small error in
y, which is often unavoidable due to measurement error, will completely destroy
the solution. In general, for an ill-posed problem, for which the stability condition does not hold, a straightforward application of standard inversion usually generates numerical nonsense. There is no way to overcome this difficulty
unless additional information about the solution is available. It is worthwhile
recalling the remark of Lanczos: “A lack of information cannot be remedied by
any mathematical trickery.”
A.2 Regularization Theory
In order to obtain a reasonable stable solution to an ill-posed problem (A.1),
the basic idea is to replace the original equation by a close equation involving
a small parameter 𝛼, such that the changed equation can be solved in a stable way and at the same time its solution is close the solution of the original
problem when 𝛼 approaches zero. This process of constructing a stable approximate solution of an ill-posed problem is called regularization. Mathematically
speaking, regularization replaces the inverse K −1 by a family R𝛼 (where 𝛼 > 0)
of approximate inverse of K in such a way that
1) R𝛼 is a continuous operator for each 𝛼 > 0, which means stability.
2) lim𝛼→0 R𝛼 Kx = x for all x ∈ X.
The family of R𝛼 is called a regularization scheme and the parameter 𝛼 is called
the regularization parameter.
From here onward, we consider K to be a linear operator. A linear ill-posed
problem is of particular interest since it appears in a variety of important applications in science and industry. For example, many deconvolution problems
are ill-posed since they have smooth or weakly singular kernels, where a Fredholm integral equation of the first kind has to be inverted. In addition, linear
ill-conditioned problems frequently arise in the iterative solution of nonlinear
systems or optimization problems.
Ill-Posed Problems and Regularization
The notion of regularization scheme is based on unperturbed data, that is,
R𝛼 y converges, as 𝛼 → 0, to x for the exact right-hand side y(= Kx). Let y𝛿 be
the measured right-hand side with a known error level
||y − y𝛿 || ≤ 𝛿
(A.2)
where || ⋅ || denotes the norm defined in the space Y . For example, if Y is ℝm or
ℂm , then its norm is the Euclidean norm, which is also known as the 𝓁 2 norm.
The regularization scheme approximates the solution x by
x𝛿𝛼 = R𝛼 y𝛿
(A.3)
The error of the approximation splits into two parts by the triangle inequality,
||x𝛿𝛼 − x|| ≤ ||R𝛼 y𝛿 − R𝛼 y|| + ||R𝛼 y − x||
≤ ||R𝛼 ||||y𝛿 − y|| + ||R𝛼 Kx − x||
≤ 𝛿||R𝛼 || + ||R𝛼 Kx − x||
(A.4)
The error consists of two parts: the first term describes the error in the data
multiplied by ||R𝛼 || and the second term denotes the approximation error
between R𝛼 and K −1 . When 𝛼 tends to zero, the first tends to infinity due to the
ill-posed nature of the problem, whereas the second term tends to zero by the
definition of regularization scheme. Every regularization scheme requires a
strategy to choose regularization parameter 𝛼 = 𝛼(𝛿) that is dependent on measurement error 𝛿 in order to keep the total error as small as possible. The corresponding parameter reflects a compromise between accuracy and stability.
For a regularization scheme, we naturally expect the regularized solution to
converge to the exact solution when the measurement error 𝛿 tends to zero.
If this condition is satisfied, the regularization scheme is said to be regular or
admissible.
The next two sections discuss specific regularization schemes and regularization parameter selection methods.
A.3 Regularization Schemes
This section introduces three regularization schemes that are frequently used
in solving linear ill-posed problems: spectral cutoff, Tikhonov regularization,
and iterative regularization. These regularization schemes have been proven to
be regular (or admissible) for linear ill-posed problems. For purpose of practical
applications, we consider a discrete linear equation system,
K ⋅x=y
where x ∈ ℂN , y ∈ ℂM , and K ∈ ℂM×N .
(A.5)
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Electromagnetic Inverse Scattering
A.3.1
Spectral Cutoff
In linear algebra, the singular value decomposition (SVD) is a factorization
of a matrix, which is the generalization of the eigen-decomposition of a positive semidefinite normal matrix to any M × N matrix. The SVD of K is represented as
K =U ⋅Σ⋅V
H
(A.6)
where the superscript H denotes the Hermitian, that is, transpose complex conjugate, and
• U is of size M × M and is composed of the left singular vectors um , which are
orthogonal and unit vectors.
• V is of size N × N and is composed of the right singular vectors 𝑣n , which
are orthogonal and unit vectors.
• Σ is of size M × N with the diagonal terms being singular values 𝜎i that are
real and placed in nonincreasing order 𝜎1 ≥ 𝜎2 ≥ · · · ≥ 0
A basic property of SVD is K ⋅ 𝑣m = 𝜎m um , which enables us to write the solution to (A.5) as
∑ 1 H
†
x=K ⋅y=
(u ⋅ y)𝑣i
(A.7)
𝜎 i
𝜎 ≠0 i
i
where
†
†
K =V ⋅Σ ⋅U
H
(A.8)
is defined as the pseudoinverse of the matrix K. Since the noise in y is inevitable,
division by small singular values leads to tremendously large errors. Thus, a natural regularization scheme is to drop off terms corresponding to small singular
values. Consequently, an approximation to the exact x reads
∑ 1 H
x1 = R1y =
(u ⋅ y)𝑣i
(A.9)
m
m
𝜎i i
𝜎 ≥𝜎
i
m
The so-defined regularization scheme R 1 is called the spectral cutoff, for which
m
1∕m is the regularization parameter. The so-obtained solution is referred to as
the truncated SVD (TSVD) solution.
