Jin Keun Seo, Department of Computational Science and Engineering, Yonsei University, KoreaProfessor 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, KoreaProfessor 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 xiList of Abbreviations xiii1 Introduction 11.1 Forward Problem 11.2 Inverse Problem 31.3 Issues in Inverse Problem Solving 41.4 Linear, Nonlinear and Linearized Problems 6References 72 Signal and System as Vectors 92.1 Vector Spaces 92.1.1 Vector Space and Subspace 92.1.2 Basis, Norm and Inner Product 112.1.3 Hilbert Space 132.2 Vector Calculus 162.2.1 Gradient 162.2.2 Divergence 172.2.3 Curl 172.2.4 Curve 182.2.5 Curvature 192.3 Taylor’s Expansion 212.4 Linear System of Equations 232.4.1 Linear System and Transform 232.4.2 Vector Space of Matrix 242.4.3 Least-Squares Solution 272.4.4 Singular Value Decomposition (SVD) 282.4.5 Pseudo-inverse 292.5 Fourier Transform 302.5.1 Series Expansion 302.5.2 Fourier Transform 322.5.3 Discrete Fourier Transform (DFT) 372.5.4 Fast Fourier Transform (FFT) 402.5.5 Two-Dimensional Fourier Transform 41References 423 Basics of Forward Problem 433.1 Understanding a PDE using Images as Examples 443.2 Heat Equation 463.2.1 Formulation of Heat Equation 463.2.2 One-Dimensional Heat Equation 483.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion 503.2.4 Boundary Conditions 513.3 Wave Equation 523.4 Laplace and Poisson Equations 563.4.1 Boundary Value Problem 563.4.2 Laplace Equation in a Circle 583.4.3 Laplace Equation in Three-Dimensional Domain 603.4.4 Representation Formula for Poisson Equation 66References 70Further Reading 704 Analysis for Inverse Problem 714.1 Examples of Inverse Problems in Medical Imaging 714.1.1 Electrical Property Imaging 714.1.2 Mechanical Property Imaging 744.1.3 Image Restoration 754.2 Basic Analysis 764.2.1 Sobolev Space 784.2.2 Some Important Estimates 814.2.3 Helmholtz Decomposition 874.3 Variational Problems 884.3.1 Lax–Milgram Theorem 884.3.2 Ritz Approach 924.3.3 Euler–Lagrange Equations 964.3.4 Regularity Theory and Asymptotic Analysis 1004.4 Tikhonov Regularization and Spectral Analysis 1044.4.1 Overview of Tikhonov Regularization 1054.4.2 Bounded Linear Operators in Banach Space 1094.4.3 Regularization in Hilbert Space or Banach Space 1124.5 Basics of Real Analysis 1164.5.1 Riemann Integrability 1164.5.2 Measure Space 1174.5.3 Lebesgue-Measurable Function 1194.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 1234.5.5 Differentiation Theory 125References 127Further Reading 1275 Numerical Methods 1295.1 Iterative Method for Nonlinear Problem 1295.2 Numerical Computation of One-Dimensional Heat Equation 1305.2.1 Explicit Scheme 1325.2.2 Implicit Scheme 1355.2.3 Crank–Nicolson Method 1365.3 Numerical Solution of Linear System of Equations 1365.3.1 Direct Method using LU Factorization 1365.3.2 Iterative Method using Matrix Splitting 1385.3.3 Iterative Method using Steepest Descent Minimization 1405.3.4 Conjugate Gradient (CG) Method 1435.4 Finite Difference Method (FDM) 1455.4.1 Poisson Equation 1455.4.2 Elliptic Equation 1465.5 Finite Element Method (FEM) 1475.5.1 One-Dimensional Model 1475.5.2 Two-Dimensional Model 1495.5.3 Numerical Examples 154References 157Further Reading 1586 CT, MRI and Image Processing Problems 1596.1 X-ray Computed Tomography 1596.1.1 Inverse Problem 1606.1.2 Basic Principle and Nonlinear Effects 1606.1.3 Inverse Radon Transform 1636.1.4 Artifacts in CT 1666.2 Magnetic Resonance Imaging 1676.2.1 Basic Principle 1676.2.2 k-Space Data 