Jin Keun Seo, Eung Je Woo

Nonlinear Inverse Problems in Imaging

• Gebundenes Buch

Produktbeschreibung

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

Produktdetails

• Verlag: Wiley & Sons
• 1. Auflage
• Seitenzahl: 374
• Erscheinungstermin: 26. Dezember 2012
• Englisch
• Abmessung: 251mm x 172mm x 22mm
• Gewicht: 706g
• ISBN-13: 9780470669426
• ISBN-10: 047066942X
• Artikelnr.: 36026509

Autorenporträt

Jin Keun Seo, Department of Computational Science and Engineering, Yonsei University, Korea Professor Seo is currently Chairman of the Computational Science and Engineering Department at Yonsei University, Korea. He has worked on a wide range of interdisciplinary areas including PDE, image processing, bio-impedance, mathematical modelling, harmonic analysis, and so on. He has published over 100 research papers in scientific journals including SIAM Journal on Applied Mathematics, Inverse Problems, IEEE Transactions on Medical Imaging, IEEE Transactions on Biomedical Engineering, Physics in Medicine and Biology, Physiological Measurement, and others. He has received distinguished research awards from the Korean Mathematical Society and Yonsei University. Since 2007, he has been an editor of Journal of Inverse Problems and Imaging. Eung Je Woo, Department of Biomedical Engineering, Kyung Hee University, Korea Professor Woo is currently with the Department of Biomedical Engineering, College of Electronics and Information, Kyung Hee University, Korea. Since 2002, he has been the director of Impedance Imaging Research Center (IIRC), Korea. For the past 20 years, he has been teaching undergraduate and graduate courses on medical instrumentation, biomedical computing, and impedance imaging. His research interests include biomedical instrumentation and imaging. From 2004 to 2010, he has been a co-organizer of international conferences on impedance imaging and electrical bioimpedance. In 2006, he served the scientific program chair of the World Congress on Medical Physics and Biomedical Engineering (WC2006). In 2009, he served the theme co-chair of IEEE EMBC09 for biomedical imaging and image processing.

