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医学图像重建教材Medical Image Reconstruction: A Conceptual Tutorial.pdf

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    Medical Image Reconstruction: A Conceptual Tutorial
    Front-matter
    Title Page
    Copyright Page
    Preface
    Table of Contens
    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 δ-function
    1.6 Worked Examples
    1.7 Summary
    Problems
    References
    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
    2.4 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 Filtered Backprojection 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
    2.7 Worked Examples
    2.8 Summary
    Problems
    References
    3 Fan-Beam Image Reconstruction
    3.1 Fan-Beam Geometry and 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
    3.5 Worked Examples
    3.6 Summary
    Problems
    References
    4 Transmission and Emission Tomography
    4.1 X-Ray Computed Tomography
    4.2 Positron Emission Tomography and Single Photon Emission Computed Tomography
    4.3 Attenuation Correction for Emission Tomography
    ∗4.4 Mathematical Expressions
    4.5 Worked Examples
    4.6 Summary
    Problems
    References
    5 3D 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 Filtered Backprojection Algorithm for Parallel Line-Integral Data
    5.4.3 3D Radon Inversion Formula
    5.4.4 3D 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
    References
    6 Iterative Reconstruction
    6.1 Solving a System of Linear Equations
    6.2 Algebraic Reconstruction Technique
    6.3 Gradient Descent Algorithms
    6.4 Maximum-Likelihood Expectation-Maximization Algorithms
    6.5 Ordered-Subset Expectation-Maximization 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
    ∗6.9 Mathematical Expressions
    6.9.1 ART
    6.9.2 Conjugate Gradient Algorithm
    6.9.3 ML-EM
    6.9.4 OS-EM
    6.9.5 Green’s One-Step Late Algorithm
    6.9.6 Matched and Unmatched Projector/Backprojector Pairs
    ∗6.10 Reconstruction Using Highly Undersampled Data with l 0 Minimization
    6.11 Worked Examples
    6.12 Summary
    Problems
    References
    7 MRI Reconstruction
    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 Worked Examples
    7.6 Summary
    Problems
    References
    Index
    Gengsheng Lawrence Zeng Medical Image Reconstruction A Conceptual Tutorial
    Gengsheng Lawrence Zeng Medical Image Reconstruction A Conceptual Tutorial With 163 Figures
    Author Prof. Dr. Gengsheng Lawrence Zeng Utah Center for Advanced Imaging Research Department of Radiology University of Utah Salt Lake City, UT 84108, USA E-mail: larry@ucair.med.utah.edu ISBN 978-7-04-020437-7 Higher Education Press, Beijing ISBN 978-3-642-05367-2 e-ISBN 978-3-642-05368-9 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2009937574 c Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Frido Steinen-Broo, EStudio Calamar, Spain Printed on acid-free paper Springer is part of Springer Science + Business Media (www.springer.com)
    This book is dedicated to Ya, Andrew, Kathy, and Megan
    Preface The first time I heard about image reconstruction was twenty years ago I came to the University of Utah as a post-doctoral fellow in the Department of Radiology. Dr. Grant Gullberg and Dr. Rolf Clackdoyle gave many lec- tures on image reconstruction and I took notes. Even today I still go back to those notes from time to time. I benefit from those notes significantly. This book is complied together with parts of those notes and some current research papers with most mathematical proofs removed. I am grateful to Dr. Gullberg and Dr. Clackdoyle for introducing me to the wonderful world of image reconstruction. I appreciate Dr. Michel Defrise, Dr. Ge Wang, and Dr. Guang-Hong Chen for their helpful suggestions. I also like to thank my colleagues in the department and in other institutions. I would especially like to thank Kathy Gullberg and Jacob Piatt for proof-reading the drafts. This tutorial text introduces the classical and modern image reconstruc- tion technologies to the general audience. It covers the 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 also discussed. Contemporary research results in exact ROI (region-of-interest) reconstruc- tion with truncated projections, Katsevich’s cone-beam filtered backprojec- tion algorithm, and reconstruction with highly undersampled data with l0-minimization are also included in this book. This book is written in an easy-to-read style, which lets the diagrams do the most talking. The readers who intend to get into medical image recon- struction will gain the general knowledge of the field in a painless way. I hope you enjoy reading it as much as I enjoy writing (and drawing) it. The first time reader can skip the more challenging materials marked by the “∗” sign without interrupting the flow of this book. Gengsheng Lawrence Zeng Salt Lake City August 2009
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