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
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 31
2.11 Reciprocity 33
2.12 Huygens? principle and extinction theorem 35
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 42
3.1.3 Fundamental theory 44
3.1.4 Analysis of resolution 48
3.1.5 Vectorial wave 49
3.2 Time reversal imaging for passive sources 51
3.2.1 Imaging by iterative time-reversal process 53
3.2.2 Imaging by the DORT method 54
3.2.3 Numerical simulations 55
3.3 Discussions 61
4 Inverse scattering problems of small scatterers 67
4.1 Forward problem: Foldy-Lax equation 68
4.2 Uniqueness theorem for inverse problem 69
4.2.1 Inverse source problem 70
4.2.2 Inverse scattering problem 71
4.3 Numerical methods 73
4.3.1 Multiple signal classification imaging 73
4.3.2 Noniterative retrieval of scattering strength 76
4.4 Inversion of vector wave equation 78
4.4.1 Forward problem 79
4.4.2 Multiple signal classification imaging 81
4.4.3 Noniterative retrieval of scattering strength tensor 88
4.4.4 Subspace imaging algorithm with enhanced resolution 90
4.5 Discussions 95
5 Linear sampling method 101
5.1 Outline of linear sampling method 102
5.2 Physical interpretation 104
5.2.1 Source distribution 104
5.2.2 Multipole radiation 106
5.3 Multipole-based linear sampling method 107
5.3.1 Description of the algorithm 107
5.3.2 Choice of the number of multipoles 108
5.3.3 Comparison with Tikhonov regularization 110
5.3.4 Numerical examples 112
5.4 Factorization method 113
5.5 Discussions 116
6 Reconstructing dielectric scatterers 119
6.1 Introduction 120
6.1.1 Uniqueness, stability, and nonlinearity 120
6.1.2 Formulation of the forward problem 122
6.1.3 Optimization approach to inverse problem 123
6.2 Noniterative inversion methods 125
6.2.1 Born approximation inversion method 125
6.2.2 Rytov approximation inversion method 126
6.2.3 Extended Born approximation inversion method 127
6.2.4 Back-propagation scheme 128
6.2.5 Numerical examples 129
6.3 Full-wave iterative inversion methods 134
6.3.1 Distorted Born iterative method 134
6.3.2 Contrast source inversion method 137
6.3.3 Contrast source extended Born method 139
6.3.4 Other iterative models 141
6.4 Subspace-based optimization method (SOM) 143
6.4.1 Gs-SOM 144
6.4.2 Twofold SOM 155
6.4.3 New fast Fourier transform SOM 159
6.4.4 SOM for vector wave 162
6.5 Discussions 164
7 Reconstructing perfect electric conductors 175
7.1 Introduction 175
7.1.1 Formulation of the forward problem 175
7.1.2 Uniqueness and stability 176
7.2 Inversion models requiring prior information 177
7.3 Inversion models without prior information 178
7.3.1 Transverse-magnetic case 179
7.3.2 Transverse-electric case 184
7.4 Mixture of PEC and dielectric scatterers 187
7.5 Discussions 193
8 Inversion for phaseless data 201
8.1 Introduction 201
8.2 Reconstructing point-like scatterers by subspace methods 203
8.2.1 Converting a nonlinear problem to a linear one 204
8.2.2 Rank of the multistatic response matrix 206
8.2.3 MUSIC localization and noniterative retrieval 207
8.3 Reconstructing point-like scatterers by compressive sensing 208
8.3.1 Introduction to compressive sensing 208
8.3.2 Solving phase-available inverse problems by CS 209
8.3.3 Solving phaseless inverse problems by CS 210
8.3.4 Applicability of CS 213
8.3.5 Numerical examples 213
8.4 Reconstructing extended dielectric scatterers 214
8.5 Discussions 217
9 Inversion with an inhomogeneous background medium 221
9 Inversion with an inhomogeneous background medium 221 microwave impedance microscopy 242
9.6 Discussions 245
10 Resolution of computational imaging 249
10.1 Diffraction-limited imaging system 249
10.2 Computational imaging 253
10.2.1 Inverse source problem 253
10.2.2 Inverse scattering problem 254
10.3 Cram,r-Rao bound 256
10.4 Resolution under Born approximation 259
10.5 Discussions 263
10.6 Summary 268
A Ill-posed problems and regularization 273
A.1 Ill-posed problems 273
A.2 Regularization theory 274
A.3 Regularization schemes 275
A.3.1 Spectral cutoff 275
A.3.2 Tikhonov regularization 276
A.3.3 Iterative regularization 277
A.4 Regularization parameter selection methods 278
A.4.1 Discrepancy principle 278
A.4.2 Generalized cross validation 279
A.4.3 L-curve method 279
A.4.4 Trial and error 279
A.5 Discussions 280
B Least squares 281
B.1 Geometric interpretation of least squares 281
B.1.1 Real space 281
B.1.2 Complex space 282
B.2 Gradient of squared residuals 282
C Conjugate gradient method 285
C.1 Solving general minimization problems 285
C.1.1 Real space 285
C.1.2 Complex space 286
C.2 Solving linear equation system 286
D Matrix-vector product by FFT procedure 289
D.1 One-dimensional case 289
D.2 Two-dimensional case 291s
