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

Table of Contents:
Preface xi List of Abbreviations xiii 1 Introduction 1 1.1 Forward Problem 1 1.2 Inverse Problem 3 1.3 Issues in Inverse Problem Solving 4 1.4 Linear, Nonlinear and Linearized Problems 6 References 7 2 Signal and System as Vectors 9 2.1 Vector Spaces 9 2.1.1 Vector Space and Subspace 9 2.1.2 Basis, Norm and Inner Product 11 2.1.3 Hilbert Space 13 2.2 Vector Calculus 16 2.2.1 Gradient 16 2.2.2 Divergence 17 2.2.3 Curl 17 2.2.4 Curve 18 2.2.5 Curvature 19 2.3 Taylor’s Expansion 21 2.4 Linear System of Equations 23 2.4.1 Linear System and Transform 23 2.4.2 Vector Space of Matrix 24 2.4.3 Least-Squares Solution 27 2.4.4 Singular Value Decomposition (SVD) 28 2.4.5 Pseudo-inverse 29 2.5 Fourier Transform 30 2.5.1 Series Expansion 30 2.5.2 Fourier Transform 32 2.5.3 Discrete Fourier Transform (DFT) 37 2.5.4 Fast Fourier Transform (FFT) 40 2.5.5 Two-Dimensional Fourier Transform 41 References 42 3 Basics of Forward Problem 43 3.1 Understanding a PDE using Images as Examples 44 3.2 Heat Equation 46 3.2.1 Formulation of Heat Equation 46 3.2.2 One-Dimensional Heat Equation 48 3.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion 50 3.2.4 Boundary Conditions 51 3.3 Wave Equation 52 3.4 Laplace and Poisson Equations 56 3.4.1 Boundary Value Problem 56 3.4.2 Laplace Equation in a Circle 58 3.4.3 Laplace Equation in Three-Dimensional Domain 60 3.4.4 Representation Formula for Poisson Equation 66 References 70 Further Reading 70 4 Analysis for Inverse Problem 71 4.1 Examples of Inverse Problems in Medical Imaging 71 4.1.1 Electrical Property Imaging 71 4.1.2 Mechanical Property Imaging 74 4.1.3 Image Restoration 75 4.2 Basic Analysis 76 4.2.1 Sobolev Space 78 4.2.2 Some Important Estimates 81 4.2.3 Helmholtz Decomposition 87 4.3 Variational Problems 88 4.3.1 Lax–Milgram Theorem 88 4.3.2 Ritz Approach 92 4.3.3 Euler–Lagrange Equations 96 4.3.4 Regularity Theory and Asymptotic Analysis 100 4.4 Tikhonov Regularization and Spectral Analysis 104 4.4.1 Overview of Tikhonov Regularization 105 4.4.2 Bounded Linear Operators in Banach Space 109 4.4.3 Regularization in Hilbert Space or Banach Space 112 4.5 Basics of Real Analysis 116 4.5.1 Riemann Integrability 116 4.5.2 Measure Space 117 4.5.3 Lebesgue-Measurable Function 119 4.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 123 4.5.5 Differentiation Theory 125 References 127 Further Reading 127 5 Numerical Methods 129 5.1 Iterative Method for Nonlinear Problem 129 5.2 Numerical Computation of One-Dimensional Heat Equation 130 5.2.1 Explicit Scheme 132 5.2.2 Implicit Scheme 135 5.2.3 Crank–Nicolson Method 136 5.3 Numerical Solution of Linear System of Equations 136 5.3.1 Direct Method using LU Factorization 136 5.3.2 Iterative Method using Matrix Splitting 138 5.3.3 Iterative Method using Steepest Descent Minimization 140 5.3.4 Conjugate Gradient (CG) Method 143 5.4 Finite Difference Method (FDM) 145 5.4.1 Poisson Equation 145 5.4.2 Elliptic Equation 146 5.5 Finite Element Method (FEM) 147 5.5.1 One-Dimensional Model 147 5.5.2 Two-Dimensional Model 149 5.5.3 Numerical Examples 154 References 157 Further Reading 158 6 CT, MRI and Image Processing Problems 159 6.1 X-ray Computed Tomography 159 6.1.1 Inverse Problem 160 6.1.2 Basic Principle and Nonlinear Effects 160 6.1.3 Inverse Radon Transform 163 6.1.4 Artifacts in CT 166 6.2 Magnetic Resonance Imaging 167 6.2.1 Basic Principle 167 6.2.2 k-Space Data 168 6.2.3 Image Reconstruction 169 6.3 Image Restoration 171 6.3.1 Role of p in (6.35) 173 6.3.2 Total Variation Restoration 175 6.3.3 Anisotropic Edge-Preserving Diffusion 180 6.3.4 Sparse Sensing 181 6.4 Segmentation 184 6.4.1 Active Contour Method 185 6.4.2 Level Set Method 187 6.4.3 Motion Tracking for Echocardiography 189 References 192 Further Reading 194 7 Electrical Impedance Tomography 195 7.1 Introduction 195 7.2 Measurement Method and Data 196 7.2.1 Conductivity and Resistance 196 7.2.2 Permittivity and Capacitance 197 7.2.3 Phasor and Impedance 198 7.2.4 Admittivity and Trans-Impedance 199 7.2.5 Electrode Contact