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

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InhaltsangabePreface 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 LeastSquares Solution 27 2.4.4 Singular Value Decomposition (SVD) 28 2.4.5 Pseudoinverse 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 TwoDimensional 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 OneDimensional Heat Equation 48 3.2.3 TwoDimensional 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 LaxMilgram Theorem 88 4.3.2 Ritz Approach 92 4.3.3 EulerLagrange 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 CrankNicolson 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 OneDimensional Model 147 5.5.2 TwoDimensional Model 149 5.5.3 Numerical Examples 154 References 157 Further Reading 158 6 CT, MRI and Image Processing Problems 159 6.1 Xray 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 kSpace 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

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