Gengsheng Lawrence Zeng. Medical Image Reconstruction. A Conceptual Tutorial

Transcription

1 Gengsheng Lawrence Zeng Medical Image Reconstruction A Conceptual Tutorial

2 Gengsheng Lawrence Zeng Medical Image Reconstruction A Conceptual Tutorial With 163 Figures

3 Author Prof. Dr. Gengsheng Lawrence Zeng Utah Center for Advanced Imaging Research Department of Radiology University of Utah Salt Lake City, UT 84108, USA ISBN Higher Education Press, Beijing ISBN e-isbn Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 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 Printedonacid-freepaper Springer is part of Springer Science + Business Media (

4 This book is dedicated to Ya, Andrew, Kathy, and Megan

5 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 lectures 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 reconstruction technologies to the general audience. It covers the topics in twodimensional (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) reconstruction with truncated projections, Katsevich s cone-beam filtered backprojection algorithm, and reconstruction with highly undersampled data with l 0 -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 reconstruction 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

6 Contents 1 Basic Principles of Tomography Tomography Projection Image Reconstruction Backprojection Mathematical Expressions Projection Backprojection The Dirac δ-function Worked Examples Summary 17 Problems 18 References 19 2 Parallel-Beam Image Reconstruction Fourier Transform Central Slice Theorem Reconstruction Algorithms Method Method Method Method 4 28 The first time reader can skip the more challenging materials marked by the sign without interrupting the flow of this book.

7 x Contents Method A Computer Simulation ROI Reconstruction with Truncated Projections 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 Algorithm Worked Examples Summary 45 Problems 46 References 46 3 Fan-Beam Image Reconstruction Fan-Beam Geometry and Point Spread Function Parallel-Beam to Fan-Beam Algorithm Conversion Short Scan Mathematical Expressions Derivation of a Filtered Backprojection Fan-Beam Algorithm A Fan-Beam Algorithm Using the Derivative and the Hilbert Transform Worked Examples Summary 63 Problems 64 References 65

8 Contents xi 4 Transmission and Emission Tomography X-Ray Computed Tomography Positron Emission Tomography and Single Photon Emission Computed Tomography Attenuation Correction for Emission Tomography Mathematical Expressions Worked Examples Summary 83 Problems 83 References D Image Reconstruction Parallel Line-Integral Data Backprojection-then-Filtering Filtered Backprojection Parallel Plane-Integral Data Cone-Beam Data Feldkamp s Algorithm Grangeat s Algorithm Katsevich s Algorithm Mathematical Expressions Backprojection-then-Filtering for Parallel Line-Integral Data Filtered Backprojection Algorithm for Parallel Line-Integral Data D Radon Inversion Formula D Backprojection-then-Filtering Algorithm for Radon Data Feldkamp s Algorithm Tuy s Relationship Grangeat s Relationship Katsevich s Algorithm Worked Examples 117

9 xii Contents 5.6 Summary 119 Problems 120 References Iterative Reconstruction Solving a System of Linear Equations Algebraic Reconstruction Technique Gradient Descent Algorithms Maximum-Likelihood Expectation-Maximization Algorithms Ordered-Subset Expectation-Maximization Algorithm Noise Handling Analytical Methods Windowing Iterative Methods Stopping Early Iterative Methods Choosing Pixels Iterative Methods Accurate Modeling Noise Modeling as a Likelihood Function Including Prior Knowledge Mathematical Expressions ART Conjugate Gradient Algorithm ML-EM OS-EM Green s One-Step Late Algorithm Matched and Unmatched Projector/Backprojector Pairs Reconstruction Using Highly Undersampled Data with l 0 Minimization Worked Examples Summary 167 Problems 168 References 170

10 Contents xiii 7 MRI Reconstruction The M The R The I To Obtain z-information Slice Selection To Obtain x-information Frequency Encoding To Obtain y-information Phase Encoding Mathematical Expressions Worked Examples Summary 190 Problems 191 References 192 Index 193