An Introduction to Mathematics of Emerging Biomedical Imaging

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Springer Science & Business Media, May 21, 2008 - Medical - 198 pages
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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.

This is the first book to highlight the most recent mathematical developments in emerging biomedical imaging techniques. The main focus is on emerging multi-physics and multi-scales imaging approaches. For such promising techniques, it provides the basic mathematical concepts and tools for image reconstruction. Further improvements in these exciting imaging techniques require continued research in the mathematical sciences, a field that has contributed greatly to biomedical imaging and will continue to do so.

The volume is suitable for a graduate-level course in applied mathematics and helps prepare the reader for a deeper understanding of research areas in biomedical imaging.

 

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Contents

44 Reconstruction from Radon Transform Samples
101
443 Direct Backprojection Method
102
444 Filtered Backprojection Reconstruction
104
445 Noise in Filtered Backprojection Reconstruction
105
Tomographic Imaging with Diffracting Sources
107
511 Mathematical Model
108
513 Static Imaging
109
514 Dynamic Imaging
110

112 Optical Tomography
13
Mathematical Tools
14
Preliminaries
17
22 Sobolev Spaces
20
23 Fourier Analysis
21
231 Shannons Sampling Theorem
23
232 Fast Fourier Transform
24
24 The TwoDimensional Radon Transform
25
25 The MoorePenrose Generalized Inverse
28
27 Compact Operators
29
28 Regularization of IllPosed Problems
30
282 The Truncated SVD
32
284 Regularization by Truncated Iterative Methods
34
29 General Image Characteristics
35
292 SignalToNoise Ratio
37
Layer Potential Techniques
42
31 The Laplace Equation
44
312 Layer Potentials
46
313 Invertibility of I KD
54
314 Neumann Function
55
315 Transmission Problem
59
32 Helmholtz Equation
62
322 Layer Potentials
63
323 Transmission Problem
65
33 Static Elasticity
70
331 Fundamental Solution
71
332 Layer Potentials
73
333 Transmission Problem
75
34 Dynamic Elasticity
80
341 Radiation Condition
81
343 Layer Potentials
82
344 Transmission Problem
83
35 Modified Stokes System
84
352 Layer Potentials
85
353 Transmission Problem
89
General Reconstruction Algorithms
93
Tomographic Imaging with NonDiffracting Sources
95
412 Imaging Equation of MRI
96
42 General Issues of Image Reconstruction
97
43 Reconstruction from Fourier Transform Samples
98
432 Basic Theory
99
515 Electrode Model
112
521 Mathematical Model
113
522 Diffraction Tomography
114
Biomagnetic Source Imaging
116
61 Mathematical Models
118
611 The Electric Forward Problem
119
62 The Inverse EEG Problem
120
63 The Spherical Model in MEG
121
Anomaly Detection Algorithms
124
Small Volume Expansions
125
71 Conductivity Problem
128
711 Formal Derivations
129
712 Polarization Tensor
131
72 Helmholtz Equation
132
721 Formal Derivations
134
731 Formal Derivations
136
732 Elastic Moment Tensor
138
74 Dynamic Elasticity
140
76 Nearly Incompressible Bodies
141
761 Formal Derivations
142
762 Viscous Moment Tensor
145
77 Diffusion Equation
147
Imaging Techniques
151
82 Multiple Signal Classification Type Algorithms
152
83 TimeDomain Imaging
156
831 Fourier and MUSICType Algorithms
157
832 TimeReversal Imaging
159
Hybrid Imaging Techniques
167
Magnetic Resonance Electrical Impedance Tomography
168
91 Mathematical Model
170
92 JSubstitution Algorithm
172
93 The Harmonic Algorithm
174
Impediography
177
102 Mathematical Model
178
103 ESubstitution Algorithm
180
Magnetic Resonance Elastography
183
112 Binary Level Set Algorithm
185
References
188
Index
197
Copyright

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Page 8 - ... Ion currents arising in the neurons of the heart and the brain produce magnetic fields outside the body that can be measured by arrays of SQUID (superconducting quantum interference device) detectors placed near the chest or head; the recording of these magnetic fields is known as magnetocardiography (MCG) or magnetoencephalography (MEG). Magnetic source imaging (MSI) is the reconstruction of the current sources in the heart or brain from these recorded magnetic fields. These fields result from...
Page 8 - ... conducting tissue layers between the central cortex and the electrodes. Cardiac electrical activity is likewise spatially complex, and involves the propagation of excitation wave fronts in the heart. Standard electrocardiographic techniques such as electrocardiography (ECG) and vectorcardiography (VCG) are very limited in their ability to provide information on regional electrical activity and to localize bioelectrical events in the heart. In fact, VCG lumps all cardiac wave fronts into a single...
Page 191 - D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Applied Mathematical Sciences, vol.
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Page 8 - ... sources in the heart or brain from these recorded magnetic fields. These fields result from the synchronous activity of tens or hundreds of thousands of neurons. Both magnetic source imaging and electrical source imaging seek to determine the location, orientation, and magnitude of current sources within the body. The magnetic field at the surface is most strongly determined by current sources directed parallel to the surface, but the electrical potentials are determined by current sources directed...
Page 9 - ... not as well be obtained from electrical potential mapping. An advantage of MSI over ESI is that all body tissues are magnetically transparent and the magnetic fields propagate to the surface without distortion. The electrical potentials at the surface, on the other hand, are distorted by variations in conductivity within the body; this is especially true in the head, where the low conductivity of the skull both distorts and hides the electrical activity of the brain. A disadvantage of MSI is...

About the author (2008)

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.

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