Integral Geometry and Tomography

Proceedings of a Joint Summer Research Conference held June 24-30, 1989

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Titles in This Series

Volume 1 Markov random fields and their 19 Proceedings of the Northwestern applications, Ross Kindermann and homotopy theory conference, Haynes J. Laurie Snell R. Miller and Stewart B. Priddy, Editors 2 Proceedings of the conference on 20 Low dimensional topology, Samuel J. integration, topology, and geometry in Lomonaco, Jr., Editor linear spaces, William H. Graves, Editor 21 Topological methods in nonlinear 3 The closed graph and P-closed functional analysis, S. P. Singh, graph properties in general topology, S. Thomeier, and B. Watson, Editors T. R. Hamlett and L. L. Herrington 22 Factorizations of b" ± 1, b = 2, 4 Problems of elastic stability and 3, 5, 6, 7, 10, 11, 12 up to high vibrations, Vadim Komkov, Editor powers, John Brillhart, D. H. Lehmer, 5 Rational constructions of modules J. L. Selfridge, Bryant Tuckerman, and for simple Lie algebras, George B. S. S. Wagstaff, Jr. Seligman 23 Chapter 9 of Ramanujan's second 6 Umbral calculus and Hopf algebras, notebook-Infinite series identities, Robert Morris, Editor transformations, and evaluations, Bruce C. Berndt and Padmini T. Joshi 7 Complex contour integral representation of cardinal spline 24 Central extensions, Galois groups, and functions, Walter Schempp ideal class groups of number fields, 8 Ordered fields and real algebraic A. Frohlich geometry, D. W. Dubois and T. Recio, 25 Value distribution theory and its Editors applications, Chung-Chun Yang, Editor 9 Papers in algebra, analysis and 26 Conference in modern analysis statistics, R. Lidl, Editor and probability, Richard Beals, 10 Operator algebras and K-theory, Anatole Beck, Alexandra Bellow, and Ronald G. Douglas and Claude Arshag Hajian, Editors Schochet, Editors 27 Microlocal analysis, M. Salah Baouendi, 11 Plane ellipticity and related problems, Richard Beals, and Linda Preiss Robert P. Gilbert, Editor Rothschild, Editors 12 Symposium on algebraic topology in 28 Fluids and plasmas: geometry and honor of Jose Adem, Samuel Gitler, dynamics, Jerrold E. Marsden, Editor Editor 29 Automated theorem proving, W. W. 13 Algebraists' homage: Papers in Bledsoe and Donald Loveland, Editors ring theory and related topics, 30 Mathematical applications· of category S. A. Amitsur, D. J. Saltman, and theory, J. W. Gray, Editor G. B. Seligman, Editors 31 Axiomatic set theory, James E. 14 Lectures on Nielsen fixed point theory, Baumgartner, Donald A. Martin, and Boju Jiang Saharon Shelah, Editors 15 Advanced analytic number theory. 32 Proceedings of the conference on Part 1: Ramification theoretic methods, Banach algebras and several complex Carlos J. Moreno variables, F. Greenleaf and D. Gulick, 16 Complex representations of GL(2, K) for Editors finite fields K, llya Piatetski-Shapiro 33 Contributions to group theory, 17 Nonlinear partial differential equations, Kenneth I. Appel, John G. Ratcliffe, and Joel A. Smeller, Editor Paul E. Schupp, Editors 18 Fixed points and nonexpansive 34 Combinatorics and algebra, Curtis mappings, Robert C. Sine, Editor Greene, Editor Titles in This Series