Inhaltsangabe

Preface xi List of Abbreviations xiii 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 References 7 2 Signal and System as
Vectors 9 2.1 Vector Spaces 9 2.1.1 Vector Space and Subspace 9 2.1.2
Basis, Norm and Inner Product 11 2.1.3 Hilbert Space 13 2.2 Vector Calculus
16 2.2.1 Gradient 16 2.2.2 Divergence 17 2.2.3 Curl 17 2.2.4 Curve 18 2.2.5
Curvature 19 2.3 Taylor's Expansion 21 2.4 Linear System of Equations 23
2.4.1 Linear System and Transform 23 2.4.2 Vector Space of Matrix 24 2.4.3
Least-Squares Solution 27 2.4.4 Singular Value Decomposition (SVD) 28 2.4.5
Pseudo-inverse 29 2.5 Fourier Transform 30 2.5.1 Series Expansion 30 2.5.2
Fourier Transform 32 2.5.3 Discrete Fourier Transform (DFT) 37 2.5.4 Fast
Fourier Transform (FFT) 40 2.5.5 Two-Dimensional Fourier Transform 41
References 42 3 Basics of Forward Problem 43 3.1 Understanding a PDE using
Images as Examples 44 3.2 Heat Equation 46 3.2.1 Formulation of Heat
Equation 46 3.2.2 One-Dimensional Heat Equation 48 3.2.3 Two-Dimensional
Heat Equation and Isotropic Diffusion 50 3.2.4 Boundary Conditions 51 3.3
Wave Equation 52 3.4 Laplace and Poisson Equations 56 3.4.1 Boundary Value
Problem 56 3.4.2 Laplace Equation in a Circle 58 3.4.3 Laplace Equation in
Three-Dimensional Domain 60 3.4.4 Representation Formula for Poisson
Equation 66 References 70 Further Reading 70 4 Analysis for Inverse Problem
71 4.1 Examples of Inverse Problems in Medical Imaging 71 4.1.1 Electrical
Property Imaging 71 4.1.2 Mechanical Property Imaging 74 4.1.3 Image
Restoration 75 4.2 Basic Analysis 76 4.2.1 Sobolev Space 78 4.2.2 Some
Important Estimates 81 4.2.3 Helmholtz Decomposition 87 4.3 Variational
Problems 88 4.3.1 Lax-Milgram Theorem 88 4.3.2 Ritz Approach 92 4.3.3
Euler-Lagrange Equations 96 4.3.4 Regularity Theory and Asymptotic Analysis
100 4.4 Tikhonov Regularization and Spectral Analysis 104 4.4.1 Overview of
Tikhonov Regularization 105 4.4.2 Bounded Linear Operators in Banach Space
109 4.4.3 Regularization in Hilbert Space or Banach Space 112 4.5 Basics of
Real Analysis 116 4.5.1 Riemann Integrability 116 4.5.2 Measure Space 117
4.5.3 Lebesgue-Measurable Function 119 4.5.4 Pointwise, Uniform, Norm
Convergence and Convergence in Measure 123 4.5.5 Differentiation Theory 125
References 127 Further Reading 127 5 Numerical Methods 129 5.1 Iterative
Method for Nonlinear Problem 129 5.2 Numerical Computation of
One-Dimensional Heat Equation 130 5.2.1 Explicit Scheme 132 5.2.2 Implicit
Scheme 135 5.2.3 Crank-Nicolson Method 136 5.3 Numerical Solution of Linear
System of Equations 136 5.3.1 Direct Method using LU Factorization 136
5.3.2 Iterative Method using Matrix Splitting 138 5.3.3 Iterative Method
using Steepest Descent Minimization 140 5.3.4 Conjugate Gradient (CG)
Method 143 5.4 Finite Difference Method (FDM) 145 5.4.1 Poisson Equation
145 5.4.2 Elliptic Equation 146 5.5 Finite Element Method (FEM) 147 5.5.1
One-Dimensional Model 147 5.5.2 Two-Dimensional Model 149 5.5.3 Numerical
Examples 154 References 157 Further Reading 158 6 CT, MRI and Image
Processing Problems 159 6.1 X-ray Computed Tomography 159 6.1.1 Inverse
Problem 160 6.1.2 Basic Principle and Nonlinear Effects 160 6.1.3 Inverse
Radon Transform 163 6.1.4 Artifacts in CT 166 6.2 Magnetic Resonance
Imaging 167 6.2.1 Basic Principle 167 6.2.2 k-Space Data 168 6.2.3 Image
Reconstruction 169 6.3 Image Restoration 171 6.3.1 Role of p in (6.35) 173
6.3.2 Total Variation Restoration 175 6.3.3 Anisotropic Edge-Preserving
Diffusion 180 6.3.4 Sparse Sensing 181 6.4 Segmentation 184 6.4.1 Active
Contour Method 185 6.4.2 Level Set Method 187 6.4.3 Motion Tracking for
Echocardiography 189 References 192 Further Reading 194 7 Electrical
Impedance Tomography 195 7.1 Introduction 195 7.2 Measurement Method and
Data 196 7.2.1 Conductivity and Resistance 196 7.2.2 Permittivity and
Capacitance 197 7.2.3 Phasor and Impedance 198 7.2.4 Admittivity and
Trans-Impedance 199 7.2.5 Electrode Contact Impedance 200 7.2.6 EIT System
201 7.2.7 Data Collection Protocol and Data Set 202 7.2.8 Linearity between
