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1 Basic Principles of Tomography 1

1.1 Tomography 1

1.2 Projection 3

1.3 Image Reconstruction 6

1.4 Backprojection 8

1.5 Mathematical Expressions 10

1.5.1 Projection 10

1.5.2 Backprojection 11

1.5.3 The Dirac δ-function 12

1.6 Worked Examples 14

1.7 Summary 17

Problems 18

References 19

2 Parallel-Beam Image Reconstruction 21

2.1 Fourier Transform 21

2.2 Central Slice Theorem 22

2.3 Reconstruction Algorithms 25

2.3.1 Method 1 25

2.3.2 Method 2 26

2.3.3 Method 3 27

2.3.4 Method 4 28

2.3.5 Method 5 28

2.4 A Computer Simulation 30

2.5 ROI Reconstruction with Truncated Projections 31

2.6 Mathematical Expressions 36

2.6.1 The Fourier Transform and Convolution 36

2.6.2 The Hilbert Transform and the Finite Hilbert Transform 36

2.6.3 Proof of the Central Slice Theorem 39

2.6.4 Derivation of the Filtered Backprojection Algorithm 40

2.6.5 Expression of the Convolution Backprojection Algorithm 41

2.6.6 Expression of the Radon Inversion Formula 41

2.6.7 Derivation of the Backprojection-then-Filtering Algorithm 41

2.7 Worked Examples 42

2.8 Summary 45

Problems 46

References 46

3 Fan-Beam Image Reconstruction 49

3.1 Fan-Beam Geometry and Point Spread Function 49

3.2 Parallel-Beam to Fan-Beam Algorithm Conversion 52

3.3 Short Scan 54

3.4 Mathematical Expressions 56

3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm 57

3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform 58

3.5 Worked Examples 60

3.6 Summary 63

Problems 64

References 65

4 Transmission and Emission Tomography 67

4.1 X-Ray Computed Tomography 67

4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography 71

4.3 Attenuation Correction for Emission Tomography 75

4.4 Mathematical Expressions 79

4.5 Worked Examples 81

4.6 Summary 83

Problems 83

References 84

5 3D Image Reconstruction 87

5.1 Parallel Line-Integral Data 87

5.1.1 Backprojection-then-Filtering 90

5.1.2 Filtered Backprojection 91

5.2 Parallel Plane-Integral Data 92

5.3 Cone-Beam Data 94

5.3.1 Feldkamp's Algorithm 95

5.3.2 Grangeat's Algorithm 96

5.3.3 Katsevich's Algorithm 97

5.4 Mathematical Expressions 101

5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data 102

5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data 103

5.4.3 3D Radon Inversion Formula 104

5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data 104

5.4.5 Feldkamp's Algorithm 105

5.4.6 Tuy's Relationship 106

5.4.7 Grangeat's Relationship 108

5.4.8 Katsevich's Algorithm 111

5.5 Worked Examples 117

5.6 Summary 119

Problems 120

References 121

6 Iterative Reconstruction 125

6.1 Solving a System of Linear Equations 125

6.2 Algebraic Reconstruction Technique 130

6.3 Gradient Descent Algorithms 131

6.4 Maximum-Likelihood Expectation-Maximization Algorithms 134

6.5 Ordered-Subset Expectation-Maximization Algorithm 135

6.6 Noise Handling 136

6.6.1 Analytical Methods—Windowing 136

6.6.2 Iterative Methods—Stopping Early 137

6.6.3 Iterative Methods——Choosing Pixels 138

6.6.4 Iterative Methods—Accurate Modeling 140

6.7 Noise Modeling as a Likelihood Function 141

6.8 Including Prior Knowledge 143

6.9 Mathematical Expressions 145

6.9.1 ART 145

6.9.2 Conjugate Gradient Algorithm 146

6.9.3 ML-EM 148

6.9.4 OS-EM 151

6.9.5 Green's One-Step Late Algorithm 151

6.9.6 Matched and Unmatched Projector/Backprojector Pairs 151

6.10 Reconstruction Using Highly Undersampled Data with l0 Minimization 153

6.11 Worked Examples 156

6.12 Summary 167

Problems 168

References 170

7 MRI Reconstruction 175

7.1 The"M" 175

7.2 The"R" 177

7.3 The"I" 180

7.3.1 To Obtain z-Information—Slice Selection 180

7.3.2 To Obtain x-Information—Frequency Encoding 182

7.3.3 To Obtain y-Information—Phase Encoding 183

7.4 Mathematical Expressions 185

7.5 Worked Examples 188

7.6 Summary 190

Problems 191

References 192

Index 193