Medical Image Reconstruction: A Conceptual Tutorial

Medical Image Reconstruction: A Conceptual Tutorial......Page 1
Front-matter......Page 2
Title Page......Page 4
Copyright Page......Page 5
Preface......Page 8
Table of Contens......Page 10
1.1 Tomography......Page 16
1.2 Projection......Page 18
1.3 Image Reconstruction......Page 21
1.4 Backprojection......Page 23
1.5.1 Projection......Page 25
1.5.2 Backprojection......Page 26
1.5.3 The Dirac δ-function......Page 27
1.6 Worked Examples......Page 29
1.7 Summary......Page 32
Problems......Page 33
References......Page 34
2.1 Fourier Transform......Page 36
2.2 Central Slice Theorem......Page 37
2.3.1 Method 1......Page 40
2.3.2 Method 2......Page 41
2.3.3 Method 3......Page 42
2.3.5 Method 5......Page 43
2.4 A Computer Simulation......Page 45
∗2.5 ROI Reconstruction with Truncated Projections......Page 46
2.6.2 The Hilbert Transform and the Finite Hilbert Transform......Page 51
2.6.3 Proof of the Central Slice Theorem......Page 54
2.6.4 Derivation of the Filtered Backprojection Algorithm......Page 55
2.6.7 Derivation of the Backprojection-then-Filtering Algorithm......Page 56
2.7 Worked Examples......Page 57
2.8 Summary......Page 60
References......Page 61
3.1 Fan-Beam Geometry and Point Spread Function......Page 64
3.2 Parallel-Beam to Fan-Beam Algorithm Conversion......Page 67
3.3 Short Scan......Page 69
∗3.4 Mathematical Expressions......Page 71
3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm......Page 72
3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform......Page 73
3.5 Worked Examples......Page 75
3.6 Summary......Page 78
Problems......Page 79
References......Page 80
4.1 X-Ray Computed Tomography......Page 82
4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography......Page 86
4.3 Attenuation Correction for Emission Tomography......Page 90
∗4.4 Mathematical Expressions......Page 94
4.5 Worked Examples......Page 96
Problems......Page 98
References......Page 99
5.1 Parallel Line-Integral Data......Page 102
5.1.1 Backprojection-then-Filtering......Page 105
5.1.2 Filtered Backprojection......Page 106
5.2 Parallel Plane-Integral Data......Page 107
5.3 Cone-Beam Data......Page 109
5.3.1 Feldkamp’s Algorithm......Page 110
5.3.2 Grangeat’s Algorithm......Page 111
5.3.3 Katsevich’s Algorithm......Page 112
∗5.4 Mathematical Expressions......Page 116
5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data......Page 117
5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data......Page 118
5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data......Page 119
5.4.5 Feldkamp’s Algorithm......Page 120
5.4.6 Tuy’s Relationship......Page 121
5.4.7 Grangeat’s Relationship......Page 123
5.4.8 Katsevich’s Algorithm......Page 126
5.5 Worked Examples......Page 132
5.6 Summary......Page 134
Problems......Page 135
References......Page 136
6.1 Solving a System of Linear Equations......Page 140
6.2 Algebraic Reconstruction Technique......Page 145
6.3 Gradient Descent Algorithms......Page 146
6.4 Maximum-Likelihood Expectation-Maximization Algorithms......Page 149
6.5 Ordered-Subset Expectation-Maximization Algorithm......Page 150
6.6.1 Analytical Methods—Windowing......Page 151
6.6.2 Iterative Methods—Stopping Early......Page 152
6.6.3 Iterative Methods—Choosing Pixels......Page 153
6.6.4 Iterative Methods—Accurate Modeling......Page 155
6.7 Noise Modeling as a Likelihood Function......Page 156
6.8 Including Prior Knowledge......Page 158
6.9.1 ART......Page 160
6.9.2 Conjugate Gradient Algorithm......Page 161
6.9.3 ML-EM......Page 163
6.9.6 Matched and Unmatched Projector/Backprojector Pairs......Page 166
∗6.10 Reconstruction Using Highly Undersampled Data with l 0 Minimization......Page 168
6.11 Worked Examples......Page 171
6.12 Summary......Page 182
Problems......Page 183
References......Page 185
7.1 The “M”......Page 190
7.2 The “R”......Page 192
7.3.1 To Obtain z-Information—Slice Selection......Page 195
7.3.2 To Obtain x-Information—Frequency Encoding......Page 197
7.3.3 To Obtain y-Information—Phase Encoding......Page 198
∗7.4 Mathematical Expressions......Page 200
7.5 Worked Examples......Page 203
7.6 Summary......Page 205
Problems......Page 206
References......Page 207
Index......Page 208