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

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 F*t,ω*2*− *F*t,ω*1

7.11.3 Weighted Difference F*t,ω*2*− **α *F*t,ω*1

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 *B**z *

9.1.2 Noise in Measured *B**z *Data

9.1.3 Measurement of **B **= (*B**x**,B**y**,B**z*)

9.2 Forward Problem and Model Construction

9.2.1 Relation between **J**, *B**z *and *σ *

9.2.2 Three Key Observations

9.2.3 Data *B**z *Traces *σ**∇**u **× ***e***z*-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 *B**z *

9.4.1 Model with Two Linearly Independent Currents

9.4.2 Uniqueness

9.4.3 Defected *B**z *Data in a Local Region

9.5 Image Reconstruction Algorithm

9.5.1 *J*-substitution Algorithm

9.5.2 Harmonic *B**z *Algorithm

9.5.3 Gradient *B**z *Decomposition and Variational *B**z *Algorithm

9.5.4 Local Harmonic *B**z *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 *B**z *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