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A comprehensive and updated overview of the theory, algorithms and applications of for electromagnetic inverse scattering problems
Xudong Chen, received the B.S. and M.S. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1999 and 2001, respectively, and the Ph.D. degree from the Massachusetts Institute of Technology, Cambridge, MA, USA, in 2005. Since then he joined the Department of Electrical and Computer Engineering, National University of Singapore, Singapore, and he is currently an Associate Professor. His research interests include mainly electromagnetic inverse problems. He has published more than 120 peer-reviewed journal papers on inverse scattering problems, material parameter retrieval, and optical encryption. The total citation of his papers is about 2,500 according to ISI Web of Science till Dec 2015. He visited the University of Paris-SUD 11 in May-June 2010 as an invited visiting Associate Professor. He was the recipient of the Young Scientist Award by the Union Radio-Scientifique Internationale (URSI) in 2010 and Engineering Young Researcher Award by FOE, National University of Singapore in 2015. He is currently an Associate Editor of the IEEE Transactions on Microwave Theory and Techniques.
Foreword xiii
Preface xv
1 Introduction 1
1.1 Introduction to Electromagnetic Inverse Scattering Problems 1
1.2 Forward Scattering Problems 2
1.3 Properties of Inverse Scattering Problems 3
1.4 Scope of the Book 6
References 9
2 Fundamentals of Electromagnetic Wave Theory 13
2.1 Maxwell's Equations 13
2.1.1 Representations in Differential Form 13
2.1.2 Time-Harmonic Forms 14
2.1.3 Boundary Conditions 15
2.1.4 Constitutive Relations 16
2.2 General Description of a Scattering Problem 16
2.3 Duality Principle 18
2.4 Radiation in Free Space 18
2.5 Volume Integral Equations for Dielectric Scatterers 20
2.6 Surface Integral Equations for Perfectly Conducting Scatterers 21
2.7 Two-Dimensional Scattering Problems 22
2.8 Scattering by Small Scatterers 24
2.8.1 Three-Dimensional Case 24
2.8.2 Two-Dimensional Case 27
2.8.3 Scattering by a Collection of Small Scatterers 28
2.8.4 Degrees of Freedom 28
2.9 Scattering by Extended Scatterers 29
2.9.1 Nonmagnetic Dielectric Scatterers 29
2.9.2 Perfectly Electrically Conducting Scatterers 31
2.10 Far-Field Approximation 32
2.11 Reciprocity 34
2.12 Huygens' Principle and Extinction Theorem 35
References 39
3 Time-Reversal Imaging 41
3.1 Time-Reversal Imaging for Active Sources 41
3.1.1 Explanation Based on Geometrical Optics 41
3.1.2 Implementation Steps 43
3.1.3 Fundamental Theory 45
3.1.4 Analysis of Resolution 48
3.1.5 Vectorial Wave 49
3.2 Time-Reversal Imaging for Passive Sources 53
3.2.1 Imaging by an Iterative Time-Reversal Process 54
3.2.2 Imaging by the DORT Method 55
3.2.3 Numerical Simulations 56
3.3 Discussions 62
References 64
4 Inverse Scattering Problems of Small Scatterers 67
4.1 Forward Problem: Foldy-Lax Equation 68
4.2 Uniqueness Theorem for the Inverse Problem 69
4.2.1 Inverse Source Problem 70
4.2.2 Inverse Scattering Problem 71
Locating Positions 72
Retrieving Scattering Strength 72
4.3 Numerical Methods 73
4.3.1 Multiple Signal Classification Imaging 73
4.3.2 Noniterative Retrieval of Scattering Strength 77
4.4 Inversion of a Vector Wave Equation 79
4.4.1 Forward Problem 79
4.4.2 Multiple Signal Classification Imaging 82
Nondegenerate Case 82
Degenerate Case 83
4.4.3 Noniterative Retrieval of Scattering Strength Tensors 88
4.4.4 Subspace Imaging Algorithm with Enhanced Resolution 90
4.5 Discussions 97
References 99
5 Linear Sampling Method 103
5.1 Outline of the Linear Sampling Method 104