Current and Voltage 204 7.3 Representation of Physical Phenomena 205 7.3.1
Derivation of Elliptic PDE 205 7.3.2 Elliptic PDE for Four-Electrode Method
206 7.3.3 Elliptic PDE for Two-Electrode Method 209 7.3.4 Min-Max Property
of Complex Potential 210 7.4 Forward Problem and Model 210 7.4.1 Continuous
Neumann-to-Dirichlet Data 211 7.4.2 Discrete Neumann-to-Dirichlet Data 212
7.4.3 Nonlinearity between Admittivity and Voltage 214 7.5 Uniqueness
Theory and Direct Reconstruction Method 216 7.5.1 Calder¿on's Approach 216
7.5.2 Uniqueness and Three-Dimensional Reconstruction: Infinite
Measurements 218 7.5.3 Nachmann's D-bar Method in Two Dimensions 221 7.6
Back-Projection Algorithm 223 7.7 Sensitivity and Sensitivity Matrix 226
7.7.1 Perturbation and Sensitivity 226 7.7.2 Sensitivity Matrix 227 7.7.3
Linearization 227 7.7.4 Quality of Sensitivity Matrix 229 7.8 Inverse
Problem of EIT 229 7.8.1 Inverse Problem of RC Circuit 229 7.8.2
Formulation of EIT Inverse Problem 231 7.8.3 Ill-Posedness of EIT Inverse
Problem 231 7.9 Static Imaging 232 7.9.1 Iterative Data Fitting Method 232
7.9.2 Static Imaging using Four-Channel EIT System 233 7.9.3 Regularization
237 7.9.4 Technical Difficulty of Static Imaging 237 7.10 Time-Difference
Imaging 239 7.10.1 Data Sets for Time-Difference Imaging 239 7.10.2
Equivalent Homogeneous Admittivity 240 7.10.3 Linear Time-Difference
Algorithm using Sensitivity Matrix 241 7.10.4 Interpretation of
Time-Difference Image 242 7.11 Frequency-Difference Imaging 243 7.11.1 Data
Sets for Frequency-Difference Imaging 243 7.11.2 Simple Difference
Ft,omega2 . Ft,omega1 244 7.11.3 Weighted Difference Ft,omega2 .
alphaFt,omega1 244 7.11.4 Linear Frequency-Difference Algorithm using
Sensitivity Matrix 245 7.11.5 Interpretation of Frequency-Difference Image
246 References 247 8 Anomaly Estimation and Layer Potential Techniques 251
8.1 Harmonic Analysis and Potential Theory 252 8.1.1 Layer Potentials and
Boundary Value Problems for Laplace Equation 252 8.1.2 Regularity for
Solution of Elliptic Equation along Boundary of Inhomogeneity 259 8.2
Anomaly Estimation using EIT 266 8.2.1 Size Estimation Method 268 8.2.2
Location Search Method 274 8.3 Anomaly Estimation using Planar Probe 281
8.3.1 Mathematical Formulation 282 8.3.2 Representation Formula 287
References 290 Further Reading 291 9 Magnetic Resonance Electrical
Impedance Tomography 295 9.1 Data Collection using MRI 296 9.1.1
Measurement of Bz 297 9.1.2 Noise in Measured Bz Data 299 9.1.3 Measurement
of B = (Bx,By,Bz) 301 9.2 Forward Problem and Model Construction 301 9.2.1
Relation between J, Bz and sigma 302 9.2.2 Three Key Observations 303 9.2.3
Data Bz Traces sigmanablau × ez Directional Change of sigma 304 9.2.4
Mathematical Analysis toward MREIT Model 305 9.3 Inverse Problem
Formulation using B or J 308 9.4 Inverse Problem Formulation using Bz 309
9.4.1 Model with Two Linearly Independent Currents 309 9.4.2 Uniqueness 310
9.4.3 Defected Bz Data in a Local Region 314 9.5 Image Reconstruction
Algorithm 315 9.5.1 J -substitution Algorithm 315 9.5.2 Harmonic Bz
Algorithm 317 9.5.3 Gradient Bz Decomposition and Variational Bz Algorithm
319 9.5.4 Local Harmonic Bz Algorithm 320 9.5.5 Sensitivity Matrix-based
Algorithm 322 9.5.6 Anisotropic Conductivity Reconstruction Algorithm 323
9.5.7 Other Algorithms 324 9.6 Validation and Interpretation 325 9.6.1
Image Reconstruction Procedure using Harmonic Bz Algorithm 325 9.6.2
Conductivity Phantom Imaging 326 9.6.3 Animal Imaging 327 9.6.4 Human
Imaging 330 9.7 Applications 331 References 332 10 Magnetic Resonance
Elastography 335 10.1 Representation of Physical Phenomena 336 10.1.1
Overview of Hooke's Law 336 10.1.2 Strain Tensor in Lagrangian Coordinates
339 10.2 Forward Problem and Model 340 10.3 Inverse Problem in MRE 342 10.4
Reconstruction Algorithms 342 10.4.1 Reconstruction of mu with the
Assumption of Local Homogeneity 344 10.4.2 Reconstruction of mu without the
Assumption of Local Homogeneity 345 10.4.3 Anisotropic Elastic Moduli
Reconstruction 349 10.5 Technical Issues in MRE 350 References 351 Further
Reading 352 Index 355