An Introduction to Mathematics of Emerging Biomedical Imaging
by Ammari, HabibFormat: Paperback
Pub. Date: 2008-07-01
Publisher(s): Springer Verlag
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Summary
Biomedical imaging is a fascinating research area to applied mathematicians. Challenging imaging problems arise and they often trigger the investigation of fundamental problems in various branches of mathematics.
Author Biography
Habib Ammari (born in June 1969) received the B.S., M.S., and Ph.D. degrees in mathematics from +ëcole Polytechnique Palaiseau in 1992, 1993, and 1995, respectively, and the Habilitation Degree from Universit+¬ Pierre et Marie Curie (Paris 6), in 1999. He is currently Director of Research at the French Center of Scientific Research (CNRS). His current research interests include biomedical imaging, electrical impedance tomography, inverse problems, and electromagnetic modelling. He has contributed over 100 peer-reviewed articles and book chapters, authored four books and edited three others. He is serving as an editor of several mathematical journals. Habib Ammari has been invited to more than 30 international conferences. He produced 10 Ph.D. students and served as adviser for 10 post-docs.
Table of Contents
| Biomedical Imaging Modalities | p. 3 |
| X-Ray Imaging and Computed Tomography | p. 3 |
| Magnetic Resonance Imaging | p. 4 |
| Electrical Impedance Tomography | p. 5 |
| T-Scan Electrical Impedance Imaging System for Anomaly Detection | p. 7 |
| Electrical and Magnetic Source Imaging | p. 7 |
| Magnetic Resonance Electrical Impedance Tomography | p. 9 |
| Impediography | p. 10 |
| Ultrasound Imaging | p. 11 |
| Microwave Imaging | p. 12 |
| Elastic Imaging | p. 12 |
| Magnetic Resonance Elastography | p. 12 |
| Optical Tomography | p. 13 |
| Mathematical Tools | |
| Preliminaries | p. 17 |
| Special Functions | p. 17 |
| Sobolev Spaces | p. 20 |
| Fourier Analysis | p. 21 |
| Shannon's Sampling Theorem | p. 23 |
| Fast Fourier Transform | p. 24 |
| The Two-Dimensional Radon Transform | p. 25 |
| The Moore-Penrose Generalized Inverse | p. 28 |
| Singular Value Decomposition | p. 28 |
| Compact Operators | p. 29 |
| Regularization of Ill-Posed Problems | p. 30 |
| Stability | p. 30 |
| The Truncated SVD | p. 32 |
| Tikhonov-Phillips Regularization | p. 32 |
| Regularization by Truncated Iterative Methods | p. 34 |
| General Image Characteristics | p. 35 |
| Spatial Resolution | p. 35 |
| Signal-To-Noise Ratio | p. 37 |
| Layer Potential Techniques | p. 43 |
| The Laplace Equation | p. 44 |
| Fundamental Solution | p. 44 |
| Layer Potentials | p. 44 |
| Invertibility of [lambda]I-K*[subscript D] | p. 54 |
| Neumann Function | p. 55 |
| Transmission Problem | p. 59 |
| Helmholtz Equation | p. 62 |
| Fundamental Solution | p. 62 |
| Layer Potentials | p. 63 |
| Transmission Problem | p. 65 |
| Static Elasticity | p. 70 |
| Fundamental Solution | p. 71 |
| Layer Potentials | p. 73 |
| Transmission Problem | p. 75 |
| Dynamic Elasticity | p. 80 |
| Radiation Condition | p. 81 |
| Fundamental Solution | p. 81 |
| Layer Potentials | p. 82 |
| Transmission Problem | p. 83 |
| Modified Stokes System | p. 84 |
| Fundamental Solution | p. 84 |
| Layer Potentials | p. 85 |
| Transmission Problem | p. 89 |
| General Reconstruction Algorithms | |
| Tomographic Imaging with Non-Diffracting Sources | p. 95 |
| Imaging Equations of CT and MRI | p. 95 |
| Imaging Equation of CT | p. 95 |
| Imaging Equation of MRI | p. 96 |
| General Issues of Image Reconstruction | p. 97 |
| Reconstruction from Fourier Transform Samples | p. 98 |
| Problem Formulation | p. 98 |
| Basic Theory | p. 99 |
| Reconstruction from Radon Transform Samples | p. 101 |
| The Inverse Radon Transform | p. 101 |
| Fourier Inversion Formula | p. 101 |
| Direct Backprojection Method | p. 102 |
| Filtered Backprojection Reconstruction | p. 104 |
| Noise in Filtered Backprojection Reconstruction | p. 105 |
| Tomographic Imaging with Diffracting Sources | p. 107 |
| Electrical Impedance Tomography | p. 107 |
| Mathematical Model | p. 108 |
| Ill-Conditioning | p. 108 |
| Static Imaging | p. 109 |
| Dynamic Imaging | p. 110 |
| Electrode Model | p. 112 |
| Ultrasound and Microwave Tomographies | p. 112 |
| Mathematical Model | p. 113 |
| Diffraction Tomography | p. 114 |
| Biomagnetic Source Imaging | p. 117 |
| Mathematical Models | p. 118 |
| The Electric Forward Problem | p. 119 |
| The Magnetic Forward Problem | p. 119 |
| The Inverse EEG Problem | p. 120 |
| The Spherical Model in MEG | p. 121 |
| Anomaly Detection Algorithms | |
| Small Volume Expansions | p. 127 |
| Conductivity Problem | p. 128 |
| Formal Derivations | p. 129 |
| Polarization Tensor | p. 131 |
| Helmholtz Equation | p. 132 |
| Formal Derivations | p. 134 |
| Static Elasticity | p. 134 |
| Formal Derivations | p. 136 |
| Elastic Moment Tensor | p. 138 |
| Dynamic Elasticity | p. 140 |
| Modified Stokes System | p. 140 |
| Nearly Incompressible Bodies | p. 141 |
| Formal Derivations | p. 142 |
| Viscous Moment Tensor | p. 145 |
| Diffusion Equation | p. 147 |
| Imaging Techniques | p. 151 |
| Projection Type Algorithms | p. 151 |
| Multiple Signal Classification Type Algorithms | p. 152 |
| Time-Domain Imaging | p. 156 |
| Fourier- and MUSIC-Type Algorithms | p. 157 |
| Time-Reversal Imaging | p. 159 |
| Hybrid Imaging Techniques | |
| Magnetic Resonance Electrical Impedance Tomography | p. 169 |
| Mathematical Model | p. 170 |
| J-Substitution Algorithm | p. 172 |
| The Harmonic Algorithm | p. 174 |
| Impediography | p. 177 |
| Physical Model | p. 177 |
| Mathematical Model | p. 178 |
| E-Substitution Algorithm | p. 180 |
| Magnetic Resonance Elastography | p. 183 |
| Mathematical Model | p. 183 |
| Binary Level Set Algorithm | p. 185 |
| References | p. 189 |
| Index | p. 197 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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