1686.2.3 Image Reconstruction 1696.3 Image Restoration 1716.3.1 Role of p in (6.35) 1736.3.2 Total Variation Restoration 1756.3.3 Anisotropic Edge-Preserving Diffusion 1806.3.4 Sparse Sensing 1816.4 Segmentation 1846.4.1 Active Contour Method 1856.4.2 Level Set Method 1876.4.3 Motion Tracking for Echocardiography 189References 192Further Reading 1947 Electrical Impedance Tomography 1957.1 Introduction 1957.2 Measurement Method and Data 1967.2.1 Conductivity and Resistance 1967.2.2 Permittivity and Capacitance 1977.2.3 Phasor and Impedance 1987.2.4 Admittivity and Trans-Impedance 1997.2.5 Electrode Contact Impedance 2007.2.6 EIT System 2017.2.7 Data Collection Protocol and Data Set 2027.2.8 Linearity between Current and Voltage 2047.3 Representation of Physical Phenomena 2057.3.1 Derivation of Elliptic PDE 2057.3.2 Elliptic PDE for Four-Electrode Method 2067.3.3 Elliptic PDE for Two-Electrode Method 2097.3.4 Min–Max Property of Complex Potential 2107.4 Forward Problem and Model 2107.4.1 Continuous Neumann-to-Dirichlet Data 2117.4.2 Discrete Neumann-to-Dirichlet Data 2127.4.3 Nonlinearity between Admittivity and Voltage 2147.5 Uniqueness Theory and Direct Reconstruction Method 2167.5.1 Calder´on’s Approach 2167.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements 2187.5.3 Nachmann’s D-bar Method in Two Dimensions 2217.6 Back-Projection Algorithm 2237.7 Sensitivity and Sensitivity Matrix 2267.7.1 Perturbation and Sensitivity 2267.7.2 Sensitivity Matrix 2277.7.3 Linearization 2277.7.4 Quality of Sensitivity Matrix 2297.8 Inverse Problem of EIT 2297.8.1 Inverse Problem of RC Circuit 2297.8.2 Formulation of EIT Inverse Problem 2317.8.3 Ill-Posedness of EIT Inverse Problem 2317.9 Static Imaging 2327.9.1 Iterative Data Fitting Method 2327.9.2 Static Imaging using Four-Channel EIT System 2337.9.3 Regularization 2377.9.4 Technical Difficulty of Static Imaging 2377.10 Time-Difference Imaging 2397.10.1 Data Sets for Time-Difference Imaging 2397.10.2 Equivalent Homogeneous Admittivity 2407.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix 2417.10.4 Interpretation of Time-Difference Image 2427.11 Frequency-Difference Imaging 2437.11.1 Data Sets for Frequency-Difference Imaging 2437.11.2 Simple Difference Ft,ω2 − Ft,ω1 2447.11.3 Weighted Difference Ft,ω2 − αFt,ω1 2447.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix 2457.11.5 Interpretation of Frequency-Difference Image 246References 2478 Anomaly Estimation and Layer Potential Techniques 2518.1 Harmonic Analysis and Potential Theory 2528.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation 2528.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity 2598.2 Anomaly Estimation using EIT 2668.2.1 Size Estimation Method 2688.2.2 Location Search Method 2748.3 Anomaly Estimation using Planar Probe 2818.3.1 Mathematical Formulation 2828.3.2 Representation Formula 287References 290Further Reading 2919 Magnetic Resonance Electrical Impedance Tomography 2959.1 Data Collection using MRI 2969.1.1 Measurement of Bz 2979.1.2 Noise in Measured Bz Data 2999.1.3 Measurement of B = (Bx,By,Bz) 3019.2 Forward Problem and Model Construction 3019.2.1 Relation between J, Bz and σ 3029.2.2 Three Key Observations 3039.2.3 Data Bz Traces σ∇u × ez Directional Change of σ 3049.2.4 Mathematical Analysis toward MREIT Model 3059.3 Inverse Problem Formulation using B or J 3089.4 Inverse Problem Formulation using Bz 3099.4.1 Model with Two Linearly Independent Currents 3099.4.2 Uniqueness 3109.4.3 Defected Bz Data in a Local Region 3149.5 Image Reconstruction Algorithm 3159.5.1 J -substitution Algorithm 3159.5.2 Harmonic Bz Algorithm 3179.5.3 Gradient Bz Decomposition and Variational Bz Algorithm 3199.5.4 Local Harmonic Bz Algorithm 3209.5.5 Sensitivity Matrix-based Algorithm 3229.5.6 Anisotropic Conductivity Reconstruction Algorithm 3239.5.7 Other Algorithms 3249.6 Validation and Interpretation 3259.