Medical Image Reconstruction (eBook)

Title Page 2
Coppyright Page 3
Preface 5
Table of Contents 6
1 Basic Principles of Tomography 11
1.1 Tomography 11
1.2 Projection 13
1.3 Image Reconstruction 16
1.4 Backprojection 18
*1.5 Mathematical Expressions 20
1.5.1 Projection 20
1.5.2 Backprojection 21
1.5.3 The Dirac d-function 22
1.6 Worked Examples 24
1.7 Summary 27
Problems 28
References 29
2 Parallel-Beam Image Reconstruction 30
2.1 Fourier Transform 30
2.2 Central Slice Theorem 31
2.3 Reconstruction Algorithms 34
2.3.1 Method 1 34
2.3.2 Method 2 35
2.3.3 Method 3 36
2.3.4 Method 4 37
2.3.5 Method 5 37
2.4 A Computer Simulation 39
*2.5 ROI Reconstruction with Truncated Projections 40
*2.6 Mathematical Expressions 45
2.6.1 The Fourier Transform and Convolution 45
2.6.2 The Hilbert Transform and the Finite Hilbert Transform 45
2.6.3 Proof of the Central Slice Theorem 48
2.6.4 Derivation of the Filtered Backprojection Algorithm 49
2.6.5 Expression of the Convolution Backprojection Algorithm 50
2.6.6 Expression of the Radon Inversion Formula 50
2.6.7 Derivation of the Backprojection-then-Filtering Algorithm 50
2.7 Worked Examples 51
2.8 Summary 54
Problems 55
References 55
3 Fan-Beam Image Reconstruction 57
3.1 Fan-Beam Geometry and Point Spread Function 57
3.2 Parallel-Beam to Fan-Beam Algorithm Conversion 60
3.3 Short Scan 62
*3.4 Mathematical Expressions 64
3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm 65
3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform 66
3.5 Worked Examples 68
3.6 Summary 71
Problems 72
References 73
4 Transmission and Emission Tomography 75
4.1 X-Ray Computed Tomography 75
4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography 79
4.3 Attenuation Correction for Emission Tomography 83
*4.4 Mathematical Expressions 87
4.5 Worked Examples 89
4.6 Summary 91
Problems 91
References 92
5 3D Image Reconstruction 94
5.1 Parallel Line-Integral Data 94
5.1.1 Backprojection-then-Filtering 97
5.1.2 Filtered Backprojection 98
5.2 Parallel Plane-Integral Data 99
5.3 Cone-Beam Data 101
5.3.1 Feldkamp’s Algorithm 102
5.3.2 Grangeat’s Algorithm 103
5.3.3 Katsevich’s Algorithm 104
*5.4 Mathematical Expressions 108
5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data 109
5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data 110
5.4.3 3D Radon Inversion Formula 111
5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data 111
5.4.5 Feldkamp’s Algorithm 112
5.4.6 Tuy’s Relationship 113
5.4.7 Grangeat’s Relationship 115
5.4.8 Katsevich’s Algorithm 118
5.5 Worked Examples 124
5.6 Summary 126
Problems 127
References 128
6 Iterative Reconstruction 131
6.1 Solving a System of Linear Equations 131
6.2 Algebraic Reconstruction Technique 136
6.3 Gradient Descent Algorithms 137
6.4 Maximum-Likelihood Expectation-Maximization Algorithms 140
6.5 Ordered-Subset Expectation-Maximization Algorithm 141
6.6 Noise Handling 142
6.6.1 Analytical Methods—Windowing 142
6.6.2 Iterative Methods—Stopping Early 143
6.6.3 Iterative Methods—Choosing Pixels 144
6.6.4 Iterative Methods—Accurate Modeling 146
6.7 Noise Modeling as a Likelihood Function 147
6.8 Including Prior Knowledge 149
*6.9 Mathematical Expressions 151
6.9.1 ART 151
6.9.2 Conjugate Gradient Algorithm 152
6.9.3 ML-EM 154
6.9.4 OS-EM 157
6.9.5 Green’s One-Step Late Algorithm 157
6.9.6 Matched and Unmatched Projector/Backprojector Pairs 157
*6.10 Reconstruction Using Highly Undersampled Data with l 0 Minimization 159
6.11 Worked Examples 162
6.12 Summary 173
Problems 174
References 176
7 MRI Reconstruction 180
7.1 The “M” 180
7.2 The “R” 182
7.3 The “I” 185
7.3.1 To Obtain z-Information—Slice Selection 185
7.3.2 To Obtain x-Information—Frequency Encoding 187
7.3.3 To Obtain y-Information—Phase Encoding 188
*7.4 Mathematical Expressions 190
7.5 Worked Examples 193
7.6 Summary 195
Problems 196
References 197
Index 198