Computed Radiation Imaging (eBook)

Front Cover 1
Computed Radiation Imaging 4
Copyright 5
Table of Contents 6
Preface 12
Chapter 1. Radiation Imaging 14
1.1 Why Radiation? 14
1.2 Imaging Modalities 15
1.3 Direct and Reconstructed Imaging 19
1.4 The Forward and Inverse Problems 20
1.5 Forward and Inverse Mapping 23
References 25
Part I: The Forward Problem 26
Chapter 2. Radiation Transport 28
2.1 Introduction 28
2.2 Variables 28
2.3 Cross Sections 30
2.4 Boltzmann Transport Equation 33
2.5 Source-Free Steady-State Problem 34
2.6 Steady-State Problem in Void 36
2.7 Point-Kernel Method 37
2.8 Charged Particles 38
References 39
Chapter 3. Measurement Models 40
3.1 Formulation 40
3.2 Scaling 40
3.3 Measured Response 41
3.4 Sensitivity 42
3.5 Variability 43
3.6 Components 43
3.7 Image Grid 46
3.8 Idealization 47
3.9 Computer Coding 48
References 49
Chapter 4. Transmission 50
4.1 Basic Model 50
4.2 Physical Relevance 53
4.3 Discretization 55
4.4 Nature of Radiation Source 56
4.5 Secondary Radiation 61
4.6 Scattering 62
4.7 Sensitivity 63
4.8 Variability 64
References 65
Chapter 5. Emission 66
5.1 Embedded Radiation 66
5.2 Induced Emission 68
5.3 Discretization 71
5.4 Sensitivity 72
5.5 Sources 72
5.6 Interfering Effects 74
References 74
Chapter 6. Scattering 76
6.1 Introduction 76
6.2 Single-Scattering Model 77
6.3 Multiple Scattering 78
6.4 Compton Scattering 79
6.5 Neutron Elastic Scattering 81
6.6 Discretization 83
6.7 Sensitivity 84
References 84
Part II: The Inverse Problem 86
Chapter 7. Features 88
7.1 Discretization 88
7.2 Well-Posed Problem 90
7.3 Existence 91
7.4 Uniqueness 91
7.5 Continuity 92
7.6 Ill-Posed Problem 92
7.7 Ill-Conditioning 93
7.A Basics of Functional Analysis 95
References 99
Chapter 8. Formulation 100
8.1 Matrix 100
8.2 Functional 101
8.3 Analytic 102
8.4 Probabilistic 105
8.A Probabilistic Basis of Maximum-Likelihood and Least-Squares Methods 107
Reference 108
Chapter 9. Preprocessing of Measurements 110
9.1 Number of Measurements 111
9.2 Frequency Analysis 117
9.3 Spatial Filtering of Noise 122
9.4 Consistency and Smoothing 126
References 135
Chapter 10. Matrix-Based Methods 138
10.1 Error Propagation 138
10.2 Singular Value Decomposition 139
10.3 Least Squares 141
10.4 Regularization Methods 141
10.5 Regularization-Parameter Determination 163
10.6 Iterative Methods 170
10.7 Nonlinear Problems 178
10.8 Software 182
References 182
Chapter 11. Functional Optimization 186
11.1 Formulation 186
11.2 Effect of Number of Measurements 188
11.3 Sensitivity to Measurement Uncertainty 189
11.4 Minimization 190
11.5 Search Methods 190
11.6 Genetic Evolution 191
11.7 Simulated Annealing 193
11.8 Neural Networks 195
References 196
Chapter 12. Analytic Methods 198
12.1 Radon Transform 198
12.2 Two-Dimensional Fourier Transforms 200
12.3 Backprojection 208
12.4 Fan-Beam Transmission Tomography 215
12.5 Cone-Beam Transmission Tomography 219
12.6 Emission Imaging 221
12.7 Scatter Imaging 226
12.8 Computer Codes 227
12.9 Wavelet Transforms 227
References 230
Chapter 13. Probabilistic Methods 232
13.1 Bayesian - Minimum Information 232
13.2 Poisson Distribution 234
13.3 Normal Distribution 236
13.4 Maximum a posteriori (MAP) 237
13.5 The Monte Carlo Method 241
References 243
Chapter 14. Incomplete Problems 244
14.1 Incompleteness 244
14.2 General Solution Methods 245
14.3 Estimation Maximization 247
14.4 Markov Random Fields 254
References 255
Chapter 15. Testing 258
15.1 Ideal Problem 258
15.2 Noisy-Ideal Problem 260
15.3 Independently-Simulated-Data Problem 263
15.4 Laboratory Problem 264
15.5 Image Quality 269
15.6 Test Objects 279
References 282
Chapter 16. Post-Processing: Image Enhancement 284
16.1 Image Convolution 285
16.2 Image Degradation 287
16.3 Frequency Filtering 290
16.4 Matrix Based 294
16.5 Statistical Methods 295
16.6 Optimization 296
16.7 Blind Deconvolution 297
References 299
Bibliography 302