Contents
1Basic principles of tomography
2Parallel-beam image reconstruction
2.5ROI reconstruction with truncated projections
2.6.1The Fourier transform and convolution
2.6.2The Hilbert transform and the finite Hilbert transform
2.6.3Proof of the central slice theorem
2.6.4Derivation of the FBP algorithm
2.6.5Expression of the convolution backprojection algorithm
2.6.6Expression of the Radon inversion formula
2.6.7Derivation of the backprojection-then-filtering algorithm
2.6.8Expression of the derivative–backprojection–Hilbert transform algorithm
2.6.9Derivation of the backprojection–derivative–Hilbert transform algorithm
3Fan-beam image reconstruction
3.1Fan-beam geometry and the point spread function
3.2Parallel-beam to fan-beam algorithm conversion
3.4.1Derivation of a filtered backprojection fan-beam algorithm
3.4.2A fan-beam algorithm using the derivative and the Hilbert transform
3.4.3Expression for the Parker weights
3.4.4Errors caused by finite bandwidth implementation
4Transmission and emission tomography
4.2Positron emission tomography and single-photon emission computed tomography
4.3Noise propagation in reconstruction
4.3.1Noise variance of emission data
4.3.2Noise variance of transmission data
4.3.3Noise propagation in an FBP algorithm
4.4Attenuation correction for emission tomography
4.4.2SPECT: Tretiak–Metz FBP algorithm for uniform attenuation
4.4.3SPECT: Inouye’s algorithm for uniform attenuation
4.5.1Expression for Tretiak–Metz FBP algorithm
4.5.2Derivation for Inouye’s algorithm
4.5.3Rullgård’s derivative-then-backprojection algorithm for uniform attenuation
4.5.4Novikov–Natterer FBP algorithm for nonuniform attenuation SPECT
5Three-dimensional image reconstruction
5.1Parallel line-integral data
5.1.1Backprojection-then-filtering
5.2Parallel plane-integral data
5.4.1Backprojection-then-filtering for parallel line-integral data
5.4.2FBP algorithm for parallel line-integral data
5.4.3Three-dimensional Radon inversion formula (FBP algorithm)
5.4.4Three-dimensional backprojection-then-filtering algorithm for Radon data
6.1Solving a system of linear equations
6.2Algebraic reconstruction technique
6.3Gradient descent algorithms
6.3.1The gradient descent algorithm
6.3.3The conjugate gradient algorithm
6.6.1Analytical methods – windowing
6.6.2Iterative methods – stopping early
6.6.3Iterative methods – choosing pixels
6.6.4Iterative methods – accurate modeling
6.7Noise modeling as a likelihood function
6.8Including prior knowledge (Bayesian)
6.9.6MAP (Green’s one-step late algorithm)
6.9.7Matched and unmatched projector/backprojector pairs
6.10Reconstruction using highly undersampled data
7.3.1To obtain z-information: slice selection
7.3.2To obtain x-information: frequency encoding
7.3.3To obtain y-information: phase encoding
7.5Image reconstruction for MRI
8Using FBP to perform iterative reconstruction
8.1The Landweber algorithm: From recursive form to non-recursive form
8.2The Landweber algorithm: From non-recursive form to closed form
8.3The Landweber algorithm: From closed form to backprojection-then-filtering algorithm
8.3.1Implementation of (ATA)–1 in the Fourier domain
8.3.2Implementation of I –(I – αATA)k in the Fourier domain
8.3.3Landweber algorithm: Backprojection-then-filtering algorithm
8.3.4Numerical examples of the window function
8.4The Landweber algorithm: The weighted FBP algorithm
8.4.1Landweber algorithm: FBP without noise weighting
8.4.2Landweber algorithm: FBP with view-based noise weighting
8.4.3Landweber algorithm: FBP with ray-based noise weighting
8.5FBP algorithm with quadratic constraints
8.5.1Example of minimum norm-constrained FBP
8.5.2Example of reference image-constrained FBP