Fundamentals of Computerized Tomography: Image Reconstruction from Projections (Google eBook)

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Springer Science & Business Media, Jul 14, 2009 - Medical - 312 pages
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This revised and updated text presents the computational and mathematical procedures underlying data collection, image reconstruction, and image display in computerized tomography. New topics: the fast calculation of a ray sum for a digitized picture, the task-oriented comparison of reconstruction algorithm performance, blob basis functions and the linogram method for image reconstruction. Features: Describes how projection data are obtained and the resulting reconstructions are used; Presents a comparative evaluation of reconstruction methods; Investigates reconstruction algorithms; Explores basis functions, functions to be optimized, norms, generalized inverses, least squares solutions, maximum entropy solutions, and most likely estimates; Discusses SNARK09, a large programming system for image reconstruction; Concludes each chapter with helpful Notes and References sections. An excellent guide for practitioners, it can also serve as a textbook for an introductory graduate course.
  

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Contents

Introduction
1
12 Probability and Random Variables
18
An Overview of the Process of CT
27
23 Data Collection for CT
30
24 Voxels Pixels and CT Numbers
31
25 The Problem of Polychromaticity
32
26 Reconstruction Algorithms
34
Physical Problems Associated with Data Collection in CT
37
87 Why So Popular?
157
Other Transform Methods for Parallel Beams
159
92 The Fourier Method of Reconstruction
161
93 Linograms
166
94 RhoFiltered Layergram
173
Filtered Backprojection for Divergent Beams
177
102 Choice of the Window Function
181
103 Point Response Function
182

32 Beam Hardening
42
33 Other Sources of Error
44
34 Scanning Modes
47
Computer Simulation of Data Collection in CT
53
42 Creation of a Phantom
54
43 A PiecewiseHomogeneous Head Phantom
56
44 Head Phantom with a Large Tumor and Local Inhomogeneities
59
45 Creation of the Ray Sums
60
46 Fast Calculation of a Ray Sum for a Digitized Picture
63
Data Collection and Reconstruction of the Head Phantom
66
52 TaskOriented Comparison of Algorithm Performance
69
53 An Illustration Using Selective Smoothing
73
54 Reconstruction from Perfect Data
78
55 Effects of Photons Statistics
83
56 Effect of Beam Hardening
86
57 The Effects of Detector Width and Scatter
91
58 Simulation of Different Scanning Modes
95
Basic Concepts of Reconstruction Algorithms
100
62 Transform Methods
106
63 Series Expansion Methods
108
64 Optimization Criteria
111
65 Blob Basis Functions
119
66 Computational Efficiency
122
Backprojection
125
72 Implementation of the Backprojection Operator
127
73 Discrete Backprojection
131
Filtered Backprojection for Parallel Beams
135
82 Derivation of the FBP Method
139
83 Implementation of the FBP Method
140
84 Fourier Transforms
143
85 Sampling and Interpolation
146
86 The Choice of Convolving and Interpolating Functions
147
104 Noise Reconstruction
186
105 Comparison of Algorithms Based on Reconstructions
188
Algebraic Reconstruction Techniques
193
112 Relaxation Methods for Solving Systems of Inequalities and Equalities
196
113 Additive ART
201
114 Tricks
205
115 Efficacy of ART
210
Quadratic Optimization Methods
217
122 Richardsons Method for Solving Systems of Equations
221
123 Smoothing Matrices
224
124 Implementation of Richardsons Methods for Image Reconstruction
226
125 A Demonstration of Quadratic Optimization
227
Truly ThreeDimensional Reconstruction
234
131 ThreeDimensional Series Expansion
236
132 Dynamically Changing 3D Phantoms and Their Projections
237
133 ThreeDimensional Reconstructions of the Dynamic Phantom
240
ThreeDimensional Display of Organs
243
142 Boundary Detection
246
143 Hidden Surface Removal
251
144 Shading
253
Mathematical Background
259
152 The Line Integral of the Relative Linear Attenuation
260
153 The Radon Inversion Formula
261
154 A Picture Is Not Uniquely Determined by a Finite Number of Its Views
265
155 Analysis of the Photon Statistics
267
156 The Integral Expression for Polychromatic Ray Sums
269
157 Proof of the Regularization Theorem
270
158 Convergence of the Relaxation Method for Inequalities
273
References
277
Index
292
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About the author (2009)

Department of Radiology, Medical Image Processing Group, Philadelphia, Pennsylvania.

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