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. The applicati...

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Main Author Zeng, Gengsheng Lawrence
Format eBook
LanguageEnglish
Published Germany De Gruyter 2017
Walter de Gruyter GmbH
Edition1
SeriesDe Gruyter Textbook
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ISBN9783110500592
3110500590
9783110500486
3110500485
DOI10.1515/9783110500592

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Abstract 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. The applications in X-ray CT, SPECT (single photon emission computed tomography), PET (positron emission tomography), and MRI (magnetic resonance imaging) are discussed. Contemporary research results in exact region-of-interest (ROI) reconstruction with truncated projections, Katsevich’s cone-beam filtered backprojection algorithm, and reconstruction with highly under-sampled data are included. The last chapter of the book is devoted to the techniques of using a fast analytical algorithm to reconstruct an image that is equivalent to an iterative reconstruction. These techniques are the author’s most recent research results. This book is intended for students, engineers, and researchers who are interested in medical image reconstruction. Written in a non-mathematical way, this book provides an easy access to modern mathematical methods in medical imaging. Table of Content:Chapter 1 Basic Principles of Tomography1.1 Tomography1.2 Projection1.3 Image Reconstruction1.4 Backprojection1.5 Mathematical ExpressionsProblemsReferencesChapter 2 Parallel-Beam Image Reconstruction2.1 Fourier Transform2.2 Central Slice Theorem2.3 Reconstruction Algorithms2.4 A Computer Simulation2.5 ROI Reconstruction with Truncated Projections2.6 Mathematical Expressions (The Fourier Transform and Convolution , The Hilbert Transform and the Finite Hilbert Transform , Proof of the Central Slice Theorem, Derivation of the Filtered Backprojection Algorithm , Expression of the Convolution Backprojection Algorithm, Expression of the Radon Inversion Formula ,Derivation of the Backprojection-then-Filtering AlgorithmProblemsReferencesChapter 3 Fan-Beam Image Reconstruction3.1 Fan-Beam Geometry and Point Spread Function3.2 Parallel-Beam to Fan-Beam Algorithm Conversion3.3 Short Scan3.4 Mathematical Expressions (Derivation of a Filtered Backprojection Fan-Beam Algorithm, A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform)ProblemsReferencesChapter 4 Transmission and Emission Tomography4.1 X-Ray Computed Tomography4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography4.3 Attenuation Correction for Emission Tomography4.4 Mathematical ExpressionsProblemsReferencesChapter 5 3D Image Reconstruction5.1 Parallel Line-Integral Data5.2 Parallel Plane-Integral Data5.3 Cone-Beam Data (Feldkamp's Algorithm, Grangeat's Algorithm, Katsevich's Algorithm)5.4 Mathematical Expressions (Backprojection-then-Filtering for Parallel Line-Integral Data, Filtered Backprojection Algorithm for Parallel Line-Integral Data, 3D Radon Inversion Formula, 3D Backprojection-then-Filtering Algorithm for Radon Data, Feldkamp's Algorithm, Tuy's Relationship, Grangeat's Relationship, Katsevich’s Algorithm)ProblemsReferencesChapter 6 Iterative Reconstruction6.1 Solving a System of Linear Equations6.2 Algebraic Reconstruction Technique6.3 Gradient Descent Algorithms6.4 Maximum-Likelihood Expectation-Maximization Algorithms6.5 Ordered-Subset Expectation-Maximization Algorithm6.6 Noise Handling (Analytical Methods, Iterative Methods, Iterative Methods)6.7 Noise Modeling as a Likelihood Function6.8 Including Prior Knowledge6.9 Mathematical Expressions (ART, Conjugate Gradient Algorithm, ML-EM, OS-EM, Green’s One-Step Late Algorithm, Matched and Unmatched Projector/Backprojector Pairs )6.10 Reconstruction Using Highly Undersampled Data with l0 MinimizationProblemsReferencesChapter 7 MRI Reconstruction7.1 The 'M'7.2 The 'R'7.3 The