It is obvious that accuracy of the approximation to the pseudoinverse requires
the integer m to be large whereas stability requires it to be small. A plot of
singular values is highly useful to study the influence of measurement error in
y on the accuracy of the TSVD solution. If singular values decay slowly to zero,
the equation is said to be mildly ill-posed. If they decay very rapidly to zero, the
equation is severely ill-posed.
However, the SVD method is restricted to small and medium sized problems,
since it becomes impractical for large-scale problems due to heavy computational cost. In this case, we need to resort to other regularization schemes.
Ill-Posed Problems and Regularization
A.3.2
Tikhonov Regularization
Tikhonov regularization is possibly the most commonly used method of regularization of ill-posed problems. A common approach to solve a linear equation
(A.5) is to determine the best fit in the sense that the defect ||K ⋅ x − y|| is minimized. If K is ill-conditioned, then the minimization problem is also ill-posed.
In order to obtain a stable solution, the following Tikhonov functional is minimized over ℂN ,
J𝛼 (x) = ||K ⋅ x − y||2 + 𝛼||x||2
(A.10)
It is worth mentioning that some references use 𝛼 2 , instead of 𝛼, in the last term
𝛼
of (A.10). The Tikhonov functional has a unique minimum x , which is also
the unique solution of the equation of zero gradient, ∇x J𝛼 = 0. The gradient of
(A.10) can be derived from the Appendix B and we eventually arrive at
H
H
K ⋅ K ⋅ x𝛼 + 𝛼x𝛼 = K ⋅ y
(A.11)
𝛼
The explicit expression of x is given by
𝛼
H
H
x = (K ⋅ K + 𝛼I)−1 ⋅ K ⋅ y
(A.12)
H
where I is an identity matrix of the same size as K ⋅ K. The regularization
H
H
scheme R𝛼 = (K ⋅ K + 𝛼I)−1 ⋅ K is called the Tikhonov regularization, for
which 𝛼 is the regularization parameter. As 𝛼 tends to zero, the approximate
†
inverse R𝛼 converge to the pseudoinverse K , whereas the solution becomes
unstable. In this sense, the solution provided by the Tikhonov regularization scheme, as a minimizer of the Tikhonov functional, keeps the residual
||K ⋅ x − y||2 small and is stabilized through the penalty term 𝛼||x||2 .
𝛼
We mention in passing that the solution x of the Tikhonov regularization
scheme has a representation through the SVD (A.6) system,
∑ 𝜎i
H
R𝛼 y =
(A.13)
(ui ⋅ y)𝑣i
2
𝜎
+
𝛼
i
i
In this sense, the Tikhonov regularization and the spectral cutoff (A.9) can be
both categorized as regularization by filtering, and they differ only in filtering
functions.
A.3.3
Iterative Regularization
This section introduces the concept that the iteration count can play the role of
regularization parameter when iterative methods are used to solve an ill-posed
problem. We apply the following iterative method to solve the linear equation
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Electromagnetic Inverse Scattering
(A.5), or equivalently, the optimization problem of minimizing ||K ⋅ x − y||2
0
x =0
m
x =x
m−1
H
− 𝜏K ⋅ (K ⋅ xm−1 − y),
m = 1, 2, …
(A.14)
For a certain range of positive 𝜏, the iteration scheme is known as Landweber
m
iteration [1] (section 2.3). The explicit form of x can be obtained from the
recursion formula (A.14).
∑
m−1
m
x =R y=𝜏
1
m
H
H
(I − 𝜏K ⋅ K)i ⋅ K ⋅ y
(A.15)
i=0
The operator R 1 defines the Landweber iteration regularization scheme, for
m
which 1∕m is the regularization parameter. We observe that high precision
(in absence of measurement noise) requires a large iteration count m whereas
stability requires a small m. This observation, together with the definition of
regularization, ensures 1∕m, instead of m, to be the regularization parameter.
In the presence of noise, both the objective function and the error of reconstruction decrease during the initial iterations, but the error of reconstruction
increases after a certain number of iterations, although the objective function
monotonically decreases. This phenomena is called semi-convergence.
We mention briefly that the Landweber iteration regularization scheme can
be also categorized as the regularization by filtering, with a more complex filtering function compared with those of Tikhonov regularization and spectral
cutoff.
A.4 Regularization Parameter Selection Methods
It is important to select the regularization parameter properly. With too little
regularization, reconstructions are unstable due to noise amplification. With
too much regularization, the accuracy of solution is often poor although it is
stable. There are two types of regularization parameter selection methods: a
priori and a posteriori methods [2] (section 4.2). An a priori selection method
requires, in addition to data noise level, a priori information about the exact
solution, such as its norm or smoothness properties that, however, will generally not be available in practical problems. This type of regularization parameter
selection method is mainly of interest to the mathematical community [12].
Thus, a posteriori selection methods are more practical since they depend on
the data noise level but not on a priori information about the exact solution.
All the following regularization parameter selection methods are a posteriori
methods.
