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Mathematical Methods in Image Reconstruction

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Mathematical Methods in Image Reconstruction S1AM Monographs on Mathematical Modeling and Computation Editor-in-Chief Joseph E. Flaherty Rensselaer Polytechnic Institute About the Series Editorial Board In 1997, SIAM began a new series on mathematical modeling and computation. Books in the series develop a focused topic from its genesis to the current state of the art; these books Ivo Babuska University of Texas at Austin • present modern mathematical developments with direct applications in science and engineering; • describe mathematical issues arising in modern applications; • develop mathematical models of topical physical, chemical, or biological systems; • present new and efficient computational tools and techniques that have direct applications in science and engineering; and • illustrate the continuing, integrated roles of mathematical, scientific, and computational investigation. Although sophisticated ideas are presented, the writing style is popular rather than formal. Texts are intended to be read by audiences with little more than a bachelor's degree in mathematics or engineering. Thus, they are suitable for use in graduate mathematics, science, and engineering courses. By design, the material is multidisciplinary. As such, we hope to foster cooperation and collaboration between mathematicians, computer scientists, engineers, and scientists. This is a difficult task because different terminology is used for the same concept in different disciplines. Nevertheless, we believe we have been successful and hope that you enjoy the texts in the series. Joseph E. Flaherty Frank Natterer and Frank Wubbeling, Mathematical Methods in Image Reconstruction Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion Michael Criebel, Thomas Dornseifer, and Tilman Neunhoeffer, Numerical Simulation in Fluid Dynamics: A Practical Introduction Khosrow Chadan, David Colton, Lassi Paivarinta, and William Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems Charles K. Chui, Wavelets: A Mathematical Tool for Signal Analysis H. Thomas Banks North Carolina State University Margaret Cheney Rensselaer Polytechnic Institute Paul Davis Worcester Polytechnic Institute Stephen H. Davis Northwestern University Jack J. Dongarra University of Tennessee at Knoxville and Oak Ridge National Laboratory Christoph Hoffmann Purdue University George M. Homsy Stanford University Joseph B. Keller Stanford University J. Tinsley Oden University of Texas at Austin James Sethian University of California at Berkeley Barna A. Szabo Washington University Mathematical Methods in Image Reconstruction Frank Natterer Frank Wubbeling Universitat Munster Munster, Germany siam Society for Industrial and Applied Mathematics Philadelphia Copyright © 2001 by the Society for Industrial and Applied Mathematics. 10987654321 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688. Library of Congress Cataloging-in-Publication Data Mathematical methods in image reconstruction / Frank Natterer...[et al.] p. cm. — (SIAM monographs on mathematical modeling and computation) Includes bibliographical references and index. ISBN 0-89871-472-9 1. Image processing—Congresses. I. Natterer, F. (Frank), 1941- II. Series. TA1637.M356 2001 621.367—dc21 00-053804 siam is a registered trademark. Contents Preface ix List of Symbols xi 1 Introduction 1.1 The Basic Example 1.2 Overview 1.3 Mathematical Preliminaries 1.3.1 Fourier analysis 1.3.2 Some integral operators 1.3.3 The Moore-Penrose generalized inverse 1.3.4 The singular value decomposition 1.3.5 Special functions 1.3.6 The fast Fourier transform 1 1 2 3 3 5 5 5 6 8 2 Integral Geometry 2.1 The Radon Transform 2.2 The Ray Transform 2.3 The Cone Beam Transform 2.4 Weighted Transforms 2.4.1 The attenuated ray transform 2.4.2 The Feig-Greenleaf transform 2.4.3 The windowed ray transform 2.5 Integration over Curved Manifolds 2.5.1 Computing an even function on S2 from its integrals over equatorial circles 2.5.2 Reduction of problems on the sphere to the Radon transform . . . . 2.5.3 Reconstruction from spherical averages 2.5.4 More general manifolds 2.6 Vector Fields 9 9 17 23 27 27 30 31 31 3 Tomography 3.1 Transmission Tomography 3.1.1 Parallel scanning geometry 41 41 41 v 32 33 34 36 36 vi Contents 3.1.2 Fan beam scanning geometry . 3.1.3 3D helical scanning 3.1.4 3D cone beam scanning 3.2 Emission Tomography 3.3 Diffraction Tomography 3.4 Magnetic Resonance Imaging 3.5 Electron Tomography 3.6 Radar 3.6.1 Synthetic aperture radar 3.6.2 Range-Doppler radar 3.7 Vector Tomography 3.7.1 Doppler tomography 3.7.2 Schlieren tomography 3.7.3 Photoelastic tomography 3.8 Seismic Tomography 3.8.1 Travel time tomography 3.8.2 Reflection tomography 3.8.3 Waveform tomography 3.9 Historical Remarks 4 Stability and Resolution 4.1 Stability 4.2 Sampling 4.3 Resolution 4.4 The FFT on Nonequispaced Grids 5 Reconstruction Algorithms 5.1 The Filtered Backprojection Algorithm 5.1.1 Standard parallel scanning 5.1.2 Parallel interlaced scanning 5.1.3 Standard fan beam scanning 5.1.4 Linear fan beam scanning 5.1.5 Fast backprojection 5.1.6 The point spread function 5.1.7 Noise in the filtered backprojection algorithm 5.1.8 Filtered backprojection for the exponential Radon transform . . . . 