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm 3259.6.2 Conductivity Phantom Imaging 3269.6.3 Animal Imaging 3279.6.4 Human Imaging 3309.7 Applications 331References 33210 Magnetic Resonance Elastography 33510.1 Representation of Physical Phenomena 33610.1.1 Overview of Hooke’s Law 33610.1.2 Strain Tensor in Lagrangian Coordinates 33910.2 Forward Problem and Model 34010.3 Inverse Problem in MRE 34210.4 Reconstruction Algorithms 34210.4.1 Reconstruction of μ with the Assumption of Local Homogeneity 34410.4.2 Reconstruction of μ without the Assumption of Local Homogeneity 34510.4.3 Anisotropic Elastic Moduli Reconstruction 34910.5 Technical Issues in MRE 350References 351Further Reading 352Index 355

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Jin Keun Seo, Department of Computational Science and Engineering, Yonsei University, KoreaProfessor 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, KoreaProfessor 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 xiList of Abbreviations xiii1 Introduction 11.1 Forward Problem 11.2 Inverse Problem 31.3 Issues in Inverse Problem Solving 41.4 Linear, Nonlinear and Linearized Problems 6References 72 Signal and System as Vectors 92.1 Vector Spaces 92.1.1 Vector Space and Subspace 92.1.2 Basis, Norm and Inner Product 112.1.3 Hilbert Space 132.2 Vector Calculus 162.2.1 Gradient 162.2.2 Divergence 172.2.3 Curl 172.2.4 Curve 182.2.5 Curvature 192.3 Taylor’s Expansion 212.4 Linear System of Equations 232.4.1 Linear System and Transform 232.4.2 Vector Space of Matrix 242.4.3 Least-Squares Solution 272.4.4 Singular Value Decomposition (SVD) 282.4.5 Pseudo-inverse 292.5 Fourier Transform 302.5.1 Series Expansion 302.5.2 Fourier Transform 322.5.3 Discrete Fourier Transform (DFT) 372.5.4 Fast Fourier Transform (FFT) 402.5.5 Two-Dimensional Fourier Transform 41References 423 Basics of Forward Problem 433.1 Understanding a PDE using Images as Examples 443.2 Heat Equation 463.2.1 Formulation of Heat Equation 463.2.2 One-Dimensional Heat Equation 483.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion 503.2.4 Boundary Conditions 513.3 Wave Equation 523.4 Laplace and Poisson Equations 563.4.1 Boundary Value Problem 563.4.2 Laplace Equation in a Circle 583.4.3 Laplace Equation in Three-Dimensional Domain 603.4.4 Representation Formula for Poisson Equation 66References 70Further Reading 704 Analysis for Inverse Problem 714.1 Examples of Inverse Problems in Medical Imaging 714.1.1 Electrical Property Imaging 714.1.2 Mechanical Property Imaging 744.1.3 Image Restoration 754.2 Basic Analysis 764.2.1 Sobolev Space 784.2.2 Some Important Estimates 814.2.3 Helmholtz Decomposition 874.3 Variational Problems 884.3.1 Lax–Milgram Theorem 884.3.2 Ritz Approach 924.3.3 Euler–Lagrange Equations 964.3.4 Regularity Theory and Asymptotic Analysis 1004.4 Tikhonov Regularization and Spectral Analysis 1044.4.1 Overview of Tikhonov Regularization 1054.4.2 Bounded Linear Operators in Banach Space 1094.4.3 Regularization in Hilbert Space or Banach Space 1124.5 Basics of Real Analysis 1164.5.1 Riemann Integrability 1164.5.2 Measure Space 1174.5.3 Lebesgue-Measurable Function 1194.