Table of contents
1 Basic Principles of Tomography
1.1 Tomography
1.2 Projection
1.3 Image Reconstruction
1.4 Backprojection
* 1.5 Mathematical Expressions
1.5.1 Projection
1.5.2 Backprojection
1.5.3 The Dirac δ-function
1.6 Worked Examples
1.7 Summary
Problems
References
2 Parallel-Beam Image Reconstruction
2.1 Fourier Transform
2.2 Central Slice Theorem
2.3 Reconstruction Algorithms
2.3.1 Method 1
2.3.2 Method 2
2.3.3 Method 3
2.3.4 Method 4
2.3.5 Method 5
2.4 A Computer Simulation
*2.5 ROI Reconstruction with Truncated Projections
*2.6 Mathematical Expressions
2.6.1 The Fourier Transform and Convolution
2.6.2 The Hilbert Transform and the Finite Hilbert Transform
2.6.3 Proof of the Central Slice Theorem
2.6.4 Derivation of the Filtered Backprojection Algorithm
2.6.5 Expression of the Convolution Backprojection Algorithm
2.6.6 Expression of the Radon Inversion Formula
2.6.7 Derivation of the Backprojection-then-Filtering Algorithm
2.7 Worked Examples
2.8 Summary
Problems
References
3 Fan-Beam Image Reconstruction
3.1 Fan-Beam Geometry and Point Spread Function
3.2 Parallel-Beam to Fan-Beam Algorithm Conversion
3.3 Short Scan
*3.4 Mathematical Expressions
3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm
3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform
3.5 Worked Examples
3.6 Summary
Problems
References
4 Transmission and Emission Tomography
4.1 X-Ray Computed Tomography
4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography
4.3 Attenuation Correction for Emission Tomography
*4.4 Mathematical Expressions
4.5 Worked Examples
4.6 Summary
Problems
References
5 3D Image Reconstruction
5.1 Parallel Line-Integral Data
5.1.1 Backprojection-then-Filtering
5.1.2 Filtered Backprojection
5.2 Parallel Plane-Integral Data
5.3 Cone-Beam Data
5.3.1 Feldkamp\'s Algorithm
5.3.2 Grangeat\'s Algorithm
5.3.3 Katsevich\'s Algorithm
*5.4 Mathematical Expressions
5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data
5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data
5.4.3 3D Radon Inversion Formula
5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data
5.4.5 Feldkamp\'s Algorithm
5.4.6 Tuy\'s Relationship
5.4.7 Grangeat\'s Relationship
5.4.8 Katsevieh\'s Algorithm
5.5 Worked Examples
5.6 Summary
Problems
References
6 Iterative Reconstruction
6.1 Solving a System of Linear Equations
6.2 Algebraic Reconstruction Technique
6.3 Gradient Descent Algorithms
6.4 Maximum-Likelihood Expectation-Maximization Algorithms
6.5 Ordered-Subset Expectation-Maximization Algorithm
6.6 Noise Handling
6.6.1 Analytical Methods——Windowing
6.6.2 Iterative Methods——Stopping Early
6.6.3 Iterative Methods——Choosing Pixels
6.6.4 Iterative Methods——Accurate Modeling
6.7 Noise Modeling as a Likelihood Function
6.8 Including Prior Knowledge
*6.9 Mathematical Expressions
6.9.1 ART
6.9.2 Conjugate Gradient Algorithm
6.9.3 ML-EM
6.9.4 OS-EM
6.9.5 Green\'s One-Step Late Algorithm
6.9.6 Matched and Unmatched Projector/Backprojector Pairs
*6.10 Reconstruction Using Highly Undersampled Data with 10 Minimization
6.11 Worked Examples
6.12 Summary
Problems
References
7 MRI Reconstruction
7.1 The \"M\"
7.2 The \"R\"
7.3 The \"T\"
7.3.1 To Obtain z-Information——Slice Selection
7.3.2 To Obtain x-Information——Frequency Encoding
7.3.3 To Obtain y-Information——Phase Encoding
*7.4 Mathematical Expressions
7.5 Worked Examples
7.6 Summary
Problems
References
