Image Reconstruction Applications in Medical Sciences
This book introduces the classical and modern image reconstruction technologies.It covers topics in two-dimensional (2D) parallel-beam and fan-beam imaging, three-dimensional (3D) parallel ray, parallel plane, and cone-beam imaging.Both analytical and iterative methods are presented.
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| Format: | E-Book |
| Sprache: | Englisch |
| Veröffentlicht: |
Germany
De Gruyter
2017
Walter de Gruyter GmbH |
| Ausgabe: | 1 |
| Schriftenreihe: | De Gruyter Textbook |
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| ISBN: | 9783110500592, 3110500590, 9783110500486, 3110500485 |
| Online-Zugang: | Volltext |
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Inhaltsangabe:
- Intro -- 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 -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. MR Ireconstruction -- 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 - !ATA)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
- 1. Basic principles of tomography --
- Contents --
- 3. Fan-beam image reconstruction --
- 4. Transmission and emission tomography --
- Preface --
- 7. MRI reconstruction --
- 8. Using FBP to perform iterative reconstruction --
- Frontmatter --
- 2. Parallel-beam image reconstruction --
- Index
- 5. Three-dimensional image reconstruction --
- 6. Iterative reconstruction --