Table of Contents

Title Page

Copyright

Preface

List of Abbreviations

Chapter 1: Introduction

1.1 Forward Problem

1.2 Inverse Problem

1.3 Issues in Inverse Problem Solving

1.4 Linear, Nonlinear and Linearized Problems

Reference

Chapter 2: Signal and System as Vectors

2.1 Vector Spaces

2.2 Vector Calculus

2.3 Taylor's Expansion

2.4 Linear System of Equations

2.5 Fourier Transform

References

Chapter 3: Basics of Forward Problem

3.1 Understanding a PDE using Images as Examples

3.2 Heat Equation

3.3 Wave Equation

3.4 Laplace and Poisson Equations

References

Further Reading

Chapter 4: Analysis for Inverse Problem

4.1 Examples of Inverse Problems in Medical Imaging

4.2 Basic Analysis

4.3 Variational Problems

4.4 Tikhonov Regularization and Spectral Analysis

4.5 Basics of Real Analysis

References

Further Reading

Chapter 5: Numerical Methods

5.1 Iterative Method for Nonlinear Problem

5.2 Numerical Computation of One-Dimensional Heat Equation

5.3 Numerical Solution of Linear System of Equations

5.4 Finite Difference Method (FDM)

5.5 Finite Element Method (FEM)

References

Further Reading

Chapter 6: CT, MRI and Image Processing Problems

6.1 X-ray Computed Tomography

6.2 Magnetic Resonance Imaging

6.3 Image Restoration

6.4 Segmentation

References

Further Reading

Chapter 7: Electrical Impedance Tomography

7.1 Introduction

7.2 Measurement Method and Data

7.3 Representation of Physical Phenomena

7.4 Forward Problem and Model

7.5 Uniqueness Theory and Direct Reconstruction Method

7.6 Back-Projection Algorithm

7.7 Sensitivity and Sensitivity Matrix

7.8 Inverse Problem of EIT

7.9 Static Imaging

7.10 Time-Difference Imaging

7.11 Frequency-Difference Imaging

References

Chapter 8: Anomaly Estimation and Layer Potential Techniques

8.1 Harmonic Analysis and Potential Theory

8.2 Anomaly Estimation using EIT

8.3 Anomaly Estimation using Planar Probe

References

Further Reading

Chapter 9: Magnetic Resonance Electrical Impedance Tomography

9.1 Data Collection using MRI

9.2 Forward Problem and Model Construction

9.3 Inverse Problem Formulation using B or J

9.4 Inverse Problem Formulation using Bz

9.5 Image Reconstruction Algorithm

9.6 Validation and Interpretation

9.7 Applications

References

Chapter 10: Magnetic Resonance Elastography

10.1 Representation of Physical Phenomena

10.2 Forward Problem and Model

10.3 Inverse Problem in MRE

10.4 Reconstruction Algorithms

10.5 Technical Issues in MRE

References

Further Reading

Index

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