Mathematics and Computer Science in Medical Imaging PDF
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Mathematics and Computer Science in Medical Imaging NATO ASI Series Advanced Science Institutes Series A series presenting the results of activities sponsored by the NA TO Science Committee, which aims at the dissemination of advanced scientific and technological knowledge, with a view to strengthening links between scientific communities. The Series is published by an international board of publishers in conjunction with the NATO Scientific Affairs Division A Life Sciences Plenum Publishing Corporation B Physics London and New York C Mathematical and D. Reidel Publishing Company Physical Sciences Dordrecht, Boston, Lancaster and Tokyo D Behavioural and Martinus Nijhoff Publishers Social Sciences Boston, The Hague, Dordrecht and Lancaster E Applied Sciences F Computer and Springer-Verlag Systems Sciences Berlin Heidelberg New York G Ecological Sciences London Paris Tokyo H Cell Biology Series F: Computer and Systems Sciences Vol. 39 Mathematics and Computer Science in Medical Imaging Edited by Max A. Viergever Delft University of Technology Department of Mathematics and Informatics P.O. Box 356,2600 AJ Delft, The Netherlands Andrew Todd-Pokropek University College London Department of Medical Physics Gower Street, London WC1 E 6BT, UK Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Published in cooperation with NATO Scientific Affairs Division Proceedings of the NATO Advanced Study Institute on Mathematics and Computer Science in Medical Imaging, held in II Ciocco, Italy, September 21 - October 4,1986. Directors Max A. Viergever, Delft, The Netherlands Andrew Todd-Pokropek, London, UK Scientific Committee Harrison H. Barrett, Tucson, USA Gabor T Herman, Philadelphia, USA Frank Natterer, Munster, FRG Robert di Paola, Villejuif, France Conference Secretary Marjoleine den Boef Sponsor NATO Scientific Affairs Division Co-sponsors Delft University of Technology, The Netherlands National Science Foundation, USA Siemens Gammasonics, The Netherlands University College London, UK ISBN-13:978-3-642-83308-3 e-ISBN-13:978-3-642-83306-9 001: 10.1007/978-3-642-83306-9 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, re-use of illustrations, recitation, broadcasting, reproduction on rnicrofilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only perrnitted under the provisions of the Gerrnan Copyright Law of Septernber 9, 1965, in its version of June 24, 1985, and a copyright fee rnust always be paid. Violations fall under the prosecution act of the Gerrnan Copyright Law. © Springer-Verlag Berlin Heidelberg 1988 Softcover reprint of the hardcover 1s t edition 1988 2145/3140-54321 0 PREFACE Medical imaging is an important and rapidly expanding area in medical science. Many of the methods employed are essentially digital, for example computerized tomography, and the subject has become increasingly influenced by develop ments in both mathematics and computer science. The mathematical problems have been the concern of a relatively small group of scientists, consisting mainly of applied mathematicians and theoretical physicists. Their efforts have led to workable algorithms for most imaging modalities. However, neither the fundamentals, nor the limitations and disadvantages of these algorithms are known to a sufficient degree to the physicists, engineers and physicians trying to implement these methods. It seems both timely and important to try to bridge this gap. This book summarizes the proceedings of a NATO Advanced Study Institute, on these topics, that was held in the mountains of Tuscany for two weeks in the late summer of 1986. At another (quite different) earlier meeting on medical imaging, the authors noted that each of the speakers had given, there, a long introduction in their general area, stated that they did not have time to discuss the details of the new work, but proceeded to show lots of clinical results, while excluding any mathematics associated with the area. The aim of the meeting reported in this book was, therefore, to do exactly the opposite: to allow as much time as needed to fully develop the fundamental ideas relevant to medical imaging, to include a full discussion of the associated mathematical problems, and to exclude (by and large) clinical results. The meeting was therefore designed primarily for physicists, engineers, computer scientists, mathematicians, and interested and informed clinicians working in the area of medical imaging. This book is aimed at a similar readership. The Advanced Study Institute was a great success. The weather was beautiful, the food was excellent, the atmosphere was warm and friendly, and the scientific level was judged to be very high. In order to extend the interest of this meeting to an audience greater than the 63 people fortunate enough to have been present in Italy, an attempt has been made to encapsulate the contents of the meeting in the form of this publication. VI At the meeting there were three general types of presentation: tutorial lectures, proffered papers, and workshops in specific areas of interest. All of the tutorial lectures are included as chapters in this book. They should represent a good introduction to various areas of importance in medical ima ging. All the proffered papers were refereed after the meeting, and some of them were selected as being of a high enough standard for inclusion in this book. These papers represent short accounts of current research in medical imaging. Unfortunately, it was not possible to retain any record of the workshops where often vigorous and heated discussions resulted. The layout of this book has been divided into two parts. The first part contains all the introductory and tutorial papers and should function as an overview of the subject matter of the meeting. This fj.rst part might serve as a text book. The second part contains papers of a more specialized nature divided into four sections; analytic reconstruction methods, iterative methods, display and evaluation, and a collection of papers grouped together under the heading of applications. The editors would like to thank all the participants, authors, the scien tific committee, and, of course, the sponsors of this meeting, and hope that this book will provide useful introductory and reference material suitable for students and workers in the expanding field of mathematics and computer science applied to medical imaging. M.A. Viergever A. Todd-Pokropek Delft, August 1987 TABLE OF CONTENTS Part 1: Introduction to and OVerview of the Field .................... . Introduction to integral transforms A. Rescigno ...................................................... 