Index
1.1 Tomography
1.2 Projection
1.3 Image Reconstruction
1.4 Backprojection
* 1.5 Mathematical Expressions
1.5.1 Projection
1.5.2 Backprojection
1.5.3 The Dirac δ-function
1.6 Worked Examples
1.7 Summary
Problems
References
2 Parallel-Beam Image Reconstruction
2.1 Fourier Transform
2.2 Central Slice Theorem
2.3 Reconstruction Algorithms
2.3.1 Method 1
2.3.2 Method 2
2.3.3 Method 3
2.3.4 Method 4
2.3.5 Method 5
2.4 A Computer Simulation
*2.5 ROI Reconstruction with Truncated Projections
*2.6 Mathematical Expressions
2.6.1 The Fourier Transform and Convolution
2.6.2 The Hilbert Transform and the Finite Hilbert Transform
2.6.3 Proof of the Central Slice Theorem
2.6.4 Derivation of the Filtered Backprojection Algorithm
2.6.5 Expression of the Convolution Backprojection Algorithm
2.6.6 Expression of the Radon Inversion Formula
2.6.7 Derivation of the Backprojection-then-Filtering Algorithm
2.7 Worked Examples
2.8 Summary
Problems
References
3 Fan-Beam Image Reconstruction
3.1 Fan-Beam Geometry and Point Spread Function
3.2 Parallel-Beam to Fan-Beam Algorithm Conversion
3.3 Short Scan
*3.4 Mathematical Expressions
3.4.1 Derivation of a Filtered Backprojection Fan-Beam Algorithm
3.4.2 A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform
3.5 Worked Examples
3.6 Summary
Problems
References
4 Transmission and Emission Tomography
4.1 X-Ray Computed Tomography
4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography
4.3 Attenuation Correction for Emission Tomography
*4.4 Mathematical Expressions
4.5 Worked Examples
4.6 Summary
Problems
References
5 3D Image Reconstruction
5.1 Parallel Line-Integral Data
5.1.1 Backprojection-then-Filtering
5.1.2 Filtered Backprojection
5.2 Parallel Plane-Integral Data
5.3 Cone-Beam Data
5.3.1 Feldkamp\'s Algorithm
5.3.2 Grangeat\'s Algorithm
5.3.3 Katsevich\'s Algorithm
*5.4 Mathematical Expressions
5.4.1 Backprojection-then-Filtering for Parallel Line-Integral Data
5.4.2 Filtered Backprojection Algorithm for Parallel Line-Integral Data
5.4.3 3D Radon Inversion Formula
5.4.4 3D Backprojection-then-Filtering Algorithm for Radon Data
5.4.5 Feldkamp\'s Algorithm
5.4.6 Tuy\'s Relationship
5.4.7 Grangeat\'s Relationship
5.4.8 Katsevieh\'s Algorithm
5.5 Worked Examples
5.6 Summary
Problems
References
6 Iterative Reconstruction
6.1 Solving a System of Linear Equations
6.2 Algebraic Reconstruction Technique
6.3 Gradient Descent Algorithms
6.4 Maximum-Likelihood Expectation-Maximization Algorithms
6.5 Ordered-Subset Expectation-Maximization Algorithm
6.6 Noise Handling
6.6.1 Analytical Methods——Windowing
6.6.2 Iterative Methods——Stopping Early
6.6.3 Iterative Methods——Choosing Pixels
6.6.4 Iterative Methods——Accurate Modeling
6.7 Noise Modeling as a Likelihood Function
6.8 Including Prior Knowledge
*6.9 Mathematical Expressions
6.9.1 ART
6.9.2 Conjugate Gradient Algorithm
6.9.3 ML-EM
6.9.4 OS-EM
6.9.5 Green\'s One-Step Late Algorithm
6.9.6 Matched and Unmatched Projector/Backprojector Pairs
*6.10 Reconstruction Using Highly Undersampled Data with 10 Minimization
6.11 Worked Examples
6.12 Summary
Problems
References
7 MRI Reconstruction
7.1 The \"M\"
7.2 The \"R\"
7.3 The \"T\"
7.3.1 To Obtain z-Information——Slice Selection
7.3.2 To Obtain x-Information——Frequency Encoding
7.3.3 To Obtain y-Information——Phase Encoding
*7.4 Mathematical Expressions
7.5 Worked Examples
7.6 Summary
Problems
References
Index