AbstractList 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. The applications in X-ray CT, SPECT (single photon emission computed tomography), PET (positron emission tomography), and MRI (magnetic resonance imaging) are discussed. Contemporary research results in exact region-of-interest (ROI) reconstruction with truncated projections, Katsevich's cone-beam filtered backprojection algorithm, and reconstruction with highly under-sampled data are included. The last chapter of the book is devoted to the techniques of using a fast analytical algorithm to reconstruct an image that is equivalent to an iterative reconstruction. These techniques are the author's most recent research results. This book is intended for students, engineers, and researchers who are interested in medical image reconstruction. Written in a non-mathematical way, this book provides an easy access to modern mathematical methods in medical imaging. Table of Content:Chapter 1 Basic Principles of Tomography1.1 Tomography1.2 Projection1.3 Image Reconstruction1.4 Backprojection1.5 Mathematical ExpressionsProblemsReferencesChapter 2 Parallel-Beam Image Reconstruction2.1 Fourier Transform2.2 Central Slice Theorem2.3 Reconstruction Algorithms2.4 A Computer Simulation2.5 ROI Reconstruction with Truncated Projections2.6 Mathematical Expressions (The Fourier Transform and Convolution , The Hilbert Transform and the Finite Hilbert Transform , Proof of the Central Slice Theorem, Derivation of the Filtered Backprojection Algorithm , Expression of the Convolution Backprojection Algorithm, Expression of the Radon Inversion Formula ,Derivation of the Backprojection-then-Filtering AlgorithmProblemsReferencesChapter 3 Fan-Beam Image Reconstruction3.1 Fan-Beam Geometry and Point Spread Function3.2 Parallel-Beam to Fan-Beam Algorithm Conversion3.3 Short Scan3.4 Mathematical Expressions (Derivation of a Filtered Backprojection Fan-Beam Algorithm, A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform)ProblemsReferencesChapter 4 Transmission and Emission Tomography4.1 X-Ray Computed Tomography4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography4.3 Attenuation Correction for Emission Tomography4.4 Mathematical ExpressionsProblemsReferencesChapter 5 3D Image Reconstruction5.1 Parallel Line-Integral Data5.2 Parallel Plane-Integral Data5.3 Cone-Beam Data (Feldkamp's Algorithm, Grangeat's Algorithm, Katsevich's Algorithm)5.4 Mathematical Expressions (Backprojection-then-Filtering for Parallel Line-Integral Data, Filtered Backprojection Algorithm for Parallel Line-Integral Data, 3D Radon Inversion Formula, 3D Backprojection-then-Filtering Algorithm for Radon Data, Feldkamp's Algorithm, Tuy's Relationship, Grangeat's Relationship, Katsevich's Algorithm)ProblemsReferencesChapter 6 Iterative Reconstruction6.1 Solving a System of Linear Equations6.2 Algebraic Reconstruction Technique6.3 Gradient Descent Algorithms6.4 Maximum-Likelihood Expectation-Maximization Algorithms6.5 Ordered-Subset Expectation-Maximization Algorithm6.6 Noise Handling (Analytical Methods, Iterative Methods, Iterative Methods)6.7 Noise Modeling as a Likelihood Function6.8 Including Prior Knowledge6.9 Mathematical Expressions (ART, Conjugate Gradient Algorithm, ML-EM, OS-EM, Green's One-Step Late Algorithm, Matched and Unmatched Projector/Backprojector Pairs )6.10 Reconstruction Using Highly Undersampled Data with l0 MinimizationProblemsReferencesChapter 7 MRI Reconstruction7.1 The 'M'7.2 The 'R'7.3 The 'I'; (To Obtain z-Information, x-Information, y-Information)7.4 Mathematical ExpressionsProblemsReferencesIndexing