Ill-Posed Problems and Regularization
A.4.1
Discrepancy Principle
The discrepancy principle states that for erroneous data the residual
||K ⋅ x − y|| should not be smaller than the noise level of measurement data
y [1] (section 2.5). That is, the regularization parameter 𝛼(𝛿) is chosen such
that
||K ⋅ R𝛼 y − y|| = 𝛾𝛿
(A.16)
for some fixed parameter 𝛾 ≥ 1. Newton’s method can be applied to solve for
𝛼. In case of a regularization scheme R 1 , where the regularization parameter
m
is discrete, with m = 1, 2, …, the integer m should be chosen as the smallest
integer satisfying
||K ⋅ R𝛼 y − y|| ≤ 𝛾𝛿
A.4.2
(A.17)
Generalized Cross Validation
The discrepancy principle is based on a deterministic model, which requires
prior knowledge of the noise level. In comparison, the generalized cross validation (GCV) is based on a stochastic framework in which it is possible to select
a proper regularization parameter in the absence of prior information about
the noise level. In this method, the regularization parameter 𝛼 is chosen as the
minimizer of the GCV formula [7] (section 7.2):
GCV (𝛼) = [
1
||(I
M
− K ⋅ R𝛼 )y||2
1
Trace(I
M
− K ⋅ R𝛼 )
]2
(A.18)
where M is the dimension of y.
A.4.3
L-Curve Method
The discrepancy principle and the generalized cross validation methods can
be understood as predictive methods that seek to provide a regularized solution that predicts the unperturbed right-hand side as accurate as possible. A
different approach is provided by the L-curve method, which seeks to balance
the norms of the regularized solution ||R𝛼 y|| and the corresponding residual
||K ⋅ R𝛼 y − y|| [6] (section 4.6). The L-curve is a log-log plot of ||R𝛼 y|| against
||K ⋅ R𝛼 y − y|| for a range of values of regularization parameter 𝛼. This curve
typically exhibits an L shape. The L-curve criterion for regularization parameter selection is to choose the parameter value corresponding to the corner of
the L-curve [13].
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Electromagnetic Inverse Scattering
A.4.4
Trial and Error
It is quite usual to select regularization parameter by trial and error where a
few different parameters 𝛼 are used and then one of them is picked as the most
reasonable result based on appropriate information on the expected solution.
In practice, before solving an inversion problem, we need to first practice on a
training set where the exact solution is known. For a given level of noise, a finite
range of regularization parameter 𝛼, which yields a solution close to the exact
one, is identified. It is of practical importance to identify a range of candidate
regularization parameters, and there is no need to identify the “best” regularization parameter. Indeed, in many inverse problems, in particular nonlinear
problems, the criteria of selecting regularization parameter themselves are not
necessarily optimal yet, and consequently there is no need to work very hard to
find a highly accurate solution.
A.5 Discussions
For a nonlinear ill-posed problem (A.1), a generalized Tikhonov functional
takes the form
F𝛼 (x, y) = 𝜌(Kx, y) + 𝛼J(x)
(A.19)
where 𝛼 is the regularization parameter, 𝜌(⋅, ⋅) is the data discrepancy functional
or fit-to-data functional, and J(⋅) is the penalty functional or regularization
functional. The data discrepancy functional quantifies how well the prediction Kx matches the observed data y. Perhaps the most familiar example is
the squared norm in the space where y is in; that is, 𝜌(Kx, y) = ||Kx − y||2 . The
role of the penalty functional is to induce stability and to incorporate a priori
information about the desired solution x. There are many choices in the penalty
functional. The standard Tikhonov penalty function is the square of Euclidean
norm in the space of the square-integrable function. For piece-wise constant
functions, a total variation (TV) penalty functional is usually adopted [14],
which reads in two-dimensional space, TV (x) = ∫ ∫ |∇x|dt1 dt2 , where x(t1 , t2 )
depends on two parameters t1 and t2 and ∇ denotes the gradient operator.
The TV penalty functional penalizes highly oscillatory solutions while allowing jumps in the regularized solution [7] (chapter 8), which is widely used in
image restoration problems in order to obtain edge-preserved images.
As a side note, there is another regularization strategy, known as multiplicative regularization, for which the functional reads
F(x, y) = 𝜌(Kx, y)M(x)
(A.20)
where the regularization parameter 𝛼 is not required. The multiplicative
function M(x) depends on the unknown x and is automatically controlled
Ill-Posed Problems and Regularization
by the optimization process itself; that is, M(x) contains tuning parameters
that depend on iteration count. An example of multiplicative regularization is
given in [10] and references therein.
Finally, we briefly mention numerical examples of solving ill-posed problems.
In this book, a numerical example of solving a linear ill-posed problem is illustrated in Section 6.2.5, where Tikhonov regularization has been used to obtain
stable solutions. Section 6.4.3 shows a numerical example where iteration count
plays the role of regularization parameter. More tutorial exercises can be found
in [15], which give readers hands-on experience in the treatment of inverse
problems.
289
291
B
Least Squares
The method of least squares is a form of mathematical regression analysis that
finds the approximate solution of overdetermined systems. The least-squares
method provides the best fit to data in the sense that the overall solution minimizes the sum of the squares of the residuals made in the results of every single
equation. This appendix covers two topics of the linear least-squares method:
the geometric interpretation of least squares and the formula of the gradient of
squared residuals. These two concepts are frequently used in this book.