5.1.9 Filtered backprojection for the attenuated Radon transform 5.2 Fourier Reconstruction 5.2.1 Standard Fourier reconstruction 5.2.2 The gridding method 5.2.3 The linogram algorithm 5.2.4 Fourier reconstruction in diffraction tomography and MRI 5.3 Iterative Methods 5.3.1 ART 5.3.2 The EM algorithm 5.3.3 Other iterative algorithms 42 43 43 44 46 51 54 55 55 56 57 57 58 58 59 59 60 60 62 63 63 65 71 78 81 81 83 87 90 93 95 96 97 99 99 100 100 102 106 108 110 110 123 118 124 Contents vii 5.4 5.5 125 127 128 129 131 133 133 134 134 136 137 Direct Algebraic Algorithms 3D Algorithms 5.5.1 The FDK approximate formula 5.5.2 Grangeat's method 5.5.3 Filtered backprojection for the cone beam transform 5.5.4 Filtered backprojection for the ray transform 5.5.5 The Radon transform in 3D 5.6 Circular Harmonic Algorithms 5.6.1 Standard parallel scanning 5.6.2 Standard fan beam scanning 5.7 ART for Nonlinear Problems 6 Problems That Have Peculiarities 6.1 Unknown Orientations 6.1.1 The geometric method 6.1.2 The moment method 6.1.3 The method of Provencher and Vogel 6.1.4 The 2D case 6.2 Incomplete Data 6.2.1 Uniqueness and stability 6.2.2 Reconstruction methods 6.2.3 Truncated projections in PET 6.2.4 Conical tilt problem in electron tomography 6.3 Discrete Tomography 6.4 Simultaneous Reconstruction of Attenuation and Activity 6.5 Local Tomography 6.6 Few Data 139 139 139 141 142 143 144 144 147 148 150 151 152 155 159 7 161 161 163 163 165 168 170 172 174 176 178 179 185 Nonlinear Tomography 7.1 Tomography with Scatter 7.2 Optical Tomography 7.2.1 The transport model 7.2.2 The diffusion model 7.2.3 The linearized problem 7.2.4 Calderon's method 7.2.5 The transport-backtransport algorithm 7.2.6 The diffusion-backdiffusion algorithm 7.3 Impedance Tomography 7.4 Ultrasound Tomography 7.4.1 Frequency domain ultrasound tomography 7.4.2 Time domain ultrasound tomography Bibliography 189 Index 209 This page intentionally left blank Preface Since the advent of computerized tomography in the seventies, many imaging techniques have emerged and have been introduced in radiology, science, and technology. Some of these techniques are now in routine use, most are still under development, and others are the subject of mainly academic research, their future usefulness in debate. This book makes an attempt to describe these techniques in a mathematical language, to provide the adequate mathematical background and the necessary mathematical tools. In particular, it gives a detailed analysis of numerical algorithms for image reconstruction. We concentrate on the developments of the last 10 to 15 years. Previous results are given without proof, except when new proofs are available. It is assumed that, or at least helpful if, the reader is familiar with the tomography literature of the eighties. The backbone of the theory of imaging is still integral geometry. We survey this field as far as is necessary for imaging purposes. Imaging techniques based on or related to integral geometry are briefly described in the section on tomography. In contrast, the section on algorithms is fairly detailed, at least in the two-dimensional (2D) case. In the threedimensional (3D) case, we derive exact and approximate inversion formulas for specific imaging devices. We describe their algorithmic implementation, which largely parallels the 2D case. The development in the field of algorithms is still quite lively, in particular in the 3D area. While some fundamental principles, such as filtered backprojection, seem to be well established, much of this section may well turn out to be just a snapshot of the present scene. General trends, such as the present revival of Fourier and iterative methods, become visible. In the last part of the book we deal with imaging techniques that are usually referred to as tomography but that are only remotely related to the straight line paradigm of tomography. These can be formulated as bilinear inverse problems of partial differential equations. We give a common framework and describe simple numerical methods based on standard iterative techniques of tomography. The book is aimed at mathematicians, engineers, physicists, and other scientists with the appropriate mathematical skills who want to understand the theoretical foundations of image reconstruction and to solve concrete problems. Often the proofs are sketchy or even missing in cases in which suitable references are easily available. We hope that the readability does not suffer from these omissions, which are necessary to keep this report at a reasonable length. A Web page for this book has been created at http://www.siam.org/books/mm05. It includes any necessary corrections, updates, and additions. Code fragments and additional papers can be found on the authors' site at http://www.inverse-problems.de/. ix

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