## An Introduction to Mathematics of Emerging Biomedical ImagingBiomedical 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 |

197 | |

### Other editions - View all

An Introduction to Mathematics of Emerging Biomedical Imaging Habib Ammari No preview available - 2009 |

An Introduction to Mathematics of Emerging Biomedical Imaging Habib Ammari No preview available - 2008 |