B.1 Geometric Interpretation of Least Squares
B.1.1 Real Space
First, we consider the case when the unknown x is a real-valued scalar and we
will solve the following linear equation,
(B.1)
ax = b
where a and b are M-tuple real-valued vectors. The pair ai and bi denotes the ith
experiment, i = 1, 2, … , M. When there are errors in a or b, the vectors a and
b are not aligned and, consequently, there is no exact solution x to (B.1). The
least-squares method provides a solution that minimizes ||ax − b||2 , the sum
of the squares of the residuals made in the results of every single equation. We
observe that, by changing the real number x, the vector ax is either stretched or
contracted, but it is always in the same line. From geometry, it is obvious that
the minimum of ||ax − b||2 occurs when ax − b is perpendicular to a, which can
be rigorously proven by the Pythagorean theorem. Thus, we have
T
a ⋅ (ax − b) = 0
(B.2)
where the superscript T denotes the transpose. The solution is given simply by
T
x = (||a||2 )−1 a ⋅ b
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
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(B.3)
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Electromagnetic Inverse Scattering
Next, we consider the case when the unknown x is an N-tuple real-valued
vector and we will solve the following linear equation,
A⋅x=b
(B.4)
where A is an M × N matrix. Along the same lines as for the case of a scalar
unknown, we easily see that the geometric interpretation states that the minimum of ||A ⋅ x − b||2 occurs when the vector A ⋅ x − b is perpendicular to the
space spanned by the columns of A, i.e., A1 , A2 , …, AN . Thus, we have
T
Ai ⋅ (A ⋅ x − b) = 0
i = 1, 2, … , N
(B.5)
Combining them together, we obtain
T
A ⋅ (A ⋅ x − b) = 0
(B.6)
The least-squares solution is given by
T
T
x = (A ⋅ A)−1 ⋅ A ⋅ b
(B.7)
†
T
T
which defines the pseudoinverse A = (A ⋅ A)−1 ⋅ A .
B.1.2 Complex Space
In the complex domain, the orthogonality of vectors a and b is understood as
H
their inner product a ⋅ b = 0, where the superscript H denotes the Hermitian;
that is, complex conjugate transpose. Thus, the least-squares solution occurs
when the orthogonality condition (B.6) is replaced by
H
A ⋅ (A ⋅ x − b) = 0
(B.8)
and consequently, the least-squares solution is given by
H
H
x = (A ⋅ A)−1 ⋅ A ⋅ b
(B.9)
†
H
H
which defines the pseudoinverse A = (A ⋅ A)−1 ⋅ A .
B.2 Gradient of Squared Residuals
For an N-tuple complex-valued vector x, define the function of squared residuals,
f (x) = ||A ⋅ x − b||2 = ΔH ⋅ Δ
(B.10)
where the residual vector is defined to be Δ = A ⋅ x − b. In solving optimizadf
tion problems, we often need to calculate the gradient of f , dx . The vector x is
Least Squares
complex (x = xR + ixI ), whereas f (x) is a scalar. Obviously, f and any component of x do not satisfy the Cauchy–Riemann equations and, consequently, f is
df
not complex differentiable. To interpret the notation of “gradient” dx , we have
to treat the real and imaginary parts separately.
T
We first introduce a useful notation. For real 𝛼, the gradient of g = 𝛼 ⋅ 𝛽 =
T
dg
𝛽 ⋅ 𝛼 with respect to 𝛼 is given by d𝛼 = 𝛽. A further step is to define the
T
T
gradient of a vector. If G = [g1 , g2 , … , gM ]T , where g1 = 𝛼 ⋅ 𝛽 1 , g2 = 𝛼 ⋅ 𝛽 2 ,
T
T
T
…, gM = 𝛼 ⋅ 𝛽 M , then G = 𝛼 ⋅ B, where B = [𝛽 1 , 𝛽 2 , … , 𝛽 M ]. We have
T
T
dg
dg
dG
= [ 1 , … , M ] = [𝛽 , … , 𝛽 ] = B; that is, d𝛼 ⋅B = B.
d𝛼
d𝛼
1
d𝛼
M
d𝛼
Let P denote the subscript R or I, and the gradient of f that is shown in (B.10)
with respect to xP is obtained by the product rule,
df
d(A ⋅ x − b)H
d(A ⋅ x − b)
=
⋅ Δ + ΔH ⋅
dxP
dxP
dxP
=
d(A ⋅ x)H
d(A ⋅ x)T
⋅Δ+
⋅ Δ∗
dxP
dxP
=
d(x )T ⋅ (A ⋅ Δ) d(x)T ⋅ (A ⋅ Δ)∗
+
dxP
dxP
H
∗
H
(B.11)
where the superscript * denotes complex conjugate. Consequently, the real and
imaginary parts can be obtained separately,
H
H
H
df
= A ⋅ Δ + (A ⋅ Δ)∗ = 2Re{A ⋅ Δ}
dxR
(B.12)
H
H
H
df
= −iA ⋅ Δ + i(A ⋅ Δ)∗ = 2Im{A ⋅ Δ}
dxI
(B.13)
Strictly speaking, the gradient of f (x) with respect to x is a 2N-dimensional real
vector,
[(
)T (
)T ]T
df
df
df
,
(B.14)
=
dx
dxR
dxI
df
If the operation on dx is limited to only vector addition and scalar-vector
multiplication, the real part and the imaginary part do not crosstalk. Consedf
quently, the 2N-dimensional real vector dx can be compactly rewritten as an
N-dimensional complex vector,
df
df
df
+i
(B.15)
=
dx dxR
dxI
This notation is analogous to the usual practice of writing the space vector r =
[x, y, z]T as r = x̂x + ŷy + ẑz.
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Electromagnetic Inverse Scattering
To conclude,
H
df
= 2A ⋅ (A ⋅ x − b)
dx
As an example, for the Tikhonov regularization functional,
J(x) = ||A ⋅ x − b||2 + 𝛼||x||2
(B.16)
(B.17)
its minimum occurs when its derivative is zero; that is,
H
2A ⋅ (A ⋅ x − b) + 2𝛼x = 0
(B.18)
which can be rewritten as
H
H
(A ⋅ A + 𝛼I) ⋅ x = A ⋅ b
(B.19)
The solution is given by
H
H
x = (A ⋅ A + 𝛼I)−1 ⋅ A ⋅ b
(B.20)
295
C
Conjugate Gradient Method
The Conjugate Gradient (CG) method is a mathematical technique for
numerically solving unconstrained optimization problems. The CG method
uses conjugate directions instead of the local gradient for going downhill and
it generally converges faster than the method of steepest descent. This book
applies the CG to minimize objective functions in inverse scattering problems
and to solve linear equation systems in an iterative way. The appendix covers
two topics: the conjugate gradient method for general minimization problems
and its application to solve linear equation systems.
