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Medical Image Reconstruction, 2nd Edition

Name: Medical Image Reconstruction, 2nd Edition
Author: Gengsheng Lawrence Zeng
ISBN: 9783111055701

by Gengsheng Lawrence Zeng

July 2023

Beginner

273 pages

8h 4m

English

De Gruyter

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Preface
1 Basic principles of tomography
1.1 Tomography1.2 Projection1.3 Image reconstruction1.4 Backprojection1.5 Mathematical expressions1.5.1 Projection1.5.2 Backprojection1.5.3 The Dirac δ-function1.6 Worked examples1.7 SummaryProblems
2 Parallel-beam image reconstruction
2.1 Fourier transform2.2 Central slice theorem2.3 Reconstruction algorithms2.3.1 Method 12.3.2 Method 22.3.3 Method 32.3.4 Method 42.3.5 Method 52.3.6 Method 62.4 A computer simulation2.5 ROI reconstruction with truncated projections2.6 Mathematical expressions2.6.1 The Fourier transform and convolution2.6.2 The Hilbert transform and the finite Hilbert transform2.6.3 Proof of the central slice theorem2.6.4 Derivation of the FBP algorithm2.6.5 Expression of the convolution backprojection algorithm2.6.6 Expression of the Radon inversion formula2.6.7 Derivation of the backprojection-then-filtering algorithm2.6.8 Expression of the derivative–backprojection–Hilbert transform algorithm2.6.9 Derivation of the backprojection–derivative–Hilbert transform algorithm2.7 Worked examples2.8 SummaryProblems
3 Fan-beam image reconstruction
3.1 Fan-beam geometry and the point spread function3.2 Parallel-beam to fan-beam algorithm conversion3.3 Short scan3.4 Mathematical expressions3.4.1 Derivation of a filtered backprojection fan-beam algorithm3.4.2 A fan-beam algorithm using the derivative and the Hilbert transform3.4.3 Expression for the Parker weights3.4.4 Errors caused by finite bandwidth implementation3.5 Worked examples3.6 SummaryProblems
4 Transmission and emission tomography
4.1 X-ray computed tomography4.2 Positron emission tomography and single-photon emission computed tomography4.3 Noise propagation in reconstruction4.3.1 Noise variance of emission data4.3.2 Noise variance of transmission data4.3.3 Noise propagation in an FBP algorithm4.4 Attenuation correction for emission tomography4.4.1 PET4.4.2 SPECT: Tretiak–Metz FBP algorithm for uniform attenuation4.4.3 SPECT: Inouye’s algorithm for uniform attenuation4.5 Mathematical expressions4.5.1 Expression for Tretiak–Metz FBP algorithm4.5.2 Derivation for Inouye’s algorithm4.5.3 Rullgård’s derivative-then-backprojection algorithm for uniform attenuation4.5.4 Novikov–Natterer FBP algorithm for nonuniform attenuation SPECT4.6 Worked examples4.7 SummaryProblems
5 Three-dimensional image reconstruction
5.1 Parallel line-integral data5.1.1 Backprojection-then-filtering5.1.2 Filtered backprojection5.2 Parallel plane-integral data5.3 Cone-beam data5.3.1 Feldkamp’s algorithm5.3.2 Grangeat’s algorithm5.3.3 Katsevich’s algorithm5.4 Mathematical expressions5.4.1 Backprojection-then-filtering for parallel line-integral data5.4.2 FBP algorithm for parallel line-integral data5.4.3 Three-dimensional Radon inversion formula (FBP algorithm)5.4.4 Three-dimensional backprojection-then-filtering algorithm for Radon data5.4.5 Feldkamp’s algorithm5.4.6 Tuy’s relationship5.4.7 Grangeat’s relationship5.4.8 Katsevich’s algorithm5.5 Radon transform and ray transform in n dimensions5.6 Worked examples5.7 SummaryProblems
6 Iterative reconstruction