Volume 35 Four-manifold theory, Cameron Gordon 53 The Selberg trace formula and related and Robion Kirby, Editors topics, Dennis A. Hejhal, Peter Sarnak, 36 Group actions on manifolds, Reinhard and Audrey Anne Terras, Editors Schultz, Editor 54 Differential analysis and infinite 37 Conference on algebraic topology in dimensional spaces, Kondagunta honor of Peter Hilton, Renzo Piccinini Sundaresan and Srinivasa Swaminathan, and Denis Sjerve, Editors Editors 38 Topics in complex analysis, Dorothy 55 Applications of algebraic K-theory to Browne Shaffer, Editor algebraic geometry and number theory, 39 Errett Bishop: Reflections on him and Spencer J. Bloch, R. Keith Dennis, Eric his research, Murray Rosenblatt, Editor M. Friedlander, and Michael R. Stein, Editors 40 Integral bases for affine Lie algebras and their universal enveloping algebras, 56 Multiparameter bifurcation theory, David Mitzman Martin Golubitsky and 41 Particle systems, random media and John Guckenheimer, Editors large deviations, Richard Durrett, Editor 57 Combinatoric& and ordered sets, 42 Classical real analysis, Daniel Ivan Rival, Editor Waterman, Editor 58.1 The Lefschetz centennial conference. 43 Group actions on rings, Susan Proceedings on algebraic geometry, Montgomery, Editor D. Sundararaman, Editor 44 Combinatorial methods in topology and 58.11 The Lefschetz centennial conference. algebraic geometry, John R. Harper Proceedings on algebraic topology, and Richard Mandelbaum, Editors S. Gitler, Editor 45 Finite groups-coming of age, John 58.111 The Lefschetz centennial conference. McKay, Editor Proceedings on differential equations, 46 Structure of the standard modules A. Verjovsky, Editor 1 for the affine Lie algebra A1 l, 59 Function estimates, J. S. Marron, Editor James Lepowsky and Mirko Prime 60 . Nonstrictly hyperbolic conservation 47 Linear algebra and its role in laws, Barbara Lee Keyfitz and systems theory, Richard A. Brualdi, Herbert C. Kranzer, Editors David H. Carlson, Biswa Nath Datta, Charles R. Johnson, and Robert J. 61 Residues and traces of differential Plemmons, Editors forms via Hochschild homology, 48 Analytic functions of one complex Joseph Lipman variable, Chung-chun Yang and 62 Operator algebras and mathematical Chi-tai Chuang, Editors physics, Palle E. T. Jorgensen and 49 Complex differential geometry and Paul S. Muhly, Editors nonlinear differential equations, 63 Integral geometry, Robert L. Bryant, Yum-Tong Siu, Editor Victor Guillemin, Sigurdur Helgason, and so Random matrices and their R. 0. Wells, Jr., Editors applications, Joel E. Cohen, 64 The legacy of Sonya Kovalevskaya, Harry Kesten, and Charles M. Newman, Linda Keen, Editor Editors 65 Logic and combinatorics, Stephen G. Nonlinear problems in geometry, 51 Simpson, Editor Dennis M. DeTurck, Editor 66 Free group rings, Narian Gupta 52 Geometry of normed linear spaces, R. G. Bartle, N. T. Peck, A. L. Peressini, 67 Current trends in arithmetical algebraic and J. J. Uhl, Editors geometry, Kenneth A. Ribet, Editor Titles in This Series

Volume 68 Differential geometry: The interface 86 Representation theory and number between pure and applied mathematics, theory in connection with the local Mladen Luksic, Clyde Martin, and Langlands conjecture, J. Ritter, Editor William Shadwick, Editors 87 Abelian group theory, Laszlo Fuchs, 69 Methods and applications of Rudiger Gobel, and Phillip Schultz, mathematical logic, Walter A. Carnielli Editors and Luiz Paulo de Alcantara, Editors 88 Invariant theory, R. Fossum, 70 Index theory of elliptic operators, W. Haboush, M. Hochster, and foliations, and operator algebras, V. Lakshmibai, Editors Jerome Kaminker, Kenneth C. Millett, 89 Graphs and algorithms, R. Bruce and Claude Schochet, Editors Richter, Editor 71 Mathematics and general relativity, 90 Singularities, Richard Randell, Editor James A. Isenberg, Editor 91 Commutative harmonic analysis, 72 Fixed point theory and its applications, David Colella, Editor R. F. Brown, Editor 92 Categories in computer science and 73 Geometry of random motion, Rick logic, John W. Gray and Andre Scedrov, Durrett and Mark A. Pinsky, Editors Editors 93 Representation theory, group rings, and 74 Geometry of group representations, coding theory, M. Isaacs, A. Lichtman, William M. Goldman and Andy R. Magid, D. Passman, S. Sehgal, N.J. A. Sloane, Editors and H. Zassenhaus, Editors 75 The finite calculus associated with 94 Measure and measurable dynamics, Bessel functions, Frank M. Cholewinski R. Daniel Mauldin, R. M. Shortt, and 76 The structure of finite algebras, Cesar E. Silva, Editors David C. Hobby and Ralph Mckenzie 95 Infinite algebraic extensions of finite 77 Number theory and its applications in fields, Joel V. Brawley and George E. China, Wang Yuan, Yang Chung-chun, Schnibben and Pan Chengbiao, Editors 96 Algebraic topology, Mark Mahowald 78 Braids, Joan S. Birman and Anatoly and Stewart Priddy, Editors Libgober, Editors 97 Dynamics and control of multibody 79 Regular differential forms, Ernst Kunz systems, J. E. Marsden, P. S. and Rolf Waldi Krishnaprasad, and J. C. Simo, Editors 80 Statistical inference from stochastic 98 Every planar map is four colorable, processes, N. U. Prabhu, Editor Kenneth Appel and Wolfgang Haken 99 The connection between infinite 81 Hamiltonian dynamical systems, dimensional and finite dimensional Kenneth R. Meyer and Donald G. Saari, dynamical systems, Basil Nicolaenko, Editors Ciprian Foias, and Roger Temam, 82 Classical groups and related topics, Editors Alexander J. Hahn, Donald G. James, 100 Current progress in hyperbolic systems: and Zhe-xian Wan, Editors Riemann problems and computations, 83 Algebraic K-theory and algebraic W. Brent Lindquist, Editor number theory, Michael R. Stein and 101 Recent developments in geometry, R. Keith Dennis, Editors S.-Y. Cheng, H. Choi, and Robert E. 84 Partition problems in topology, Greene, Editors Stevo Todorcevic 102 Primes associated to an ideal, 85 Banach space theory, Bor-Luh Lin, Stephen McAdam Editor 103 Coloring theories, Steve Fisk Titles in This Series

