An Introduction to Mathematics of... | Buy

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.

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.

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
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