C.1 Solving General Minimization Problems
C.1.1
Real Space
First, we consider the case when the unknown x is an N-tuple real-valued vector
and we will minimize the objective function f (x). The implementation steps of
the CG algorithm are given as follows [7] (section 3.2).
Step 1: Initial iteration step, n = 0:
Choose the initial guess x0 ;
Calculate the gradient (Fréchet derivative) g 0 = ∇f (x0 ) at x0 ;
Initialize the search directions, p0 = −g 0 ;
Step 2:Line search for the scalar
Line search for the scalar 𝛼n that minimizes f (xn + 𝛼n pn );
Update solution: xn+1 = xn + 𝛼n pn ;
Update the search direction:
Calculate the gradient g n+1 = ∇f (xn+1 )
Calculate the coefficient 𝛽n , following either the Fletcher–Reeves formula
𝛽n =
∥g n+1 ∥2
∥g n
∥2
or the Polak–Ribière formula 𝛽n =
Update the search direction pn+1 = −g n+1 + 𝛽n pn
T
g n+1 ⋅(g n+1 −g n )
∥g n ∥2
;
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Electromagnetic Inverse Scattering
Step 3: Is a predetermined termination condition satisfied, such as whether the
gradient falls below some predetermined value?
If yes: stop iteration.
If no: n ∶= n + 1 and go to Step 2.
C.1.2
Complex Space
For an N-tuple complex-valued vector x = xR + ixI , the CG method outlined in
Section C.1.1 in fact applies to a 2N-dimensional real vector,
[(xR )T , (xI )T ]T
(C.1)
As discussed in Appendix B.2, if the operation is limited to only vector addition and scalar-vector multiplication, the real part and the imaginary part do
not cross talk and consequently the N-dimensional complex-value notation is
equivalent to the 2N-dimensional real-value notation. After a careful check up,
we find that the only step in the CG method that cannot use the N-dimensional
T
complex-value notation is the numerator in the Polak–Ribière formula g n+1 ⋅
(g n+1 − g n ). In 2N-dimensional real-value notation, it reads
Re[g n+1 ]T ⋅ Re[g n+1 − g n ] + Im[g n+1 ]T ⋅ Im[g n+1 − g n ]
(C.2)
If we force ourselves to use the N-dimensional complex-value notation, then
(C.2) is equal to
H
Re[g n+1 ⋅ (g n+1 − g n )]
(C.3)
To summarize, for a complex-valued vector x, the only change we need to make
is to replace the numerator in the Polak–Ribière formula by (C.3).
C.2 Solving Linear Equation Systems
In the complex space, we solve the linear equation system,
A⋅x=b
(C.4)
which is equivalent to the minimization problem
1
(C.5)
f (x) = ∥ A ⋅ x − b∥2
2
Thus, we are able to apply the CG method to minimize (C.5). The following
three analytical results are obtained [16] (section 4.11), due to the special property of the linear equation system:
Conjugate Gradient Method
H
• g n = ∇f (xn ) = A ⋅ (A ⋅ xn − b), which is taken directly from Appendix B.2.
• The scalar 𝛼n is actually just the least-squares solution, 𝛼n = −
H
(A⋅pn )H ⋅(A⋅xn −b)
• Since g n+1 ⋅ g n = 0, the Polak–Ribière formula reduces to 𝛽n =
is exactly the same as the Fletcher–Reeves formula.
∥A⋅pn ∥2
∥g n+1 ∥2
∥g n ∥2
.
, which
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299
D
Matrix-Vector Product by the FFT Procedure
Electromagnetic problems posed in terms of integral equations with
convolutional kernels can sometimes be discretized to yield matrices with
discrete convolutional symmetries. Such examples include scattering problems
by a one-dimensional lattice of uniform cells, a rectangle of uniform cells, or
a cuboid of uniform cells embedded in a homogeneous background medium.
Uniform cells are necessary to make good use the property of translational
invariance of a background Green’s function. For a matrix-vector product,
with size of the matrix being N × N, the FFT procedure has a computational
complexity of O(N log N), which is lower than the counterpart O(N 2 ) of the
traditional approach (section 4.12 of [16] and [17]).
D.1 One-Dimensional Case
In one-dimensional scattering problems, electric field E may present the form
of discrete convolution between kernel-induced function G and the current J,
Em =
N
∑
Gm−n Jn ,
m = 1, 2, … , N
(D.1)
n=1
where Gm−n denotes the interaction between the mth and the nth cell. Due to
the property of translational invariance of the background Green’s function,
any two cells that are different by m − n in terms of the indices of cells have an
interaction coefficient Gm−n . Written in the form of matrix-vector multiplication, (D.1) appears as
E =G⋅J
(D.2)
where the element of the N × N matrix G is given by
Gm,n = Gm−n ,
m, n = 1, 2, … , N
(D.3)
It is well known that the discrete convolution can be efficiently implemented
by the FFT and inverse FFT algorithm if the convolution is of type circular
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
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300
Electromagnetic Inverse Scattering
discrete convolution, also known as cyclic discrete convolution. If the elements
of the sequence G repeat with period N, that is, Gn−N = Gn , n = 1, 2, … , N − 1,
the operation (D.1) is known as the circular discrete convolution. For a circular
discrete convolution, the matrix G in (D.1) would exhibit the property that its
rows are circular right shifts of the elements of the preceding row.
