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Nonlinear Inverse Problems in Imaging

Book Description

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

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Preface
  5. List of Abbreviations
  6. Chapter 1: Introduction
    1. 1.1 Forward Problem
    2. 1.2 Inverse Problem
    3. 1.3 Issues in Inverse Problem Solving
    4. 1.4 Linear, Nonlinear and Linearized Problems
    5. Reference
  7. Chapter 2: Signal and System as Vectors
    1. 2.1 Vector Spaces
    2. 2.2 Vector Calculus
    3. 2.3 Taylor's Expansion
    4. 2.4 Linear System of Equations
    5. 2.5 Fourier Transform
    6. References
  8. Chapter 3: Basics of Forward Problem
    1. 3.1 Understanding a PDE using Images as Examples
    2. 3.2 Heat Equation
    3. 3.3 Wave Equation
    4. 3.4 Laplace and Poisson Equations
    5. References
    6. Further Reading
  9. Chapter 4: Analysis for Inverse Problem
    1. 4.1 Examples of Inverse Problems in Medical Imaging
    2. 4.2 Basic Analysis
    3. 4.3 Variational Problems
    4. 4.4 Tikhonov Regularization and Spectral Analysis
    5. 4.5 Basics of Real Analysis
    6. References
    7. Further Reading
  10. Chapter 5: Numerical Methods
    1. 5.1 Iterative Method for Nonlinear Problem
    2. 5.2 Numerical Computation of One-Dimensional Heat Equation
    3. 5.3 Numerical Solution of Linear System of Equations
    4. 5.4 Finite Difference Method (FDM)
    5. 5.5 Finite Element Method (FEM)
    6. References
    7. Further Reading
  11. Chapter 6: CT, MRI and Image Processing Problems
    1. 6.1 X-ray Computed Tomography
    2. 6.2 Magnetic Resonance Imaging
    3. 6.3 Image Restoration
    4. 6.4 Segmentation
    5. References
    6. Further Reading
  12. Chapter 7: Electrical Impedance Tomography
    1. 7.1 Introduction
    2. 7.2 Measurement Method and Data
    3. 7.3 Representation of Physical Phenomena
    4. 7.4 Forward Problem and Model
    5. 7.5 Uniqueness Theory and Direct Reconstruction Method
    6. 7.6 Back-Projection Algorithm
    7. 7.7 Sensitivity and Sensitivity Matrix
    8. 7.8 Inverse Problem of EIT
    9. 7.9 Static Imaging
    10. 7.10 Time-Difference Imaging
    11. 7.11 Frequency-Difference Imaging
    12. References
  13. Chapter 8: Anomaly Estimation and Layer Potential Techniques
    1. 8.1 Harmonic Analysis and Potential Theory
    2. 8.2 Anomaly Estimation using EIT
    3. 8.3 Anomaly Estimation using Planar Probe
    4. References
    5. Further Reading
  14. Chapter 9: Magnetic Resonance Electrical Impedance Tomography
    1. 9.1 Data Collection using MRI
    2. 9.2 Forward Problem and Model Construction
    3. 9.3 Inverse Problem Formulation using B or J
    4. 9.4 Inverse Problem Formulation using Bz
    5. 9.5 Image Reconstruction Algorithm
    6. 9.6 Validation and Interpretation
    7. 9.7 Applications
    8. References
  15. Chapter 10: Magnetic Resonance Elastography
    1. 10.1 Representation of Physical Phenomena
    2. 10.2 Forward Problem and Model
    3. 10.3 Inverse Problem in MRE
    4. 10.4 Reconstruction Algorithms
    5. 10.5 Technical Issues in MRE
    6. References
    7. Further Reading
  16. Index