5.2 Physical Interpretation 106
5.2.1 Source Distribution 106
5.2.2 Multipole Radiation 108
5.3 Multipole-Based Linear Sampling Method 109
5.3.1 Description of the Algorithm 109
5.3.2 Choice of the Number of Multipoles 110
5.3.3 Comparison with Tikhonov Regularization 113
5.3.4 Numerical Examples 114
5.4 Factorization Method 116
5.5 Discussions 118
References 119
6 Reconstructing Dielectric Scatterers 123
6.1 Introduction 124
6.1.1 Uniqueness, Stability, and Nonlinearity 124
6.1.2 Formulation of the Forward Problem 126
6.1.3 Optimization Approach to the Inverse Problem 127
6.2 Noniterative Inversion Methods 129
6.2.1 Born Approximation Inversion Method 130
6.2.2 Rytov Approximation Inversion Method 130
6.2.3 Extended Born Approximation Inversion Method 131
6.2.4 Back-Propagation Scheme 133
6.2.5 Numerical Examples 134
6.3 Full-Wave Iterative Inversion Methods 139
6.3.1 Distorted Born Iterative Method 139
6.3.2 Contrast Source Inversion Method 142
6.3.3 Contrast Source Extended Born Method 144
6.3.4 Other Iterative Models 146
6.4 Subspace-Based Optimization Method (SOM) 149
6.4.1 Gs-SOM 149
6.4.2 Twofold SOM 161
6.4.3 New Fast Fourier Transform SOM 164
6.4.4 SOM for the Vector Wave 169
6.5 Discussions 171
References 174
7 Reconstructing Perfect Electric Conductors 183
7.1 Introduction 183
7.1.1 Formulation of the Forward Problem 183
7.1.2 Uniqueness and Stability 184
7.2 Inversion Models Requiring Prior Information 185
7.3 Inversion Models Without Prior Information 186
7.3.1 Transverse-Magnetic Case 187
7.3.2 Transverse-Electric Case 192
7.4 Mixture of PEC and Dielectric Scatterers 196
7.5 Discussions 202
References 203
8 Inversion for Phaseless Data 207
8.1 Introduction 207
8.2 Reconstructing Point-Like Scatterers by Subspace Methods 209
8.2.1 Converting a Nonlinear Problem to a Linear One 210
8.2.2 Rank of the Multistatic Response Matrix 212
8.2.3 MUSIC Localization and Noniterative Retrieval 213
8.3 Reconstructing Point-Like Scatterers by Compressive Sensing 214
8.3.1 Introduction to Compressive Sensing 214
8.3.2 Solving Phase-Available Inverse Problems by CS 215
8.3.3 Solving Phaseless Inverse Problems by CS 216
8.3.4 Applicability of CS 218
8.3.5 Numerical Examples 219
8.4 Reconstructing Extended Dielectric Scatterers 220
8.5 Discussions 223
References 224
9 Inversion with an Inhomogeneous Background Medium 227
9.1 Introduction 227
9.2 Integral Equation Approach via Numerical Green's Function 229
9.3 Differential Equation Approach 235
9.4 Homogeneous Background Approach 240
9.5 Examples of Three-Dimensional Problems 243
9.5.1 Confocal Laser Scanning Microscope 246
9.5.2 Near-Field Scanning Microwave Impedance Microscopy 249
9.6 Discussions 252
References 254
10 Resolution of Computational Imaging 257
10.1 Diffraction-Limited Imaging System 257
10.2 Computational Imaging 261
10.2.1 Inverse Source Problem 261
10.2.2 Inverse Scattering Problem 262
10.3 Cramér-Rao Bound 264
10.4 Resolution under the Born Approximation 268
10.5 Discussions 272
10.6 Summary 277
References 278
Appendices A Ill-Posed Problems and Regularization 281
A. 1 Ill-Posed Problems 281
A. 2 Regularization Theory 282
A. 3 Regularization Schemes 283
A. 4 Regularization Parameter Selection Methods 286
A. 5 Discussions 288
B Least Squares 291
B.1 Geometric Interpretation of Least Squares 291
B.2 Gradient of Squared Residuals 292
C conjugate Gradient Method 295
C.1 Solving General Minimization Problems 295
C.2 Solving Linear Equation Systems 296
D Matrix-Vector Product by the FFT Procedure 299
D. 1 One-Dimensional Case 299
D. 2 Two-Dimensional Case 300
Appendix References 301
Index 303
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