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by Gengsheng Lawrence Zeng
Image Reconstruction
Cover
Title Page
Copyright
Preface
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
Bibliography
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
2.3.6 Method 6
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 FBP 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
2.6.8 Expression of the derivative–backprojection–Hilbert transform algorithm
2.6.9 Derivation of the backprojection–derivative–Hilbert transform algorithm
2.7 Worked examples
2.8 Summary
Problems
Bibliography
3 Fan-beam image reconstruction
3.1 Fan-beam geometry and the 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
3.4.3 Expression for the Parker weights
3.4.4 Errors caused by finite bandwidth implementation
3.5 Worked examples
3.6 Summary
Problems
Bibliography
4 Transmission and emission tomography
4.1 X-ray computed tomography
4.2 Positron emission tomography and single-photon emission computed tomography
4.3 Noise propagation in reconstruction
4.3.1 Noise variance of emission data
4.3.2 Noise variance of transmission data
4.3.3 Noise propagation in an FBP algorithm
4.4 Attenuation correction for emission tomography
4.4.1 PET
4.4.2 SPECT: Tretiak–Metz FBP algorithm for uniform attenuation
4.4.3 SPECT: Inouye’s algorithm for uniform attenuation
4.5 Mathematical expressions
4.5.1 Expression for Tretiak–Metz FBP algorithm
4.5.2 Derivation for Inouye’s algorithm
4.5.3 Rullgård’s derivative-then-backprojection algorithm for uniform attenuation
4.5.4 Novikov–Natterer FBP algorithm for nonuniform attenuation SPECT
4.6 Worked examples
4.7 Summary
Problems
Bibliography
5 Three-dimensional 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 FBP algorithm for parallel line-integral data
5.4.3 Three-dimensional Radon inversion formula (FBP algorithm)
5.4.4 Three-dimensional 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 Katsevich’s algorithm
5.5 Worked examples
5.6 Summary
Problems
Bibliography
6 Iterative reconstruction
6.1 Solving a system of linear equations
6.2 Algebraic reconstruction technique
6.3 Gradient descent algorithms
6.3.1 The gradient descent algorithm
6.3.2 The Landweber algorithm
6.3.3 The conjugate gradient algorithm
6.4 ML-EM algorithms
6.5 OS-EM 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 (Bayesian)
6.9 Mathematical expressions
6.9.1 ART
6.9.2 The Landweber algorithm
6.9.3 CG algorithm
6.9.4 ML-EM
6.9.5 OS-EM
6.9.6 MAP (Green’s one-step late algorithm)
6.9.7 Matched and unmatched projector/backprojector pairs
6.10 Reconstruction using highly undersampled data
6.11 Worked examples
6.12 Summary
Problems
Bibliography
7 MRI reconstruction
7.1 The “M”
7.2 The “R”
7.3 The “I”
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 Image reconstruction for MRI
7.5.1 Fourier reconstruction
7.5.2 Iterative reconstruction
7.6 Worked examples
7.7 Summary
Problems
Bibliography
8 Using FBP to perform iterative reconstruction
8.1 The Landweber algorithm: From recursive form to non-recursive form
8.2 The Landweber algorithm: From non-recursive form to closed form
8.3 The Landweber algorithm: From closed form to backprojection-then-filtering algorithm
8.3.1 Implementation of (ATA)–1 in the Fourier domain
8.3.2 Implementation of I –(I – αATA)k in the Fourier domain
8.3.3 Landweber algorithm: Backprojection-then-filtering algorithm
8.3.4 Numerical examples of the window function
8.4 The Landweber algorithm: The weighted FBP algorithm
8.4.1 Landweber algorithm: FBP without noise weighting
8.4.2 Landweber algorithm: FBP with view-based noise weighting
8.4.3 Landweber algorithm: FBP with ray-based noise weighting
8.5 FBP algorithm with quadratic constraints
8.5.1 Example of minimum norm-constrained FBP
8.5.2 Example of reference image-constrained FBP
8.6 Convolution backprojection
8.7 Non-quadratic constraints
8.8 A viewpoint from calculus of variations
8.9 Summary
Problems
Bibliography
Index
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Front Matter
Gengsheng Lawrence Zeng
Image Reconstruction
De Gruyter Graduate
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