Physical description 
xiv, 359 pages : illustrations ; 25 cm 
Bibliography 
Includes bibliographical references and index. 
Contents 
1 Introduction 1  1.1 Forward Problem 1  1.2 Inverse Problem 3  1.3 Issues in Inverse Problem Solving 4  1.4 Linear, Nonlinear and Linearized Problems 6  2 Signal and System as Vectors 9  2.1 Vector Spaces 9  2.2 Vector Calculus 16  2.3 Taylor's Expansion 21  2.4 Linear System of Equations 23  2.5 Fourier Transform 30  3 Basics of Forward Problem 43  3.1 Understanding a PDE using Images as Examples 44  3.2 Heat Equation 46  3.3 Wave Equation 52  3.4 Laplace and Poisson Equations 56  4 Analysis for Inverse Problem 71  4.1 Examples of Inverse Problems in Medical Imaging 71  4.2 Basic Analysis 76  4.3 Variational Problems 88  4.4 Tikhonov Regularization and Spectral Analysis 104  4.5 Basics of Real Analysis 116  5 Numerical Methods 129  5.1 Iterative Method for Nonlinear Problem 129  5.2 Numerical Computation of OneDimensional Heat Equation 130  5.3 Numerical Solution of Linear System of Equations 136  5.4 Finite Difference Method (FDM) 145  5.5 Finite Element Method (FEM) 147  6 CT, MRI and Image Processing Problems 159  6.1 Xray Computed Tomography 159  6.2 Magnetic Resonance Imaging 167  6.3 Image Restoration 171  6.4 Segmentation 184  7 Electrical Impedance Tomography 195  7.1 Introduction 195  7.2 Measurement Method and Data 196  7.3 Representation of Physical Phenomena 205  7.4 Forward Problem and Model 210  7.5 Uniqueness Theory and Direct Reconstruction Method 216  7.6 BackProjection Algorithm 223  7.7 Sensitivity and Sensitivity Matrix 226  7.8 Inverse Problem of EIT 229  7.9 Static Imaging 232  7.10 TimeDifference Imaging 239  7.11 FrequencyDifference Imaging 243  8 Anomaly Estimation and Layer Potential Techniques 251  8.1 Harmonic Analysis and Potential Theory 252  8.2 Anomaly Estimation using EIT 266  8.3 Anomaly Estimation using Planar Probe 281  9 Magnetic Resonance Electrical Impedance Tomography 295  9.1 Data Collection using MRI 296  9.2 Forward Problem and Model Construction 301  9.3 Inverse Problem Formulation using B or J 308  9.4 Inverse Problem Formulation using B<sub>z</sub> 309  9.5 Image Reconstruction Algorithm 315  9.6 Validation and Interpretation 325  9.7 Applications 331  10 Magnetic Resonance Elastography 335  10.1 Representation of Physical Phenomena 336  10.2 Forward Problem and Model 340  10.3 Inverse Problem in MRE 342  10.4 Reconstruction Algorithms 342  10.5 Technical Issues in MRE 350. 
Summary 
"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, selfcontained coverage includes basic concepts, models, computational methods, numerical simulations, examples, and case studies. Provides a stepbystep 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"Publisher. 
Other author 
Woo, E. J. (Eung Je)

Subject 
Image processing  Mathematics.


Crosssectional imaging  Mathematics.


Inverse problems (Differential equations)


Nonlinear theories.

ISBN 
9780470669426 (hardback) 

047066942X (hardback) 