Unfortunately, (D.1) is not a circular discrete convolution, which can be easily
seen since the second row of G is not a circular right shift of the elements of the
first row. The reason is that the interaction between cells 1 and N is different
from that between cells 2 and 1; that is, the elements of the sequence G does
not repeat with period N. Nevertheless, it is well known that any noncircular
discrete convolution can be extended to a larger-size circular discrete convolution. The sequence Ge , extended from G, repeats its elements with period
2N − 1. Written in a vector, the elements in a period are
e
G = [G0 , G1 , G2 , … , GN−1 , G1−N , G2−N , … , G−1 ]T .
(D.4)
In order not to change the value of Em in (D.1), where N is replaced by 2N − 1,
the vector J should be padded with N − 1 zeros so as to eliminate the contribue
tion of the added N − 1 elements in Ge . Consequently, the J , expanded from J
by padding zeros, is a vector of size 2N − 1,
e
J = [J1 , J2 , … , JN , 0, 0, … 0]T .
(D.5)
The extended circular discrete convolution can be written in the form of
matrix-vector multiplication,
e
e
E =G ⋅J
e
(D.6)
e
where E is of size 2N − 1, with only the first N elements being the original vale
ues of E. The G is of size (2N − 1) × (2N − 1). We can easily verify that its rows
are circular right shifts of the elements of the preceding row and, in addition,
e
the first column of it is just the vector G in (D.4).
According to the discrete convolution theorem, the Fourier transform of a
circulate discrete convolution is the point-wise product of Fourier transforms.
e
Thus, the E can be efficiently calculated by the FFT and inverse FFT algorithm,
e
e
e
E = FFT−1 [FFT(G ). ∗ FFT(J )]
(D.7)
where . ∗ denotes point-wise multiplication. Finally, the sought-after E is
e
extracted from the first N elements of E .
D.2 Two-Dimensional Case
The procedures outlined in the previous section are easily generalized to two
or three dimensions. A two-dimensional discrete convolution is an operation
Matrix-Vector Product by the FFT Procedure
of the form
Ei,j =
M N
∑
∑
Gi−m,j−n Jm,n ,
i = 1, 2, … , M, j = 1, 2, … , N
(D.8)
m=1 n=1
e
To create a circular discrete convolution, the extended parameters are J of size
(2M − 1) × (2N − 1),
{
J(p, q), if 1 ≤ p ≤ M and 1 ≤ q ≤ N
J e (p, q) =
(D.9)
0,
if M < p ≤ 2M − 1 or N < q ≤ 2N − 1
e
and G of size (2M − 1) × (2N − 1)
Ge (p, q) = Gp′ ,q′ ,
(D.10)
{
{
p − 1,
if 1 ≤ p ≤ M
q − 1, if 1 ≤ q ≤ N
where p′ =
, q′ =
p − 2M, M < p ≤ 2M−1
q − 2N, N < q ≤ 2N −1
e
The extended E is of size (2M − 1) × (2N − 1) and can be calculated by the
two-dimensional FFT and inverse FFT algorithms,
e
e
e
E = FFT2 −1 [FFT2 (G ). ∗ FFT2 (J )]
(D.11)
where FFT2 denotes two-dimensional fast Fourier transform. Finally, the
e
sought-after E is extracted as the upper left sub matrix of size M × N from E .
Appendix References
1 Kirsch, A. (1996) An introduction to the mathematical theory of inverse
problem, Springer, New York.
2 Colton, D. and Kress, R. (1998) Inverse acoustic and electromagnetic scatter-
ing theory, 2nd Edn. Springer-Verlag, Berlin, Germany.
3 Neumaier, A. (1998) Solving ill-conditioned and singular linear systems: A
tutorial on regularization. SIAM Review, 40 (3), 636–666.
4 Isakov, V. (2006) Inverse problems for partial differential equations, vol. 127,
Springer Science and Business Media.
5 Engl, H.W., Hanke, M., and Neubauer, A. (1996) Regularization of inverse
problems, vol. 375, Kluwer.
6 Hansen, P.C. (1998) Rank-deficient and discrete ill-posed problems: numeri-
cal aspects of linear inversion, SIAM.
7 Vogel, C. (2002) Computational methods for inverse problems, SIAM.
8 Nakamura, G. and Potthast, R. (2015) Inverse modeling, IOP Publishing.
9 Mojabi, P. and LoVetri, J. (2009) Overview and classification of some reg-
ularization techniques for the Gauss-Newton inversion method applied to
301
302
Electromagnetic Inverse Scattering
10
11
12
13
14
15
16
17
inverse scattering problems. IEEE Transactions on Antennas and Propagation, 57 (9), 2658–2665.
Abubakar, A., van den Berg, P.M., and Mallorqui, J.J. (2002) Imaging of
biomedical data using a multiplicative regularized contrast source inversion
method. IEEE Transactions on Microwave Theory and Techniques, 50 (7),
1761–1771.
Hadamard, J. (1923) Lectures on Cauchy’s problem in linear partial differential equations, Yale University Press.
Cheng, J. and Yamamoto, M. (2000) One new strategy for a priori choice of
regularizing parameters in Tikhonov’s regularization. Inverse Probl., 16 (4),
L31.
Belge, M., Kilmer, M.E., and Miller, E.L. (2002) Efficient determination of
multiple regularization parameters in a generalized l-curve framework.
Inverse Probl., 18 (4), 1161.
