Nonlinear inverse problems in imaging

This book provides researchers and engineers in the imaging field with the skills they need to effectively deal with nonlinear inverse problems associated with different imaging modalities, including impedance imaging, optical tomography, elastography, and electrical source imaging. Focusing on numerically implementable methods, the book bridges the gap between theory and applications, helping readers tackle problems in applied mathematics and engineering. Complete, self-contained coverage includes basic concepts, models, computational methods, numerical simulations, examples, and case studies.

Provides a step-by-step progressive treatment of topics for ease of understanding. Discusses the underlying physical phenomena as well as implementation details of image reconstruction algorithms as prerequisites for finding solutions to non linear inverse problems with practical significance and value. Includes end of chapter problems, case studies and examples with solutions throughout the book. Companion website will provide further examples and solutions, experimental data sets, open problems, teaching material such as PowerPoint slides and software including MATLAB m files.

Essential reading for Graduate students and researchers in imaging science working across the areas of applied mathematics, biomedical engineering, and electrical engineering and specifically those involved in nonlinear imaging techniques, impedance imaging, optical tomography, elastography, and electrical source imaging

Author Notes

Jin Keun Seo, Department of Computational Science and Engineering, Yonsei University, Korea
Professor Seo is currently Chairman of the Computational Science and Engineering Department at Yonsei University, Korea. He has worked on a wide range of interdisciplinary areas including PDE, image processing, bio-impedance, mathematical modelling, harmonic analysis, and so on. He has published over 100 research papers in scientific journals including SIAM Journal on Applied Mathematics , Inverse Problems, IEEE Transactions on Medical Imaging , IEEE Transactions on Biomedical Engineering , Physics in Medicine and Biology , Physiological Measurement , and others. He has received distinguished research awards from the Korean Mathematical Society and Yonsei University. Since 2007, he has been an editor of Journal of Inverse Problems and Imaging .

Eung Je Woo, Department of Biomedical Engineering, Kyung Hee University, Korea
Professor Woo is currently with the Department of Biomedical Engineering, College of Electronics and Information, Kyung Hee University, Korea. Since 2002, he has been the director of Impedance Imaging Research Center (IIRC), Korea. For the past 20 years, he has been teaching undergraduate and graduate courses on medical instrumentation, biomedical computing, and impedance imaging. His research interests include biomedical instrumentation and imaging. From 2004 to 2010, he has been a co-organizer of international conferences on impedance imaging and electrical bioimpedance. In 2006, he served the scientific program chair of the World Congress on Medical Physics and Biomedical Engineering (WC2006). In 2009, he served the theme co-chair of IEEE EMBC09 for biomedical imaging and image processing.