Volume 104 Accessible categories: The foundations of categorical model theory, Michael Makkai and Robert Pare 105 Geometric and topological invariants of elliptic operators, Jerome Kaminker, Editor 106 Logic and computation, Wilfried Sieg, Editor 107 Harmonic analysis and partial differential equations, Mario Milman and Tomas Schonbek, Editors 108 Mathematics of nonlinear science, Melvyn S. Berger, Editor 109 Combinatorial group theory, Benjamin Fine, Anthony Gaglione, and Francis C. Y. Tang, Editors 110 Lie algebras and related topics, Georgia Benkart and J. Marshall Osborn, Editors 111 Finite geometries and combinatorial designs, Earl S. Kramer and Spyros S. Magliveras, Editors 112 Statistical analysis of measurement error models and applications, Philip J. Brown and Wayne A. Fuller, Editors 113 Integral geometry and tomography, Eric Grinberg and Eric Todd Quinto, Editors Integral Geometry and Tomography ONTEMPORARY ATHEMATICS

113

Integral Geometry and Tomography

Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference held June 24-30, 1989, with support from the National Science Foundation

Eric Grinberg and Eric Todd Quinto, Editors

AMERICAN MATHEMATICAL SOCIETY • PROVIDENCE, RHODE ISLAND EDITORIAL BOARD

Richard W. Beals, managing editor Sylvain E. Cappell Jan Mycielski Jonathan Goodman Linda Preiss Rothschild Craig Huneke Michael E. Taylor

The Joint Summer Research Conference on Integral Geometry and Tomography was held at Humboldt State University, Arcata, California, on June 24-30, 1989. Research was partially supported by NSF Grant OMS 8613199 and ARO Grant DAAL 03-89-G-0038.

1980 Mathematics Subject Classification (1985 Revision). Primary 44-XX, 92-XX.

Library of Congress Cataloging-in-Publication Data Integral geometry and tomography: proceedings of the AMS-IMS-SIAM joint summer research conference, held June 24-30, 1989/Eric Grinberg and Eric Todd Quinto, editors. p. cm.-(Contemporary mathematics; 113) Includes bibliographical references. ISBN 0-8218-5120-9 1. Geometry, Integral-Congresses. 2. Tomography-Congresses. I. Grinberg, Eric, 1958-. II. Quinto, Eric Todd, 1951-. Ill. American Mathematical Society. IV. Institute of Mathematical Statistics. V. Society for Industrial and Applied Mathematics. VI. Series: Contemporary mathematics (American Mathematical Society); v. 113. QA649.157 1990 90-19434 516.3'62-dc20 CIP

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Preface Xl ERIC GRINBERG AND ERIC ToDD QuiNTO

Admissible complexes for the combinatorial radon transform. A Progress Report ETHAN D. BoLKER, ERIC GRINBERG, AND JosEPH P. S. KuNG

On generalized radon transforms with unknown measures JAN BOMAN 5

Paley-Wiener theorems on rank one symmetric spaces of non compact type WILLIAM 0. BRAY AND DONALD C. SOLMON 17 Inversion of the X-ray transform: continuous vs. discrete ENRICO CASADIO T ARABUSI 31 The mathematics and physics of radiation dose planning using X-rays A. M. CORMACK AND E. T. QUINTO 41 The radon transform and tensor products LEON EHRENPREIS 57 An application of a multidimensional sampling theorem to computed tomography ADEL F ARID ANI 65 The radon transform for symmetric forms on real projective spaces HUBERT GOLDSCHMIDT 81 Bi-invariant differential operators on the complex motion group and the range of the d-plane transform on en FULTON B. GONZALEZ 97 Processes of flats induced by higher-dimensional processes II PAUL R. Goo DEY AND RALPH HowARD 111