Rudin, L.I., Osher, S., and Fatemi, E. (1992) Nonlinear total variation
based noise removal algorithms. Physica D: Nonlinear Phenomena, 60
(1), 259–268.
Hansen, P.C. (2010) Discrete inverse problems: insight and algorithms,
SIAM.
Peterson, A.F., Ray, S.L., and Mittra, R. (1998) Computational methods for
electromagnetics, IEEE Press, New York.
Su, C.C. (1987) Calculation of electromagnetic scattering from a dielectric
cylinder using the conjugate gradient method and FFT. IEEE Transactions
on Antennas and Propagation, 35 (12), 1418–1425.
303
Index
a
acoustic wave 48
active source 16, 36, 41 see also
primary source
addition theorem 108, 198
analytical continuation 98, 117, 118,
262
angular spectrum representation 32,
33
b
back-propagation scheme 133, 134
basis function 29–31, 150, 193, 231,
244
Bayesian approach 8, 215
Born approximation (BA) 118, 130,
131, 216, 268–272
boundary conditions (BC) 3, 15, 21
see also Dirichlet BC;
impedance BC; Neumann BC;
radiation BC
boundary element method (BEM)
229
boundary integral (BI) 228, 243
c
Calderón’s problem 124
coherent wave 224, 258, 260,
277
compact 151, 163
compressive/compressed sensing (CS)
214–220
computational complexity
FFT 30, 164, 299
general iterative approach 128
matrix inversion 30
matrix-vector product 160, 161,
163, 299
SVD 160, 161, 163
computed tomography (CT) 131
conducting medium/conductor 16
confocal laser scanning microscopy
(CLSM) 246–248
conjugate gradient (CG) 30, 129, 140,
295–297
constitutive relation 16
continuity of solution 3, 4 see also
stability of solution
contrast 127, 263
contrast current density 126
contrast source extend Born (CS-EB)
144–146
contrast source inversion (CSI)
142–144, 155
convolution 19, 260, 270, 299–301
coupled dipole method (CDM) 30,
147
covariance matrix 63, 74
Cramér-Rao bound (CRB) 79,
264–268
Computational Methods for Electromagnetic Inverse Scattering, First Edition. Xudong Chen.
© 2018 John Wiley & Sons Singapore Pte. Ltd. Published 2018 by John Wiley & Sons Singapore Pte. Ltd.
304
Index
d
data equation 127, 198, 221
decomposition of the time-reversal
operator (DORT) 55, 74
degenerate scatterer 82
degree of freedom 28, 57, 125, 128,
173
depolarization factor 24 see also
polarization strength
deterministic optimization algorithm
128
diagonal tensor approximation (DTA)
132, 148
dielectric material 3, 16
diffraction tomography (DT)
131
diffuse optical tomography (DOT)
224
direct problem 1 see also forward
problem
Dirichlet BC 3, 203
discrete dipole approximation (DDA)
30 see also coupled dipole
method (CDM)
distorted Born iterative method
(DBIM) 139–142, 173
distorted-wave Born approximation
(DWBA) 139
domain of interest (DOI) 1, 71, 103,
127
double-layer potential 39
duality principle 18, 26
dual space method 146, 185
e
edge-based vector elements 244
eigen-dipole 92
eigen-source 57
eigenvalue 54–56, 84, 93
decomposition 54–56
eigenvector 54–56, 84, 93
electric current dipole 24
electric field integral equation (EFIE)
21–23, 30, 193
electric impedance tomography (EIT)
124, 173
evanescent wave 33, 44, 259
Ewald limiting sphere 270, 273
Ewald–Oseen extinction theorem see
extinction theorem
Ewald’s sphere of reflection 269
existence of solution 3, 4, 281, 282
extended Born approximation (EBA)
131–133, 145
extended scatterer 6, 29, 98, 117, 149,
220
extinction theorem 39
f
factorization method (FM) 116–118,
186, 196
far field 32–34, 46, 50, 104, 118, 263,
272
integral equation 104, 116
MSR matrix 117
operator 117
pattern 104
fast Fourier transform (FFT) 30, 128,
160, 164, 166, 299–301
field-type equations 127
finite-aperture data see
limited-aperture data
finite difference (FD) 228, 236, 267
finite element method (FEM) 228,
231, 243
Fisher information matrix (FIM) 79,
265
Foldy–Lax equation 28, 68, 80, 217
forward problem 1, 2
Fourier components/spectrum 148,
163, 164, 185, 259–262,
268–273
Fourier transform (FT) 32, 45,
259–262, 268–273, 299–301
frequency hopping 173
full-aperture data 48, 56, 146, 186
full-wave nonlinear algorithm 6, 8,
139, 229, 277
Index
g
Galerkin FEM 232, 244
Gauss–Newton 128
generalized eigenvalue decomposition
93
Green’s function
dyadic 19, 34, 35, 261
eigen-function expansion 273
inhomogeneous background 34,
35, 139, 229, 245
scalar 19, 22, 48
vector 72, 82, 117
Green’s theorem/formula 45, 106,
107, 230
Gs subspace-based optimization
method (Gs-SOM) 149
h
Hadamard product 211, 221
Heaviside step function 190
Herglotz wave function 117
heuristic optimization algorithm 128
Huygens’ principle 35–39, 236, 244
inverse Fourier transform (IFT) 32,
259–262
inverse source problem 7, 41, 70, 151,
215, 261
invisibility/cloaking 4, 124
Ipswich database 171
l
least-squares 77, 89, 110, 133,
291–294
level set 187
Levenberg–Marquardt 128, 152