Preface	p. xi
List of Abbreviations	p. xiii
1 Introduction	p. 1
1.1 Forward Problem	p. 1
1.2 Inverse Problem	p. 3
1.3 Issues in Inverse Problem Solving	p. 4
1.4 Linear, Nonlinear and Linearized Problems	p. 6
References	p. 7
2 Signal and System as Vectors	p. 9
2.1 Vector Spaces	p. 9
2.1.1 Vector Space and Subspace	p. 9
2.1.2 Basis, Nonn and Inner Product	p. 11
2.7.3 Hilbert Space	p. 13
2.2 Vector Calculus	p. 16
2.2.1 Gradient	p. 16
2.2.2 Divergence	p. 17
2.2.3 Curl	p. 17
2.2.4 Carve	p. 18
2.2.5 Curvature	p. 19
2.3 Taylor's Expansion	p. 21
2.4 Linear System of Equations	p. 23
2.4.1 Linear System and Transform	p. 23
2.4.2 Vector Space of Matrix	p. 24
2.4.3 Least-Squares Solution	p. 27
2.4.4 Singular Value Decomposition (SVD)	p. 28
2.4.5 Pseudo-inverse	p. 29
2.5 Fourier Transform	p. 30
2.5.1 Series Expansion	p. 30
2.5.2 Fourier Transform	p. 32
2.5.3 Discrete Fourier Transform (DFT)	p. 37
2.5.4 Fast Fourier Transform (FFT)	p. 40
2.5.5 Two-Dimensional Fourier Transform	p. 41
References	p. 42
3 Basics of Forward Problem	p. 43
3.1 Understanding a PDE using Images as Examples	p. 44
3.2 Heat Equation	p. 46
3.2.1 Formulation of Heat Equation	p. 46
3.2.2 One-Dimensional Heat Equation	p. 48
3.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion	p. 50
3.2.4 Boundary Conditions	p. 51
3.3 Wave Equation	p. 52
3.4 Laplace and Poisson Equations	p. 56
3.4.1 Boundary Value Problem	p. 56
3.4.2 Laplace Equation in a Circle	p. 58
3.4.3 Laplace Equation in Three-Dimensional Domain	p. 60
3.4.4 Representation Formula for Poisson Equation	p. 66
References	p. 70
Further Reading	p. 70
4 Analysis for Inverse Problem	p. 71
4.1 Examples of Inverse Problems in Medical Imaging	p. 71
4.1.1 Electrical Property Imaging	p. 71
4.1.2 Mechanical Property Imaging	p. 74
4.1.3 Image Restoration	p. 75
4.2 Basic Analysis	p. 76
4.2.1 Sobolev Space	p. 78
4.2.2 Some Important Estimates	p. 81
4.2.3 Helmholtz Decomposition	p. 87
4.3 Variational Problems	p. 88
4.3.1 Lax-Milgram Theorem	p. 88
4.3.2 Ritz Approach	p. 92
4.3.3 Euler-Lagrange Equations	p. 96
4.3.4 Regularity Theory and Asymptotic Analysis	p. 100
4.4 Tikhonov Regularization and Spectral Analysis	p. 104
4.4.1 Overview of Tikhonov Regularization	p. 105
4.4.2 Bounded Linear Operators in Banach Space	p. 109
4.4.3 Regularization in Hilbert Space or Banach Space	p. 112
4.5 Basics of Real Analysis	p. 116
4.5.1 Riemann Integrability	p. 116
4.5.2 Measure Space	p. 117
4.5.3 Lebesgue-Measurable Function	p. 119
4.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure	p. 123
4.5.5 Differentiation Theory	p. 125
References	p. 127
Further Reading	p. 127
5 Numerical Methods	p. 129
5.1 Iterative Method for Nonlinear Problem	p. 129
5.2 Numerical Computation of One-Dimensional Heat Equation	p. 130
5.2.1 Explicit Scheme	p. 132
5.2.2 Implicit Scheme	p. 135
5.2.3 Crank-Nicolson Method	p. 136
5.3 Numerical Solution of Linear System of Equations	p. 136
5.3.1 Direct Method using LU Factorization	p. 136
5.3.2 Iterative Method using Matrix Splitting	p. 138
5.3.3 Iterative Method using Steepest Descent Minimization	p. 140
5.3.4 Conjugate Gradient (CG) Method	p. 143
5.4 Finite Difference Method (FDM)	p. 145
5.4.1 Poisson Equation	p. 145
5.4.2 Elliptic Equation	p. 146
5.5 Finite Element Method (FEM)	p. 147
5.5.7 One-Dimensional Model	p. 147
5.5.2 Two-Dimensional Model	p. 149
5.5.3 Numerical Examples	p. 154
References	p. 157
Further Reading	p. 158
6 CT, MRI and Image Processing Problems	p. 159
6.1 X-ray Computed Tomography	p. 159
6.1.1 Inverse Problem	p. 160
6.1.2 Basic Principle and Nonlinear Effects	p. 160
6.1.3 Inverse Radon Transform	p. 163
6.1.4 Artifacts in CT	p. 166
6.2 Magnetic Resonance Imaging	p. 167
6.2.1 Basic Principle	p. 167
6.2.2 k-Space Data	p. 168
6.2.3 Image Reconstruction	p. 169
6.3 Image Restoration	p. 171
6.3.1 Role of p in (6.35)	p. 173
6.3.2 Total Variation Restoration	p. 175
6.3.3 Anisotropic Edge-Preserving Diffusion	p. 180
6.3.4 Sparse Sensing	p. 181
6.4 Segmentation	p. 184
6.4.1 Active Contour Method	p. 185
6.4.2 Level Set Method	p. 187
6.4.3 Motion Tracking for Echocardiography	p. 189
References	p. 192
Further Reading	p. 194
7 Electrical Impedance Tomography	p. 195
7.1 Introduction	p. 195
7.2 Measurement Method and Data	p. 196
7.2.1 Conductivity and Resistance	p. 196
7.2.2 Permittivity and Capacitance	p. 197
7.2.3 Phasor and Impedance	p. 198
7.2.4 Admittivity and Trans-Impedance	p. 199
7.2.5 Electrode Contact Impedance	p. 200
7.2.6 EIT System	p. 201