6.1 Solving a system of linear equations6.2 Algebraic reconstruction technique6.3 Gradient descent algorithms6.3.1 The gradient descent algorithm6.3.2 The Landweber algorithm6.3.3 The conjugate gradient algorithm6.4 ML-EM algorithms6.5 OS-EM algorithm6.6 Noise handling6.6.1 Analytical methods: windowing6.6.2 Iterative methods: stopping early6.6.3 Iterative methods: choosing pixels6.6.4 Iterative methods: accurate modeling6.7 Noise modeling as a likelihood function6.8 Including prior knowledge (Bayesian)6.9 Mathematical expressions6.9.1 ART6.9.2 The Landweber algorithm6.9.3 CG algorithm6.9.4 ML-EM6.9.5 OS-EM6.9.6 MAP (Green’s one-step-late algorithm)6.9.7 Matched and unmatched projector/backprojector pairs6.10 Reconstruction using highly undersampled data6.11 Worked examples6.12 SummaryProblems
7 MRI reconstruction
7.1 The “M”7.2 The “R”7.3 The “I”7.3.1 To obtain z-information: slice selection7.3.2 To obtain x-information: frequency encoding7.3.3 To obtain y-information: phase encoding7.4 Mathematical expressions7.5 Image reconstruction for MRI7.5.1 Fourier reconstruction7.5.2 Iterative reconstruction7.6 Worked examples7.7 SummaryProblems
8 Using FBP to perform iterative reconstruction
8.1 The Landweber algorithm: from recursive form to nonrecursive form8.2 The Landweber algorithm: from nonrecursive form to closed form8.3 The Landweber algorithm: from closed form to backprojection-then-filtering algorithm8.3.1 Implementation of (AT A)–1 in the Fourier domain8.3.2 Implementation of I – (I – αATA)k in the Fourier domain8.3.3 Landweber algorithm: backprojection-then-filtering algorithm8.3.4 Numerical examples of the window function8.4 The Landweber algorithm: the weighted FBP algorithm8.4.1 Landweber algorithm: FBP without noise weighting8.4.2 Landweber algorithm: FBP with view-based noise weighting8.4.3 Landweber algorithm: FBP with ray-based noise weighting8.5 FBP algorithm with quadratic constraints8.5.1 Example of minimum-norm constrained FBP8.5.2 Example of reference-image constrained FBP8.6 Convolution backprojection8.7 Nonquadratic constraints8.8 A viewpoint from calculus of variations8.9 Noise-weighted FBP algorithm for uniformly attenuated SPECT projections8.10 Other applications of the Landwever-FBP algorithm8.10.1 The Landweber-FBP version of the iterative FBP algorithm8.10.2 Estimation of the initial image’s contributions to the iterative Landweber reconstruction8.10.3 FBP implementation of the immediately after-backprojection filtering8.10.4 Real-time selection of iteration number8.11 SummaryProblems
9 Machine learning
9.1 Analytic algorithms versus iterative algorithms versus machine learning algorithms9.2 Basic principles of machine learning9.2.1 Adaptive linear neuron (ADALINE)9.2.2 Universal approximation9.2.3 A deep model for a continuous univariate function9.2.4 When the input is an n-tuple9.2.5 An n-tuple to m-tuple mapping9.3 Image reconstruction as an n-tuple to m-tuple mapping9.4 Convolutional neural network (CNN) as a matched filter for identification9.5 Convolutional neural network (CNN) as a nonlinear denoise filter9.6 Super-resolution9.7 Training9.8 Attention9.9 Perspectives9.10 SummaryProblem

Index

Content preview from Medical Image Reconstruction, 2nd Edition

4 Transmission and emission tomography

This book considers real imaging systems in this chapter. If the radiation source is outside the patient, the imaging system acquires transmission data. If the radiation sources are inside the patient, the imaging system acquires the emission data. For transmission scans, the image to be obtained is a map (or distribution) of the attenuation coefficients inside the patient. For the emission scans, the image to be obtained is the distribution of the injected isotopes within the patient. Even for emission scans, an additional transmission scan is sometimes required in order to compensate for the attenuation effect of the emission photons. Some attenuation compensation methods for emission imaging are discussed ...

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Medical Image Reconstruction, 2nd Edition

by Gengsheng Lawrence Zeng

4 Transmission and emission tomography

Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.