limited-aperture data 44, 48, 118,
119
linear sampling method (LSM) 103,
186, 196
Lippmann–Schwinger equation 126,
263, 273
l0 norm 215
l1 norm 215
L2 -norm 105
locally plane wave 3, 33
local shape function (LSF) 186
low-pass filter 164, 260, 270, 271
i
ill-posed problem 3, 71, 125, 130, 185,
208, 215, 281–289
impedance BC 3, 203
impenetrable scatterers 3, 6, 184, 191
incoherence
sampling 214, 216
wave/source 224, 258, 261
induced source 7, 41, 53
monopole/dipole 24–29, 68, 82, 92,
210, 212
multipole 108, 109
surface 21, 23, 183, 184
volumetric 17, 23, 36, 107, 126,
240, 263
injectivity 69–73, 98, 150
Institut Fresnel database 171, 201
interior transmission problem 4, 119,
124
inverse crime 174
m
magnetic current density 17
magnetic current dipole 25
magnetic field integral equation (MFIE)
21–23, 193
Maxwell’s equations 2, 13, 14
medium problem 3
method of moments (MoM) 29, 31,
150, 188
microwave impedance microscopy
(MIM) 249–252
modified gradient method 143, 238
monochromatic wave 3, 14, 45 see
also time-harmonic wave
monopole 27, 28, 46–48, 90, 104
multiple scattering 5–7, 28, 53, 68,
99, 214–216, 240, 273–277
multiple signal classification (MUSIC)
73, 74, 82, 117, 213
305
306
Index
multipole-based linear sampling
method (MLSM) 109
multipole expansion 108, 197
multistatic response (MSR) matrix
53, 69, 212
n
near field 47, 119, 249
Neumann BC 3, 203
Neumann series 274
Newton–Kantorovich (NK) 140–142
noise subspace 74, 91, 155
nondegenerate scatterer 82
noniterative method 6, 77, 88, 129,
214
nonmagnetic 16, 21, 28, 82, 91
nonradiating current 143, 155, 262
NP-hard (non-deterministic
polynomial-time hard) 154,
215
o
obstacle problem 3, 184
optical microscopy 257–261
overdetermined problem 77, 282, 291
p
partial-aperture data see
limited-aperture data
passive source 17, 53 see also induced
source
penetrable scatterers 3, 6, 187
perfect electric conductor (PEC) 3, 16
permeability 16
permittivity 3, 5, 16
phase conjugation mirror (PCM) 42
phaseless data 6, 207
PhaseLift 154
phase retrieval (PR) 208
point-like scatterer 6, 53, 63, 67, 117,
149, 209, 214 see also small
scatterer
point spread function (PSF) 253, 258,
260, 270, 271
polarization strength/tenor 24–28,
30, 80, 210
primary source 16, 72, 109, 230, 277
see also active source
propagation wave see travelling wave
pseudoinverse of matrix 77, 89, 110,
284, 292
q
quadratically constrained quadratic
problem (QCQP) 154
quadratic function 142, 143, 154
qualitative method 6, 103, 186, 196
quantitative method 6, 103, 123, 196
quasi-analytical (QA) approximation
132, 148
quasi-Newton method 128
r
radiation BC 3, 17, 19, 24, 46, 50
radiation operator 129
range of matrix/operator 72–74, 117,
118
rank of matrix 72, 82–84, 117, 212,
220
Rayleigh scattering 24, 67, 80
Rayleigh’s criterion 257, 258
Rayleigh theorem 84
reciprocity 34, 46, 47, 50, 140, 173
refractive index 124, 230, 258
relative permittivity 16
resolution 47–49, 51, 62, 63, 76, 90,
99, 119, 257–277
restricted isometry property (RIP)
215
Rytov approximation (RA) 130
s
scattering potential 263 see also
contrast
scattering strength 68, 221 see also
polarization strength
Index
secondary source 17, 41 see also
induced source
Shannon–Nyquist sampling theorem
214
signal subspace 74, 91, 155
signal-to-noise ratio (SNR) 76, 267,
275
single-layer potential 39
singular value decomposition (SVD)
56, 73, 91, 105, 151, 284
small scatterer 3, 6, 24, 67, 254
see also point-like scatterer
Sommerfeld radiation condition 19,
24, 46 see also radiation BC
sound-hard BC 3
sound-soft BC 3, 184, 185
source-type equation 127
source-type integral equation (STIE)
143, 155
sparsity 71, 214–216, 220
stability estimate 5, 125, 185, 208
stability of solution 4–5, 125, 130,
135, 185, 208, 274, 281–289
state equation 127, 189, 198, 221
stationary phase 33
stochastic optimization algorithm
128
subspace-based optimization method
(SOM) 149
super-resolution 49, 63, 262, 272–277
t
test function in MUSIC 82
testing function 29–31, 150, 230, 244
through-wall-imaging (TWI) 241
Tikhonov regularization 105, 130,
285, 288, 294
time-harmonic wave 3, 14, 25, 74
time-reversal invariance 45
Maxwell equations 51
time-reversal mirror (TRM) 41–43
time-reversal operator (TRO) 54
T-matrix 147, 197, 198
total variation (TV) 174, 247, 288
transmission eigenvalue 4, 119,
124
transverse electric (TE) 22, 192
transverse magnetic (TM) 22, 187
travelling wave 33, 44, 259
truncated SVD 161, 284
two-dimensional scattering problem
22
twofold subspace-based optimization
method (TSOM) 161–164
u
underdetermined problem 215,
216
uniqueness of solution 3, 22, 69–73,
124, 184, 196, 207–209, 281
v
virtual image
42
w
well-posed problem 3, 132, 215
well-resolved targets 54–56
Weyl identity 44, 259, 272
307