- Cover
- Title Page
- Copyright
- Preface
- Contents
- 1 Basic principles of tomography
- 2 Parallel-beam image reconstruction
- 2.1 Fourier transform
- 2.2 Central slice theorem
- 2.3 Reconstruction algorithms
- 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
- 4 Transmission and emission tomography
- 5 Three-dimensional image reconstruction
- 5.1 Parallel line-integral data
- 5.2 Parallel plane-integral data
- 5.3 Cone-beam data
- 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
- 6.4 ML-EM algorithms
- 6.5 OS-EM algorithm
- 6.6 Noise handling
- 6.7 Noise modeling as a likelihood function
- 6.8 Including prior knowledge (Bayesian)
- 6.9 Mathematical expressions
- 6.10 Reconstruction using highly undersampled data
- 6.11 Worked examples
- 6.12 Summary
- Problems
- Bibliography
- 7 MRI reconstruction
- 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
- 8.4 The Landweber algorithm: The weighted FBP algorithm
- 8.5 FBP algorithm with quadratic constraints
- 8.6 Convolution backprojection
- 8.7 Non-quadratic constraints
- 8.8 A viewpoint from calculus of variations
- 8.9 Summary
- Problems
- Bibliography
- Index
In Loving Memory of My Parents
Yuqing Lu and Yiduo Zeng
To My Family
Ya Li
Andrew Fang
Kathy Fang
Megan Zeng
-
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