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 1234.5.5 Differentiation Theory 125References 127Further Reading 1275 Numerical Methods 1295.1 Iterative Method for Nonlinear Problem 1295.2 Numerical Computation of One-Dimensional Heat Equation 1305.2.1 Explicit Scheme 1325.2.2 Implicit Scheme 1355.2.3 Crank–Nicolson Method 1365.3 Numerical Solution of Linear System of Equations 1365.3.1 Direct Method using LU Factorization 1365.3.2 Iterative Method using Matrix Splitting 1385.3.3 Iterative Method using Steepest Descent Minimization 1405.3.4 Conjugate Gradient (CG) Method 1435.4 Finite Difference Method (FDM) 1455.4.1 Poisson Equation 1455.4.2 Elliptic Equation 1465.5 Finite Element Method (FEM) 1475.5.1 One-Dimensional Model 1475.5.2 Two-Dimensional Model 1495.5.3 Numerical Examples 154References 157Further Reading 1586 CT, MRI and Image Processing Problems 1596.1 X-ray Computed Tomography 1596.1.1 Inverse Problem 1606.1.2 Basic Principle and Nonlinear Effects 1606.1.3 Inverse Radon Transform 1636.1.4 Artifacts in CT 1666.2 Magnetic Resonance Imaging 1676.2.1 Basic Principle 1676.2.2 k-Space Data 1686.2.3 Image Reconstruction 1696.3 Image Restoration 1716.3.1 Role of p in (6.35) 1736.3.2 Total Variation Restoration 1756.3.3 Anisotropic Edge-Preserving Diffusion 1806.3.4 Sparse Sensing 1816.4 Segmentation 1846.4.1 Active Contour Method 1856.4.2 Level Set Method 1876.4.3 Motion Tracking for Echocardiography 189References 192Further Reading 1947 Electrical Impedance Tomography 1957.1 Introduction 1957.2 Measurement Method and Data 1967.2.1 Conductivity and Resistance 1967.2.2 Permittivity and Capacitance 1977.2.3 Phasor and Impedance 1987.2.4 Admittivity and Trans-Impedance 1997.2.5 Electrode Contact Impedance 2007.2.6 EIT System 2017.2.7 Data Collection Protocol and Data Set 2027.2.8 Linearity between Current and Voltage 2047.3 Representation of Physical Phenomena 2057.3.1 Derivation of Elliptic PDE 2057.3.2 Elliptic PDE for Four-Electrode Method 2067.3.3 Elliptic PDE for Two-Electrode Method 2097.3.4 Min–Max Property of Complex Potential 2107.4 Forward Problem and Model 2107.4.1 Continuous Neumann-to-Dirichlet Data 2117.4.2 Discrete Neumann-to-Dirichlet Data 2127.4.3 Nonlinearity between Admittivity and Voltage 2147.5 Uniqueness Theory and Direct Reconstruction Method 2167.5.1 Calder´on’s Approach 2167.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements 2187.5.3 Nachmann’s D-bar Method in Two Dimensions 2217.6 Back-Projection Algorithm 2237.7 Sensitivity and Sensitivity Matrix 2267.7.1 Perturbation and Sensitivity 2267.7.2 Sensitivity Matrix 2277.7.3 Linearization 2277.7.4 Quality of Sensitivity Matrix 2297.8 Inverse Problem of EIT 2297.8.1 Inverse Problem of RC Circuit 2297.8.2 Formulation of EIT Inverse Problem 2317.8.3 Ill-Posedness of EIT Inverse Problem 2317.9 Static Imaging 2327.9.1 Iterative Data Fitting Method 2327.9.2 Static Imaging using Four-Channel EIT System 2337.9.3 Regularization 2377.9.4 Technical Difficulty of Static Imaging 2377.10 Time-Difference Imaging 2397.10.1 Data Sets for Time-Difference Imaging 2397.10.2 Equivalent Homogeneous Admittivity 2407.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix 2417.10.4 Interpretation of Time-Difference Image 2427.11 Frequency-Difference Imaging 2437.11.1 Data Sets for Frequency-Difference Imaging 2437.11.2 Simple Difference Ft,ω2 − Ft,ω1 2447.11.3 Weighted Difference Ft,ω2 − αFt,ω1 2447.