Impedance 200 7.2.6 EIT System 201 7.2.7 Data Collection Protocol and Data Set 202 7.2.8 Linearity between Current and Voltage 204 7.3 Representation of Physical Phenomena 205 7.3.1 Derivation of Elliptic PDE 205 7.3.2 Elliptic PDE for Four-Electrode Method 206 7.3.3 Elliptic PDE for Two-Electrode Method 209 7.3.4 Min–Max Property of Complex Potential 210 7.4 Forward Problem and Model 210 7.4.1 Continuous Neumann-to-Dirichlet Data 211 7.4.2 Discrete Neumann-to-Dirichlet Data 212 7.4.3 Nonlinearity between Admittivity and Voltage 214 7.5 Uniqueness Theory and Direct Reconstruction Method 216 7.5.1 Calder´on’s Approach 216 7.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements 218 7.5.3 Nachmann’s D-bar Method in Two Dimensions 221 7.6 Back-Projection Algorithm 223 7.7 Sensitivity and Sensitivity Matrix 226 7.7.1 Perturbation and Sensitivity 226 7.7.2 Sensitivity Matrix 227 7.7.3 Linearization 227 7.7.4 Quality of Sensitivity Matrix 229 7.8 Inverse Problem of EIT 229 7.8.1 Inverse Problem of RC Circuit 229 7.8.2 Formulation of EIT Inverse Problem 231 7.8.3 Ill-Posedness of EIT Inverse Problem 231 7.9 Static Imaging 232 7.9.1 Iterative Data Fitting Method 232 7.9.2 Static Imaging using Four-Channel EIT System 233 7.9.3 Regularization 237 7.9.4 Technical Difficulty of Static Imaging 237 7.10 Time-Difference Imaging 239 7.10.1 Data Sets for Time-Difference Imaging 239 7.10.2 Equivalent Homogeneous Admittivity 240 7.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix 241 7.10.4 Interpretation of Time-Difference Image 242 7.11 Frequency-Difference Imaging 243 7.11.1 Data Sets for Frequency-Difference Imaging 243 7.11.2 Simple Difference Ft,ω2 − Ft,ω1 244 7.11.3 Weighted Difference Ft,ω2 − αFt,ω1 244 7.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix 245 7.11.5 Interpretation of Frequency-Difference Image 246 References 247 8 Anomaly Estimation and Layer Potential Techniques 251 8.1 Harmonic Analysis and Potential Theory 252 8.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation 252 8.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity 259 8.2 Anomaly Estimation using EIT 266 8.2.1 Size Estimation Method 268 8.2.2 Location Search Method 274 8.3 Anomaly Estimation using Planar Probe 281 8.3.1 Mathematical Formulation 282 8.3.2 Representation Formula 287 References 290 Further Reading 291 9 Magnetic Resonance Electrical Impedance Tomography 295 9.1 Data Collection using MRI 296 9.1.1 Measurement of Bz 297 9.1.2 Noise in Measured Bz Data 299 9.1.3 Measurement of B = (Bx,By,Bz) 301 9.2 Forward Problem and Model Construction 301 9.2.1 Relation between J, Bz and σ 302 9.2.2 Three Key Observations 303 9.2.3 Data Bz Traces σ∇u × ez Directional Change of σ 304 9.2.4 Mathematical Analysis toward MREIT Model 305 9.3 Inverse Problem Formulation using B or J 308 9.4 Inverse Problem Formulation using Bz 309 9.4.1 Model with Two Linearly Independent Currents 309 9.4.2 Uniqueness 310 9.4.3 Defected Bz Data in a Local Region 314 9.5 Image Reconstruction Algorithm 315 9.5.1 J -substitution Algorithm 315 9.5.2 Harmonic Bz Algorithm 317 9.5.3 Gradient Bz Decomposition and Variational Bz Algorithm 319 9.5.4 Local Harmonic Bz Algorithm 320 9.5.5 Sensitivity Matrix-based Algorithm 322 9.5.6 Anisotropic Conductivity Reconstruction Algorithm 323 9.5.7 Other Algorithms 324 9.6 Validation and Interpretation 325 9.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm 325 9.6.2 Conductivity Phantom Imaging 326 9.6.3 Animal Imaging 327 9.6.4 Human Imaging 330 9.7 Applications 331 References 332 10 Magnetic Resonance Elastography 335 10.1 Representation of Physical Phenomena 336 10.1.1 Overview of Hooke’s Law 336 10.1.2 Strain Tensor in Lagrangian Coordinates 339 10.2 Forward Problem and Model 340 10.3 Inverse Problem in MRE 342 10.4 Reconstruction Algorithms 342 10.4.1 Reconstruction of μ with the Assumption of Local Homogeneity 344 10.4.2 Reconstruction of μ without the Assumption of Local Homogeneity 345 10.4.3 Anisotropic Elastic Moduli Reconstruction 349 10.5 Technical Issues in MRE 350 References 351 Further Reading 352 Index 355