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. The applications in X-ray CT, SPECT (single photon emission computed tomography), PET (positron emission tomography), and MRI (magnetic resonance imaging) are discussed. Contemporary research results in exact region-of-interest (ROI) reconstruction with truncated projections, Katsevich’s cone-beam filtered backprojection algorithm, and reconstruction with highly under-sampled data are included. The last chapter of the book is devoted to the techniques of using a fast analytical algorithm to reconstruct an image that is equivalent to an iterative reconstruction. These techniques are the author’s most recent research results. This book is intended for students, engineers, and researchers who are interested in medical image reconstruction. Written in a non-mathematical way, this book provides an easy access to modern mathematical methods in medical imaging. Table of Content:Chapter 1 Basic Principles of Tomography1.1 Tomography1.2 Projection1.3 Image Reconstruction1.4 Backprojection1.5 Mathematical ExpressionsProblemsReferencesChapter 2 Parallel-Beam Image Reconstruction2.1 Fourier Transform2.2 Central Slice Theorem2.3 Reconstruction Algorithms2.4 A Computer Simulation2.5 ROI Reconstruction with Truncated Projections2.6 Mathematical Expressions (The Fourier Transform and Convolution , The Hilbert Transform and the Finite Hilbert Transform , Proof of the Central Slice Theorem, Derivation of the Filtered Backprojection Algorithm , Expression of the Convolution Backprojection Algorithm, Expression of the Radon Inversion Formula ,Derivation of the Backprojection-then-Filtering AlgorithmProblemsReferencesChapter 3 Fan-Beam Image Reconstruction3.1 Fan-Beam Geometry and Point Spread Function3.2 Parallel-Beam to Fan-Beam Algorithm Conversion3.3 Short Scan3.4 Mathematical Expressions (Derivation of a Filtered Backprojection Fan-Beam Algorithm, A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform)ProblemsReferencesChapter 4 Transmission and Emission Tomography4.1 X-Ray Computed Tomography4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography4.3 Attenuation Correction for Emission Tomography4.4 Mathematical ExpressionsProblemsReferencesChapter 5 3D Image Reconstruction5.1 Parallel Line-Integral Data5.2 Parallel Plane-Integral Data5.3 Cone-Beam Data (Feldkamp's Algorithm, Grangeat's Algorithm, Katsevich's Algorithm)5.4 Mathematical Expressions (Backprojection-then-Filtering for Parallel Line-Integral Data, Filtered Backprojection Algorithm for Parallel Line-Integral Data, 3D Radon Inversion Formula, 3D Backprojection-then-Filtering Algorithm for Radon Data, Feldkamp's Algorithm, Tuy's Relationship, Grangeat's Relationship, Katsevich’s Algorithm)ProblemsReferencesChapter 6 Iterative Reconstruction6.1 Solving a System of Linear Equations6.2 Algebraic Reconstruction Technique6.3 Gradient Descent Algorithms6.4 Maximum-Likelihood Expectation-Maximization Algorithms6.5 Ordered-Subset Expectation-Maximization Algorithm6.6 Noise Handling (Analytical Methods, Iterative Methods, Iterative Methods)6.7 Noise Modeling as a Likelihood Function6.8 Including Prior Knowledge6.9 Mathematical Expressions (ART, Conjugate Gradient Algorithm, ML-EM, OS-EM, Green’s One-Step Late Algorithm, Matched and Unmatched Projector/Backprojector Pairs )6.10 Reconstruction Using Highly Undersampled Data with l0 MinimizationProblemsReferencesChapter 7 MRI Reconstruction7.1 The 'M'7.2 The 'R'7.3 The
Author Zeng, Gengsheng Lawrence
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Snippet This book introduces the classical and modern image reconstruction technologies. It covers topics in two-dimensional (2D) parallel-beam and fan-beam imaging,...
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MEDICAL / Diagnostic Imaging / Radiography
Medical sciences
Subtitle Applications in Medical Sciences
TableOfContents 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 --
Title Image Reconstruction
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