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix 2457.11.5 Interpretation of Frequency-Difference Image 246References 2478 Anomaly Estimation and Layer Potential Techniques 2518.1 Harmonic Analysis and Potential Theory 2528.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation 2528.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity 2598.2 Anomaly Estimation using EIT 2668.2.1 Size Estimation Method 2688.2.2 Location Search Method 2748.3 Anomaly Estimation using Planar Probe 2818.3.1 Mathematical Formulation 2828.3.2 Representation Formula 287References 290Further Reading 2919 Magnetic Resonance Electrical Impedance Tomography 2959.1 Data Collection using MRI 2969.1.1 Measurement of Bz 2979.1.2 Noise in Measured Bz Data 2999.1.3 Measurement of B = (Bx,By,Bz) 3019.2 Forward Problem and Model Construction 3019.2.1 Relation between J, Bz and σ 3029.2.2 Three Key Observations 3039.2.3 Data Bz Traces σ∇u × ez Directional Change of σ 3049.2.4 Mathematical Analysis toward MREIT Model 3059.3 Inverse Problem Formulation using B or J 3089.4 Inverse Problem Formulation using Bz 3099.4.1 Model with Two Linearly Independent Currents 3099.4.2 Uniqueness 3109.4.3 Defected Bz Data in a Local Region 3149.5 Image Reconstruction Algorithm 3159.5.1 J -substitution Algorithm 3159.5.2 Harmonic Bz Algorithm 3179.5.3 Gradient Bz Decomposition and Variational Bz Algorithm 3199.5.4 Local Harmonic Bz Algorithm 3209.5.5 Sensitivity Matrix-based Algorithm 3229.5.6 Anisotropic Conductivity Reconstruction Algorithm 3239.5.7 Other Algorithms 3249.6 Validation and Interpretation 3259.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm 3259.6.2 Conductivity Phantom Imaging 3269.6.3 Animal Imaging 3279.6.4 Human Imaging 3309.7 Applications 331References 33210 Magnetic Resonance Elastography 33510.1 Representation of Physical Phenomena 33610.1.1 Overview of Hooke’s Law 33610.1.2 Strain Tensor in Lagrangian Coordinates 33910.2 Forward Problem and Model 34010.3 Inverse Problem in MRE 34210.4 Reconstruction Algorithms 34210.4.1 Reconstruction of μ with the Assumption of Local Homogeneity 34410.4.2 Reconstruction of μ without the Assumption of Local Homogeneity 34510.4.3 Anisotropic Elastic Moduli Reconstruction 34910.5 Technical Issues in MRE 350References 351Further Reading 352Index 355

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Jin Keun Seo, Department of Computational Science and Engineering, Yonsei University, KoreaProfessor 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, KoreaProfessor 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 xiList of Abbreviations xiii1 Introduction 11.1 Forward Problem 11.2 Inverse Problem 31.3 Issues in Inverse Problem Solving 41.4 Linear, Nonlinear and Linearized Problems 6References 72 Signal and System as Vectors 92.1 Vector Spaces 92.1.1 Vector Space and Subspace 92.1.2 Basis, Norm and Inner Product 112.1.3 Hilbert Space 132.2 Vector Calculus 162.2.1 Gradient 162.2.2 Divergence 172.2.3 Curl 172.2.4 Curve 182.2.5 Curvature 192.3 Taylor’s Expansion 212.4 Linear System of Equations 232.4.1 Linear System and Transform 232.4.2 Vector Space of Matrix 242.4.3 Least-Squares Solution 272.4.4 Singular Value Decomposition (SVD) 282.4.5 Pseudo-inverse 292.5 Fourier Transform 302.5.1 Series Expansion 302.5.2 Fourier Transform 322.5.3 Discrete Fourier Transform (DFT) 372.5.4 Fast Fourier Transform (FFT) 402.5.5 Two-Dimensional Fourier Transform 41References 423 Basics of Forward Problem 433.1 Understanding a PDE using Images as Examples 443.2 Heat Equation 463.2.1 