3 Introduction to discrete reconstruction methods in medical imaging M.A. Viergever ................................................... 43 Image structure J.J. Koenderink 67 Fundamentals of the Radon transform H.H. Barrett 105 Regularization techniques in medical imaging F. Natterer ...................................................... 127 Statistical methods in pattern recognition C. R. Appledorn ................................................... 143 Image data compression techniques: A survey A. Todd-Pokropek ................................................. 167 From 2D to 3D representation G.T. Herman ...................................................... 197 VLSI-intensive graphics systems H. Fuchs ......................................................... 221 Knowledge based interpretation of medical images J. Fox, N. Walker ................................................ 241 Part 2: Selected Topics ..........................•.....•.............. 267 2.1 Analytic Reconstruction Methods 269 The attenuated Radon transform F. Natterer ••.•..•......•.•.•.••••••••••••••••••••..••..••.. 271 Inverse imaging with strong multiple scattering S. Leeman, V.C. Roberts, P.E. Chandler, L.A. Ferrari 279 2.2 Iterative Methods 291 Possible criteria for choosing the number of iterations in some iterative reconstruction methods M. Defrise .........................•........................ 293 Initial performance of block-iterative reconstruction algorithms G.T. Herman, H. Levkowitz ..............................•.... 305 Maximum likelihood reconstruction in PET and TOFPET C.T. Chen, C.E. Metz, X. Hu ................................. 319 Maximum likelihood reconstruction for SPECT using Monte Carlo simulation C.E. Floyd, S.H. Manglos, R.J. Jaszczak, R.E. Coleman ....... 331 X-ray coded source tomosynthesis I. Magnin ...........................•...................•... 339 VIII Some mathematical aspects of electrical impedance tomography W.R. Breckon, M.K. Pidcock •...•..........•.....•............. 351 2.3 Display and Evaluation ..........•........•.•.•.•............. 363 Hierarchical figure-based shape description for medical imaging S.M. Pizer, W.R. Oliver, J.M. Gauch, S.H. Bloomberg .......... 365 GIHS: A generalized color model and its use for the represen tation of multiparameter medical images H. Levkowitz, G.T. Herman ..•....................•........•... 389 The evaluation of image processing algorithms for use in medical imaging M. de Belder ..••................•..................•......... 401 Focal lesions in medical images: A detection problem J.M. Thijssen ..•........•.•.....................•........•... 415 2.4 Applications................................................. 439 Time domain phase: A new tool in medical ultrasound imaging D.A. Seggie, S. Leeman, G.M. Doherty ........................• 441 Performance of echographic equipment and potentials for tissue characterization J.M. Thijssen, B.J. Oosterveld ..............•............•... 455 Development of a model to predict the potential accuracy of vessel blood flow measurements from dynamic angiographic recordings D.J. Hawkes, A.C.F. Colchester, J.N.H. Brunt, D.A.G. Wicks, G. H. du Boulay, A. Wallis .................•.....•............ 469 The quantitative imaging potential of the HIDAC positron camera D.W. Townsend, P.E. Frey, G. Reich, A. Christin, H.J. Tochon- Danguy, G. Schaller, A. Donath, A. Jeavons ................... 479 The use of cluster analysis and constrained optimisation techniques in factor analysis of dynamic structures A.S. Houston 491 Detection of elliptical contours J.A.K. Blokland, A.M. Vossepoel, A.R. Bakker, E.K.J. Pauwels. 505 Optimal non-linear filters for images with non-Gaussian differential distributions M. Fuderer ...•.•..............................•...•........•. 517 Participants ...........•..........•....•.............................. 527 Subject index ...................•.•.............•..................... 533 Part 1 Introduction to and Overview of the Field INTRODUCTION TO INTEGRAL TRANSFORMS Aldo Rescigno Yale University, New Haven, U.S.A. and University of Ancona, Italy ABSTRACT This note is the summary of a series of lectures presented for a number of years by the author to graduate students interested in Mathematical Modeling in Biology. Its aim is essentially practical; proofs are given when they help understanding the essence of a problem, otherwise the approach is most ly intuitive. The operational calculus is presented from an algebraic point of view; starting from the convolution integral, the extension from func tions to operators is defined as analogous to the extension from natural numbers to rational numbers. Laplace and Fourier transforms are presented as special cases. The note concludes showing the connection between Fourier transforms and Fourier series. 1. CONVOLUTION INTEGRAL Let g(x) .dx measure the intensity of the signal emitted by a one-dimensional source from the small interval (x, x+dx), f(u) the density of the signal received at the point of coordinate u of a one-dimensional observer, and h(u,x) the transmittance along the straight line connecting the points u, x. Then r-+= f(u) J g(x)h(u,x)dx (1) --= or, with an appropriate choice of the u and x axes, f(u) ~ f-+= g(x)h(u-x)dx. (2) --= If g(x) o for x < 0 and h(u-x) o for x > u, then fU f(u) = g(x)h(u-x)dx. (3) o This is the well-known convolution integral. NATO ASI Series, Vol. F39 Mathematics and Computer Science in Medical Imaging Edited by M. A. Viergever and A. E. Todd-Pokropek © Springer-Verlag Berlin Heidelberg 1988
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