Fundamentals of computerized tomography : image reconstruction from projections

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.

1 Introduction	p. 1
1.1 Image Reconstruction from Projections	p. 1
1.2 Probability and Random Variables	p. 18
2 An Overview of the Process of CT	p. 27
2.1 What Are We Trying to Do?	p. 27
2.2 Traditional Tomography	p. 27
2.3 Data Collection for CT	p. 30
2.4 Voxels, Pixels, and CT Numbers	p. 31
2.5 The Problem of Polychromaticity	p. 32
2.6 Reconstruction Algorithms	p. 34
3 Physical Problems Associated with Data Collection in CT	p. 37
3.1 Photon Statistics	p. 37
3.2 Beam Hardening	p. 42
3.3 Other Sources of Error	p. 44
3.4 Scanning Modes	p. 47
4 Computer Simulation of Data Collection in CT	p. 53
4.1 Pictures and Digitization	p. 53
4.2 Creation of a Phantom	p. 54
4.3 A Piecewise-Homogeneous Head Phantom	p. 56
4.4 Head Phantom with a Large Tumor and Local Inhomogeneities	p. 59
4.5 Creation of the Ray Sums	p. 60
4.6 Fast Calculation of a Ray Sum for a Digitized Picture	p. 63
5 Data Collection and Reconstruction of the Head Phantom	p. 67
5.1 Methods of Picture Comparison	p. 67
5.2 Task-Oriented Comparison of Algorithm Performance	p. 69
5.3 An Illustration Using Selective Smoothing	p. 73
5.4 Reconstruction from Perfect Data	p. 78
5.5 Effects of Photons Statistics	p. 83
5.6 Effect of Beam Hardening	p. 86
5.7 The Effects of Detector Width and Scatter	p. 91
5.8 Simulation of Different Scanning Modes	p. 95
6 Basic Concepts of Reconstruction Algorithms	p. 101
6.1 Problem Statement	p. 101
6.2 Transform Methods	p. 106
6.3 Series Expansion Methods	p. 108
6.4 Optimization Criteria	p. 111
6.5 Blob Basis Functions	p. 119
6.6 Computational Efficiency	p. 122
7 Backprojection	p. 125
7.1 Continuous Backprojection	p. 125
7.2 Implementation of the Backprojection Operator	p. 127
7.3 Discrete Backprojection	p. 131
8 Filtered Backprojection for Parallel Beams	p. 135
8.1 Convolutions, Hilbert Transforms, Regularization	p. 135
8.2 Derivation of the FBP Method	p. 139
8.3 Implementation of the FBP Method	p. 140
8.4 Fourier Transforms	p. 143
5.5 Sampling and Interpolation	p. 146
8.6 The Choice of Convolving and Interpolating Functions	p. 147
8.7 Why So Popular?	p. 157
9 Other Transform Methods for Parallel Beams	p. 159
9.1 Two-Dimensional Fourier Transforms	p. 159
9.2 The Fourier Method of Reconstruction	p. 161
9.3 Linograms	p. 166
9.4 Rho-Filtered Layergram	p. I73
10 Filtered Backprojection for Divergent Beams	p. 177
10.1 The Divergent Beam FBP Algorithm	p. 177
10.2 Choice of the Window Function	p. 181
10.3 Point Response Function	p. 182
10.4 Noise Reconstruction	p. 186
10.5 Comparison of Algorithms Based on Reconstructions	p. 188
11 Algebraic Reconstruction Techniques	p. 193
11.1 What is Art?	p. 193
11.2 Relaxation Methods for Solving Systems of Inequalities and Equalities	p. 196
11.3 Additive Art	p. 201
11.4 Tricks	p. 205
11.5 Efficacy of Art	p. 210
12 Quadratic Optimization Methods	p. 217
12.1 Mathematical Background to Quadratic Optimization	p. 217
12.2 Richardson's Method for Solving Systems of Equations	p. 221
12.3 Smoothing Matrices	p. 224
12.4 Implementation of Richardson's Methods for Image Reconstruction	p. 226
12.5 A Demonstration of Quadratic Optimization	p. 227
13 Truly Three-Dimensional Reconstruction	p. 235
13.1 Three-Dimensional Series Expansion	p. 236
13.2 Dynamically Changing 3D Phantoms and Their Projections	p. 237
13.3 Three-Dimensional Reconstructions of the Dynamic Phantom	p. 240
14 Three-Dimensional Display of Organs	p. 243
14.1 The Basic Approach	p. 243
14.2 Boundary Detection	p. 246
14.3 Hidden Surface Removal	p. 251
14.4 Shading	p. 253
15 Mathematical Background	p. 259
15.1 The Dimensionality of the Linear Attenuation Coefficient	p. 259
15.2 The Line Integral of the Relative Linear Attenuation	p. 260
15.3 The Radon Inversion Formula	p. 261
15.4 A Picture is Not Uniquely Determined by a Finite Number of Its Views	p. 265
15.5 Analysis of the Photon Statistics	p. 267
15.6 The Integral Expression for Polychromatic Ray Sums	p. 269
15.7 Proof of the Regularization Theorem	p. 270
15.8 Convergence of the Relaxation Method for Inequalities	p. 273
References	p. 277
Index	p. 293

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