IX X CONTENTS

Microlocal techniques in integral geometry ALLAN GREENLEAF AND GuNTHER UHLMANN 121 Backscattering comes to the rescue F. A. GRUNBAUM 137 The totally-geodesic radon transform on constant curvature spaces SIGURDUR HELGASON 141 An application for a limited solid angle X-ray transform STEVEN H. IZEN 151 On a conjectured projection inequality of Petty ERWIN LUTWAK 171 A generalized radon transform in wideband radar PETER MAASS 183 Summability and approximate reconstruction from radon transform data W. R. MADYCH 189 An inverse problem for a transport equation and integral geometry FRANK NATTERER 221 Invariant radon transforms on a symmetric space JEREMY ORLOFF 233 Nonuniqueness and the null space of the divergent beam X-ray trans- form DONALD C. SaLMON 243 Preface

The 1989 AMS summer conference on integral geometry and tomogra- phy at Humboldt State University, Arcata, California, was a lively successor to the 1984 AMS integral geometry conference. This later gathering saw a greater representation of applied mathematics as well as combinatorics. The interaction among conferees was remarkable: theorems announced during one lecture were used a few days later in another. Participants shared ideas, experiences, and data. Results in seemingly different fields were found to be manifestations of the same abstract principles. The papers in these proceed- ings represent lectures presented at the conference, results emanating from conference activities and reports from researchers who were not able to at- tend the conference but whose influence was significant nonetheless. All the papers in this volume were submitted in final form, and no versions will be submitted for publication elsewhere.

The pure mathematical articles in these proceedings are in areas as diverse as combinatorics, geometric inequalities, microlocal analysis, group theory, and harmonic analysis. Balker, Grinberg, and Kung explore the parallels be- tween the combinatorial Radon transform and the admissibility problem for the continuous Radon transform. The interplay between n-dimensional data and induced k-dimensional data occurs throughout integral geometry and to- mography. Goodey and Howard study this theme in the context of stochas- tic geometry: they investigate the k-dimensional processes induced by an n-dimensional stationary process. Lutwak gives some formulations of affine isoperimetric inequalities using a number of integral-geometric functionals. The techniques of microlocal analysis have become increasingly popular and successful in dealing with Radon transforms. Boman uses pseudodifferen- tial operators to understand the identification problem for X-ray transforms with general measures. Greenleaf and Uhlmann give a survey of the relevant microlocal machinery and show how it can be applied to admissibility and Sobolev estimate problems.

The interplay between Lie group theory, harmonic analysis, and Radon transforms is well represented in the proceedings. Ehrenpreis discusses the notion of Radon transform and its relationship to geometry and linear and nonlinear harmonic analysis. Helgason gives inversion and support theorems for the totally geodesic Radon transform on spaces of constant curvature.

XI xii PREFACE

Goldschmidt studies the Radon transform for symmetric forms on real pro- jective spaces. Tarabusi inverts the X-ray transform on both the real hyper- bolic plane and on locally finite trees. Gonzalez characterizes the range of the complex d-plane transform by a single partial differential operator. Bray and Solmon prove range theorems for the horocycle transform, and Orloff gives a support theorem for the K invariant horocycle transform on noncompact rank-one symmetric space using integral equations techniques.

Over the years tomography has broadened to include topics such as radia- tion dose planning [Cormack and Quinto], radar [MaaB], and partial differen- tial equations [Natterer]. The interplay between fields is remarkable. MaaB uses ideas in [Ehrenpreis] to prove injectivity of a transform in wideband radar; Natterer uses information about the attenuated Radon transform to prove uniqueness and stability for an inverse problem for the transport equa- tion; and Cormack and Quinto investigate the mathematics and physics of radiation dose planning using dual Radon transforms. Significant progress has occurred in X-ray tomography as well. Gri.i.nbaum solves a finite tomo- graphic problem using a model that incorporates scattering as well as absorp- tion. Faridani gives efficient X-ray sampling geometries using a new sampling theorem for multi-dimensional band-limited functions. Izen develops an in- version method for severely limited X-ray tomographic data in space, and Madych develops convolution formulas for complete and incomplete data problems. Solmon proves a necessary and sufficient condition for a function to be in the null space of the finite divergent beam transform.

The conference organizers, Leon Ehrenpreis, Fulton Gonzalez, Eric Grin- berg, and Todd Quinto, would like to thank the American Mathematical Society, the National Science Foundation, and the U.S. Army for their finan- cial support, without which this conference would not have been possible. In particular, we would like to thank Carole Kohanski and Donna Harmon, our liaisons with the AMS, for their patient help during the conference and their quick and efficient work producing these proceedings. The editors of these proceedings, Eric Grinberg and Todd Quinto, would like to thank the authors for submitting articles of such high quality and the referees for prompt, in- sightful reviews. Finally, we would like to thank the conference participants, who made the bureaucratic work worthwhile, and without whom the proceed- ings would have been rather slim.

Eric Grinberg and Eric Todd Quinto June 20, 1990 ......

ISBN 0-8218-5120-9