An Introduction to Mathematics of Emerging Biomedical Imaging (eBook)

Habib Ammari (born in June 1969) received the B.S., M.S., and Ph.D. degrees in mathematics from École Polytechnique Palaiseau in 1992, 1993, and 1995, respectively, and the Habilitation Degree from Université 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.

Preface 6
Contents 7
Introduction 11
1 Biomedical Imaging Modalities 13
1.1 X-Ray Imaging and Computed Tomography 13
1.2 Magnetic Resonance Imaging 14
1.3 Electrical Impedance Tomography 15
1.4 T-Scan Electrical Impedance Imaging System for Anomaly Detection 17
1.5 Electrical and Magnetic Source Imaging 17
1.6 Magnetic Resonance Electrical Impedance Tomography 19
1.7 Impediography 20
1.8 Ultrasound Imaging 21
1.9 Microwave Imaging 22
1.10 Elastic Imaging 22
1.11 Magnetic Resonance Elastography 22
1.12 Optical Tomography 23
Part I Mathematical Tools 25
2 Preliminaries 27
2.1 Special Functions 27
2.2 Sobolev Spaces 30
2.3 Fourier Analysis 31
2.4 The Two-Dimensional Radon Transform 35
2.5 The Moore-Penrose Generalized Inverse 38
2.6 Singular Value Decomposition 38
2.7 Compact Operators 39
2.8 Regularization of Ill-Posed Problems 40
2.9 General Image Characteristics 45
Bibliography and Discussion 51
3 Layer Potential Techniques 53
3.1 The Laplace Equation 54
3.2 Helmholtz Equation 72
3.3 Static Elasticity 80
3.4 Dynamic Elasticity 90
3.5 Modi.ed Stokes System 94
Bibliography and Discussion 102
Part II General Reconstruction Algorithms 104
4 Tomographic Imaging with Non-Diffracting Sources 105
4.1 Imaging Equations of CT and MRI 105
4.2 General Issues of Image Reconstruction 107
4.3 Reconstruction from Fourier Transform Samples 108
4.4 Reconstruction from Radon Transform Samples 111
Bibliography and Discussion 116
5 Tomographic Imaging with Diffracting Sources 117
5.1 Electrical Impedance Tomography 117
5.2 Ultrasound and Microwave Tomographies 122
Bibliography and Discussion 125
6 Biomagnetic Source Imaging 127
6.1 Mathematical Models 128
6.2 The Inverse EEG Problem 130
6.3 The Spherical Model in MEG 131
Bibliography and Discussion 132
Part III Anomaly Detection Algorithms 135
7 Small Volume Expansions 137
7.1 Conductivity Problem 138
7.2 Helmholtz Equation 142
7.3 Static Elasticity 144
7.4 Dynamic Elasticity 150
7.5 Modified Stokes System 150
7.6 Nearly Incompressible Bodies 151
7.7 Diffusion Equation 157
Bibliography and Discussion 158
8 Imaging Techniques 161
8.1 Projection Type Algorithms 161
8.2 Multiple Signal Classification Type Algorithms 162
8.3 Time-Domain Imaging 166
Bibliography and Discussion 174
Part IV Hybrid Imaging Techniques 178
9 Magnetic Resonance Electrical Impedance Tomography 179
9.1 Mathematical Model 180
9.2 J-Substitution Algorithm 182
9.3 The Harmonic Algorithm 184
Bibliography and Discussion 185
10 Impediography 187
10.1 Physical Model 187
10.2 Mathematical Model 188
10.3 E-Substitution Algorithm 190
Bibliography and Discussion 192
11 Magnetic Resonance Elastography 193
11.1 Mathematical Model 193
11.2 Binary Level Set Algorithm 195
Bibliography and Discussion 196
References 199
Index 207