7.2.7 Data Collection Protocol and Data Set	p. 202
7.2.8 Linearity between Current and Voltage	p. 204
7.3 Representation of Physical Phenomena	p. 205
7.3.1 Derivation of Elliptic PDE	p. 205
7.3.2 Elliptic PDE for Four-Electrode Method	p. 206
7.3.3 Elliptic PDE for Two-Electrode Method	p. 209
7.3.4 Min-Max Property of Complex Potential	p. 210
7.4 Forward Problem and Model	p. 210
7.4.1 Continuous Neumann-to-Dirichlet Data	p. 211
7.4.2 Discrete Neumann-to-Dirichlet Data	p. 212
7.4.3 Nonlinearity between Admittivity and Voltage	p. 214
7.5 Uniqueness Theory and Direct Reconstruction Method	p. 216
7.5.1 Calderón's Approach	p. 216
7.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements	p. 218
7.5.3 Nachmann's D-bar Method in Two Dimensions	p. 221
7.6 Back-Projection Algorithm	p. 223
7.7 Sensitivity and Sensitivity Matrix	p. 226
7.7.1 Perturbation and Sensitivity	p. 226
7.7.2 Sensitivity Matrix	p. 227
7.7.3 Linearization	p. 227
7.7.4 Quality of Sensitivity Matrix	p. 229
7.8 Inverse Problem of EIT	p. 229
7.8.1 Inverse Problem of RC Circuit	p. 229
7.8.2 Formulation of EIT Inverse Problem	p. 231
7.8.3 Ill-Posedness of EIT Inverse Problem	p. 231
7.9 Static Imaging	p. 232
7.9.1 Iterative Data Fitting Method	p. 232
7.9.2 Static Imaging using Four-Channel EIT System	p. 233
7.9.3 Regularization	p. 237
7.9.4 Technical Difficulty of Static Imaging	p. 237
7.10 Time-Difference Imaging	p. 239
7.10.1 Data Sets for Time-Difference Imaging	p. 239
7.10.2 Equivalent Homogeneous Admittivity	p. 240
7.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix	p. 241
7.10.4 Interpretation of Time-Difference Image	p. 242
7.11 Frequency-Difference Imaging	p. 243
7.11.1 Data Sets for Frequency-Difference Imaging	p. 243
7.11.2 Simple Difference F t,¿2 - F t,¿1	p. 244
7.11.3 Weighted Difference F t,¿2 - ¿F t,¿1	p. 244
7.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix	p. 245
7.11.5 Interpretation of Frequency-Difference Image	p. 246
References	p. 247
8 Anomaly Estimation and Layer Potential Techniques	p. 251
8.1 Harmonic Analysis and Potential Theory	p. 252
8.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation	p. 252
8.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity	p. 259
8.2 Anomaly Estimation using EIT	p. 266
8.2.1 Size Estimation Method	p. 268
8.2.2 Location Search Method	p. 21A
8.3 Anomaly Estimation using Planar Probe	p. 281
8.3.1 Mathematical Formulation	p. 282
8.3.2 Representation Formula	p. 287
References	p. 290
Further Reading	p. 291
9 Magnetic Resonance Electrical Impedance Tomography	p. 295
9.1 Data Collection using MRI	p. 296
9.1.1 Measurement of B z	p. 297
9.1.2 Noise in Measured B z Data	p. 299
9.1.3 Measurement of B = (B x , B y , B z )	p. 301
9.2 Forward Problem and Model Construction	p. 301
9.2.1 Relation between J, B z and ¿	p. 302
9.2.2 Three Key Observations	p. 303
9.2.3 Data B z Traces ¿∇u ¿e z Directional Change of ¿	p. 304
9.2.4 Mathematical Analysis toward MREIT Model	p. 305
9.3 Inverse Problem Formulation using B or J	p. 308
9.4 Inverse Problem Formulation using B z	p. 309
9.4.1 Model with Two Linearly Independent Currents	p. 309
9.4.2 Uniqueness	p. 310
9.4.3 Defected B z Data in a Local Region	p. 314
9.5 Image Reconstruction Algorithm	p. 315
9.5.1 J-substitution Algorithm	p. 315
9.5.2 Harmonic B z Algorithm	p. 317
9.5.3 Gradient B z Decomposition and Variational B z Algorithm	p. 319
9.5.4 Local Harmonic B z Algorithm	p. 320
9.5.5 Sensitivity Matrix-based Algorithm	p. 322
9.5.6 Anisotropic Conductivity Reconstruction Algorithm	p. 323
9.5.7 Other Algorithms	p. 324
9.6 Validation and Interpretation	p. 325
9.6.1 Image Reconstruction Procedure using Harmonic B z Algorithm	p. 325
9.6.2 Conductivity Phantom Imaging	p. 326
9.6.3 Animal Imaging	p. 327
9.6.4 Human Imaging	p. 330
9.7 Applications	p. 331
References	p. 332
10 Magnetic Resonance Elastography	p. 335
10.1 Representation of Physical Phenomena	p. 336
10.1.1 Overview of Hooke's Law	p. 336
10.1.2 Strain Tensor in Lagrangian Coordinates	p. 339
10.2 Forward Problem and Model	p. 340
10.3 Inverse Problem in MRE	p. 342
10.4 Reconstruction Algorithms	p. 342
10.4.1 Reconstruction of ¿ with the Assumption of Local Homogeneity	p. 344
10.4.2 Reconstruction of ¿ without the Assumption of Local Homogeneity	p. 345
10.4.3 Anisotropic Elastic Moduli Reconstruction	p. 349
10.5 Technical Issues in MRE	p. 350
References	p. 351
Further Reading	p. 352
Index	p. 355

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