Formulation of Heat Equation 463.2.2 One-Dimensional Heat Equation 483.2.3 Two-Dimensional Heat Equation and Isotropic Diffusion 503.2.4 Boundary Conditions 513.3 Wave Equation 523.4 Laplace and Poisson Equations 563.4.1 Boundary Value Problem 563.4.2 Laplace Equation in a Circle 583.4.3 Laplace Equation in Three-Dimensional Domain 603.4.4 Representation Formula for Poisson Equation 66References 70Further Reading 704 Analysis for Inverse Problem 714.1 Examples of Inverse Problems in Medical Imaging 714.1.1 Electrical Property Imaging 714.1.2 Mechanical Property Imaging 744.1.3 Image Restoration 754.2 Basic Analysis 764.2.1 Sobolev Space 784.2.2 Some Important Estimates 814.2.3 Helmholtz Decomposition 874.3 Variational Problems 884.3.1 Lax–Milgram Theorem 884.3.2 Ritz Approach 924.3.3 Euler–Lagrange Equations 964.3.4 Regularity Theory and Asymptotic Analysis 1004.4 Tikhonov Regularization and Spectral Analysis 1044.4.1 Overview of Tikhonov Regularization 1054.4.2 Bounded Linear Operators in Banach Space 1094.4.3 Regularization in Hilbert Space or Banach Space 1124.5 Basics of Real Analysis 1164.5.1 Riemann Integrability 1164.5.2 Measure Space 1174.5.3 Lebesgue-Measurable Function 1194.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure 1234.5.5 Differentiation Theory 125References 127Further Reading 1275 Numerical Methods 1295.1 Iterative Method for Nonlinear Problem 1295.2 Numerical Computation of One-Dimensional Heat Equation 1305.2.1 Explicit Scheme 1325.2.2 Implicit Scheme 1355.2.3 Crank–Nicolson Method 1365.3 Numerical Solution of Linear System of Equations 1365.3.1 Direct Method using LU Factorization 1365.3.2 Iterative Method using Matrix Splitting 1385.3.3 Iterative Method using Steepest Descent Minimization 1405.3.4 Conjugate Gradient (CG) Method 1435.4 Finite Difference Method (FDM) 1455.4.1 Poisson Equation 1455.4.2 Elliptic Equation 1465.5 Finite Element Method (FEM) 1475.5.1 One-Dimensional Model 1475.5.2 Two-Dimensional Model 1495.5.3 Numerical Examples 154References 157Further Reading 1586 CT, MRI and Image Processing Problems 1596.1 X-ray Computed Tomography 1596.1.1 Inverse Problem 1606.1.2 Basic Principle and Nonlinear Effects 1606.1.3 Inverse Radon Transform 1636.1.4 Artifacts in CT 1666.2 Magnetic Resonance Imaging 1676.2.1 Basic Principle 1676.2.2 k-Space Data 1686.2.3 Image Reconstruction 1696.3 Image Restoration 1716.3.1 Role of p in (6.35) 1736.3.2 Total Variation Restoration 1756.3.3 Anisotropic Edge-Preserving Diffusion 1806.3.4 Sparse Sensing 1816.4 Segmentation 1846.4.1 Active Contour Method 1856.4.2 Level Set Method 1876.4.3 Motion Tracking for Echocardiography 189References 192Further Reading 1947 Electrical Impedance Tomography 1957.1 Introduction 1957.2 Measurement Method and Data 1967.2.1 Conductivity and Resistance 1967.2.2 Permittivity and Capacitance 1977.2.3 Phasor and Impedance 1987.2.4 Admittivity and Trans-Impedance 1997.2.5 Electrode Contact Impedance 2007.2.6 EIT System 2017.2.7 Data Collection Protocol and Data Set 2027.2.8 Linearity between Current and Voltage 2047.3 Representation of Physical Phenomena 2057.3.1 Derivation of Elliptic PDE 2057.3.2 Elliptic PDE for Four-Electrode Method 2067.3.3 Elliptic PDE for Two-Electrode Method 2097.3.4 Min–Max Property of Complex Potential 2107.4 Forward Problem and Model 2107.4.1 Continuous Neumann-to-Dirichlet Data 2117.4.2 Discrete Neumann-to-Dirichlet Data 2127.4.3 Nonlinearity between Admittivity and Voltage 2147.5 Uniqueness Theory and Direct Reconstruction Method 2167.5.1 Calder´on’s Approach 2167.