1;Title Page;2 2;Coppyright Page;3 3;Preface;5 4;Table of Contents;6 5;1 Basic Principles of Tomography;11 5.1;1.1 Tomography;11 5.2;1.2 Projection;13 5.3;1.3 Image Reconstruction;16 5.4;1.4 Backprojection;18 5.5;*1.5 Mathematical Expressions;20 5.5.1;1.5.1 Projection;20 5.5.2;1.5.2 Backprojection;21 5.5.3;1.5.3 The Dirac d-function;22 5.6;1.6 Worked Examples;24 5.7;1.7 Summary;27 5.8;Problems;28 5.9;References;29 6;2 Parallel-Beam Image Reconstruction;30 6.1;2.1 Fourier Transform;30 6.2;2.2 Central Slice Theorem;31 6.3;2.3 Reconstruction Algorithms;34 6.3.1;2.3.1 Method 1;34 6.3.2;2.3.2 Method 2;35 6.3.3;2.3.3 Method 3;36 6.3.4;2.3.4 Method 4;37 6.3.5;2.3.5 Method 5;37 6.4;2.4 A Computer Simulation;39 6.5;*2.5 ROI Reconstruction with Truncated Projections;40 6.6;*2.6 Mathematical Expressions;45 6.6.1;2.6.1 The Fourier Transform and Convolution;45 6.6.2;2.6.2 The Hilbert Transform and the Finite Hilbert Transform;45 6.6.3;2.6.3 Proof of the Central Slice Theorem;48 6.6.4;2.6.4 Derivation of the Filtered Backprojection Algorithm;49 6.6.5;2.6.5 Expression of the Convolution Backprojection Algorithm;50 6.6.6;2.6.6 Expression of the Radon Inversion Formula;50 6.6.7;2.6.7 Derivation of the Backprojection-then-Filtering Algorithm;50 6.7;2.7 Worked Examples;51 6.8;2.8 Summary;54 6.9;Problems;55 6.10;References;55 7;3 Fan-Beam Image Reconstruction;57 7.1;3.1 Fan-Beam Geometry and Point Spread Function;57 7.2;3.2 Parallel-Beam to Fan-Beam Algorithm Conversion;60 7.3;3.3 Short Scan;62 7.4;*3.4 Mathematical Expressions;64 7.4.1;3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm;65 7.4.2;3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform;66 7.5;3.5 Worked Examples;68 7.6;3.6 Summary;71 7.7;Problems;72 7.8;References;73 8;4 Transmission and Emission Tomography;75 8.1;4.1 X-Ray Computed Tomography;75 8.2;4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography;79 8.3;4.3 Attenuation Correction for Emission Tomography;83 8.4;*4.4 Mathematical Expressions;87 8.5;4.5 Worked Examples;89 8.6;4.6 Summary;91 8.7;Problems;91 8.8;References;92 9;5 3D Image Reconstruction;94 9.1;5.1 Parallel Line-Integral Data;94 9.1.1;5.1.1 Backprojection-then-Filtering;97 9.1.2;5.1.2 Filtered Backprojection;98 9.2;5.2 Parallel Plane-Integral Data;99 9.3;5.3 Cone-Beam Data;101 9.3.1;5.3.1 Feldkamp's Algorithm;102 9.3.2;5.3.2 Grangeat's Algorithm;103 9.3.3;5.3.3 Katsevich's Algorithm;104 9.4;*5.4 Mathematical Expressions;108 9.4.1;5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data;109 9.4.2;5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data;110 9.4.3;5.4.3 3D Radon Inversion Formula;111 9.4.4;5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data;111 9.4.5;5.4.5 Feldkamp's Algorithm;112 9.4.6;5.4.6 Tuy's Relationship;113 9.4.7;5.4.7 Grangeat's Relationship;115 9.4.8;5.4.8 Katsevich's Algorithm;118 9.5;5.5 Worked Examples;124 9.6;5.6 Summary;126 9.7;Problems;127 9.8;References;128 10;6 Iterative Reconstruction;131 10.1;6.1 Solving a System of Linear Equations;131 10.2;6.2 Algebraic Reconstruction Technique;136 10.3;6.3 Gradient Descent Algorithms;137 10.4;6.4 Maximum-Likelihood Expectation-Maximization Algorithms;140 10.5;6.5 Ordered-Subset Expectation-Maximization Algorithm;141 10.6;6.6 Noise Handling;142 10.6.1;6.6.1 Analytical Methods-Windowing;142 10.6.2;6.6.2 Iterative Methods-Stopping Early;143 10.6.3;6.6.3 Iterative Methods-Choosing Pixels;144 10.6.4;6.6.4 Iterative Methods-Accurate Modeling;146 10.7;6.7 Noise Modeling as a Likelihood Function;147 10.8;6.8 Including Prior Knowledge;149 10.9;*6.9 Mathematical Expressions;151 10.9.1;6.9.1 ART;151 10.9.2;6.9.2 Conjugate Gradient Algorithm;152 10.9.3;6.9.3 ML-EM;154 10.9.4;6.9.4 OS-EM;157 10.9.5;6.9.5 Green's One-Step Late Algorithm;157 10.9.6;6.9.6 Matched and Unmatched Projector/Backprojector Pairs;157 10.10;*6.10 Reconstruction Using Highly Undersampled Data with l 0 Minimization;159 10.11;6.11 Worked Examp