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Preface
List of Abbreviations
1 Introduction
1.1 Forward Problem
1.2 Inverse Problem
1.3 Issues in Inverse Problem Solving
1.4 Linear, Nonlinear and Linearized Problems
2 Signal and System as Vectors
2.1 Vector Space
2.1.1 Vector Space and Subspace
2.1.2 Basis, Norm and Inner Product
2.1.3 Hilbert Space
2.2 Vector Calculus
2.2.1 Gradient
2.2.2 Divergence
2.2.3 Curl
2.2.4 Curve
2.2.5 Curvature
2.3 Taylor’s Expansion
2.4 Linear System of Equations
2.4.1 Linear System and Transform
2.4.2 Vector Space of Matrix
2.4.3 Least Square Solution
2.4.4 Singular Value Decomposition (SVD)
2.4.5 Pseudo-inverse
2.5 Fourier Transform
2.5.1 Series Expansion
2.5.2 Fourier Transform
2.5.3 Discrete Fourier Transform (DFT)
2.5.4 Fast Fourier Transform (FFT)
2.5.5 Two-dimensional Fourier Transform
References
3 Basics for Forward Problem
3.1 Understanding PDE using Images as Examples
3.2 Heat Equation
3.2.1 Formulation of Heat Equation
3.2.2 One-dimensional Heat Equation
3.2.3 Two-dimensional Heat Equation and Isotropic Diffusion
3.2.4 Boundary Conditions
3.3 Wave Equation
3.4 Laplace and Poisson Equations
3.4.1 Boundary Value Problem
3.4.2 Laplace Equation in a Circle
3.4.3 Laplace Equation in Three-dimensional Domain
3.4.4 Representation Formula for Poisson Equation
References
4 Analysis for Inverse Problem
4.1 Examples of Inverse Problems in Medical Imaging
4.1.1 Electrical Property Imaging
4.1.2 Mechanical Property Imaging
4.1.3 Image Restoration
4.2 Basic Analysis
4.2.1 Sobolev Space
4.2.2 Some Important Estimates
4.2.3 Helmholtz Decomposition
4.3 Variational Problems
4.3.1 Lax-Milgram Theorem
4.3.2 Ritz Approach
4.3.3 Euler-Lagrange Equations
4.3.4 Regularity Theory and Asymptotic Analysis
4.4 Tikhonov Regularization and Spectral Analysis
4.4.1 Overview of Tikhonov Regularization
4.4.2 Bounded Linear Operators in Banach Space
4.4.3 Regularization in Hilbert Space or Banach Space
4.5 Basics of Real Analysis
4.5.1 Riemann Integrable
4.5.2 Measure Space
4.5.3 Lebesgue Measurable Function
4.5.4 Pointwise, Uniform, Norm Convergence and Convergence in Measure
4.5.5 Differentiation Theory
References
5 Numerical Methods
5.1 Iterative Method for Nonlinear Problem
5.2 Numerical Computation of One-dimensional Heat equation
5.2.1 Explicit Scheme
5.2.2 Implicit Scheme
5.2.3 Crank-Nicolson Method
5.3 Numerical Solution of Linear System of Equations
5.3.1 Direct Method using LU Factorization
5.3.2 Iterative Method using Matrix Splitting
5.3.3 Iterative Method using Steepest Descent Minimization
5.3.4 Conjugate Gradient (CG) Method
5.4 Finite Difference Method (FDM)
5.4.1 Poisson Equation
5.4.2 Elliptic Equation
5.5 Finite Element Method (FEM)
5.5.1 One-dimensional Model
5.5.2 Two-dimensional Model
5.5.3 Numerical Examples
References
6 CT, MRI and Image Processing Problems
6.1 X-ray CT
6.1.1 Inverse Problem
6.1.2 Basic Principle and Nonlinear Effects
6.1.3 Inverse Radon Transform
6.1.4 Artifacts in CT
6.2 MRI
6.2.1 Basic Principle
6.2.2 K-space Data
6.2.3 Image Reconstruction
6.3 Image Restoration
6.3.1 Role of p in (6.35)
6.3.2 Total Variation Restoration
6.3.3 Anisotropic Edge-preserving Diffusion
6.3.4 Sparse Sensing
6.4 Segmentation
6.4.1 Active Contour Method
6.4.2 Level Set Method
6.4.3 Motion Tracking for Echocardiography
References
7 Electrical Impedance Tomography
7.1 Introduction
7.2 Measurement Method and Data
7.2.1 Conductivity and Resistance
7.2.2 Permittivity and Capacitance
7.2.3 Phasor and Impedance
7.2.4 Admittivity and Trans-impedance
7.2.5 Electrode Contact Impedance
7.2.6 EIT System
7.2.7 Data Collection Protocol and Data Set