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite Measurements 2187.5.3 Nachmann’s D-bar Method in Two Dimensions 2217.6 Back-Projection Algorithm 2237.7 Sensitivity and Sensitivity Matrix 2267.7.1 Perturbation and Sensitivity 2267.7.2 Sensitivity Matrix 2277.7.3 Linearization 2277.7.4 Quality of Sensitivity Matrix 2297.8 Inverse Problem of EIT 2297.8.1 Inverse Problem of RC Circuit 2297.8.2 Formulation of EIT Inverse Problem 2317.8.3 Ill-Posedness of EIT Inverse Problem 2317.9 Static Imaging 2327.9.1 Iterative Data Fitting Method 2327.9.2 Static Imaging using Four-Channel EIT System 2337.9.3 Regularization 2377.9.4 Technical Difficulty of Static Imaging 2377.10 Time-Difference Imaging 2397.10.1 Data Sets for Time-Difference Imaging 2397.10.2 Equivalent Homogeneous Admittivity 2407.10.3 Linear Time-Difference Algorithm using Sensitivity Matrix 2417.10.4 Interpretation of Time-Difference Image 2427.11 Frequency-Difference Imaging 2437.11.1 Data Sets for Frequency-Difference Imaging 2437.11.2 Simple Difference Ft,ω2 − Ft,ω1 2447.11.3 Weighted Difference Ft,ω2 − αFt,ω1 2447.11.4 Linear Frequency-Difference Algorithm using Sensitivity Matrix 2457.11.5 Interpretation of Frequency-Difference Image 246References 2478 Anomaly Estimation and Layer Potential Techniques 2518.1 Harmonic Analysis and Potential Theory 2528.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation 2528.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity 2598.2 Anomaly Estimation using EIT 2668.2.1 Size Estimation Method 2688.2.2 Location Search Method 2748.3 Anomaly Estimation using Planar Probe 2818.3.1 Mathematical Formulation 2828.3.2 Representation Formula 287References 290Further Reading 2919 Magnetic Resonance Electrical Impedance Tomography 2959.1 Data Collection using MRI 2969.1.1 Measurement of Bz 2979.1.2 Noise in Measured Bz Data 2999.1.3 Measurement of B = (Bx,By,Bz) 3019.2 Forward Problem and Model Construction 3019.2.1 Relation between J, Bz and σ 3029.2.2 Three Key Observations 3039.2.3 Data Bz Traces σ∇u × ez Directional Change of σ 3049.2.4 Mathematical Analysis toward MREIT Model 3059.3 Inverse Problem Formulation using B or J 3089.4 Inverse Problem Formulation using Bz 3099.4.1 Model with Two Linearly Independent Currents 3099.4.2 Uniqueness 3109.4.3 Defected Bz Data in a Local Region 3149.5 Image Reconstruction Algorithm 3159.5.1 J -substitution Algorithm 3159.5.2 Harmonic Bz Algorithm 3179.5.3 Gradient Bz Decomposition and Variational Bz Algorithm 3199.5.4 Local Harmonic Bz Algorithm 3209.5.5 Sensitivity Matrix-based Algorithm 3229.5.6 Anisotropic Conductivity Reconstruction Algorithm 3239.5.7 Other Algorithms 3249.6 Validation and Interpretation 3259.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm 3259.6.2 Conductivity Phantom Imaging 3269.6.3 Animal Imaging 3279.6.4 Human Imaging 3309.7 Applications 331References 33210 Magnetic Resonance Elastography 33510.1 Representation of Physical Phenomena 33610.1.1 Overview of Hooke’s Law 33610.1.2 Strain Tensor in Lagrangian Coordinates 33910.2 Forward Problem and Model 34010.3 Inverse Problem in MRE 34210.4 Reconstruction Algorithms 34210.4.1 Reconstruction of μ with the Assumption of Local Homogeneity 34410.4.2 Reconstruction of μ without the Assumption of Local Homogeneity 34510.4.3 Anisotropic Elastic Moduli Reconstruction 34910.5 Technical Issues in MRE 350References 351Further Reading 352Index 355

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