7.2.8 Linearity between Current and Voltage
7.3 Representation of Physical Phenomena
7.3.1 Derivation of Elliptic PDE
7.3.2 Elliptic PDE for Four-electrode Method
7.3.3 Elliptic PDE for Two-electrode Method
7.3.4 Min-max Property of Complex Potential
7.4 Forward Problem and Model
7.4.1 Continuous Neumann-to-Dirichlet Data
7.4.2 Discrete Neumann-to-Dirichlet Data
7.4.3 Nonlinearity between Admittivity and Voltage
7.5 Uniqueness Theory and Direct Reconstruction Method
7.5.1 Calder´on’s Approach
7.5.2 Uniqueness and Three-dimensional Reconstruction: Infinite Measurements
7.5.3 Nachmann’s D-bar Method in Two Dimension
7.6 Backprojection Algorithm
7.7 Sensitivity and Sensitivity Matrix
7.7.1 Perturbation and Sensitivity
7.7.2 Sensitivity Matrix
7.7.3 Linearization
7.7.4 Quality of Sensitivity Matrix
7.8 Inverse Problem of EIT
7.8.1 Inverse Problem of RC Circuit
7.8.2 Formulation of EIT Inverse Problem
7.8.3 Ill-posedness of EIT Inverse Problem
7.9 Static Imaging
7.9.1 Iterative Data Fitting Method
7.9.2 Static Imaging using 4-channel EIT System
7.9.3 Regularization
7.9.4 Technical Difficulty of Static Imaging
7.10 Time-difference Imaging
7.10.1 Data Sets for Time-difference Imaging
7.10.2 Equivalent Homogeneous Admittivity
7.10.3 Linear Time-difference Algorithm using Sensitivity Matrix
7.10.4 Interpretation of Time-difference Image
7.11 Frequency-difference Imaging
7.11.1 Data Sets for Frequency-difference Imaging
7.11.2 Simple Difference Ft,ω2− Ft,ω1
7.11.3 Weighted Difference Ft,ω2− α Ft,ω1
7.11.4 Linear Frequency-difference Algorithm using Sensitivity Matrix
7.11.5 Interpretation of Frequency-difference Image
References
8 Anomaly Estimation and Layer Potential Techniques
8.1 Harmonic Analysis and Potential Theory
8.1.1 Layer Potentials and Boundary Value Problems for Laplace Equation
8.1.2 Regularity for Solution of Elliptic Equation along Boundary of Inhomogeneity
8.2 Anomaly Estimation using EIT
8.2.1 Size Estimation Method
8.2.2 Location Search Method
8.3 Anomaly Estimation using Planar Probe
8.3.1 Mathematical Formulation
8.3.2 Representation Formula
References
9 Magnetic Resonance Electrical Impedance Tomography
9.1 Data Collection using MRI
9.1.1 Measurement of Bz
9.1.2 Noise in Measured Bz Data
9.1.3 Measurement of B = (Bx,By,Bz)
9.2 Forward Problem and Model Construction
9.2.1 Relation between J, Bz and σ
9.2.2 Three Key Observations
9.2.3 Data Bz Traces σ∇u × ez-directional Change of σ
9.2.4 Mathematical Analysis toward MREIT Model
9.3 Inverse Problem Formulation using B or J
9.4 Inverse Problem Formulation using Bz
9.4.1 Model with Two Linearly Independent Currents
9.4.2 Uniqueness
9.4.3 Defected Bz Data in a Local Region
9.5 Image Reconstruction Algorithm
9.5.1 J-substitution Algorithm
9.5.2 Harmonic Bz Algorithm
9.5.3 Gradient Bz Decomposition and Variational Bz Algorithm
9.5.4 Local Harmonic Bz Algorithm
9.5.5 Sensitivity Matrix Based Algorithm
9.5.6 Anisotropic Conductivity Reconstruction Algorithm
9.5.7 Other Algorithms
9.6 Validation and Interpretation
9.6.1 Image Reconstruction Procedure using Harmonic Bz Algorithm
9.6.2 Conductivity Phantom Imaging
9.6.3 Animal Imaging
9.6.4 Human Imaging
9.7 Applications
References
10 Magnetic Resonance Elastography
10.1 Representation of Physical Phenomena
10.1.1 Overview of Hooke’s Law
10.1.2 Strain Tensor in Lagrangian Coordinates
10.2 Forward Problem and Model
10.3 Inverse Problem in MRE
10.4 Reconstruction Algorithms
10.4.1 Reconstruction of μ with the Assumption of Local Homogeneity
10.4.2 Reconstruction of μ without the Assumption of Local Homogeneity
10.4.3 Anisotropic Elastic Moduli Reconstruction
10.5 Technical Issues in MRE
References