Reconstructive Integral Geometry

170 Pages·2004·5.642 MB·English

by  Victor Palamodov (auth.)

Monographs in Mathematics
Vol. 98
Managing Editors:
H.Amann
Universitiit Zurich, Switzerland
J.-P. Bourguignon
IHES, Bures-sur-Yvette, France
K. Grove
University of Maryland, College Park, USA
P.-L. Lions
Universite de Paris-Dauphine, France
Associate Editors:
H. Araki, Kyoto University
F. Brezzi, Universita di Pavia
K.c. Chang, Peking University
N. Hitchin, University of Warwick
H. Hofer, Courant Institute, New York
H. Knorrer, ETH Zurich
K. Masuda, University of Tokyo
D. Zagier, Max-Planck-Institut Bonn
Victor Palamodov
Reconstructive
Integral
Geometry
Springer Basel AG
Author:
Victor Palamodov
School of Mathematics
Tel Aviv University
RamatAviv
Tel Aviv 69978
Israel
e-mail: [email protected]
2000 Mathematics Subject Classification: primary 44AI2, 53C65, 65R32; secondary 35C15,
35NIO, 92C55
A CIP catalogue record for this book is available from the
Library of Congress, Washington D.C., USA
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationa1bibliografie; detailed
bibliographic data is available in the Internet at <http://dnb.ddb.de>
ISBN 978-3-0348-9629-0 ISBN 978-3-0348-7941-5 (eBook)
DOI 10.1007/978-3-0348-7941-5
This work is subject to copyright. AlI rights are reserved, whether the whole or part of the material
is concemed, specifically the rights of translation, reprinting, re-use of illustrations, recitation,
broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind
of use permission of the copyright owner must be obtained.
© 2004 Springer Basel AG
Originally published by Birkhlluser Verlag in 2004
Softcover reprint of the hardcover 1s t edition 2004
Printed on acid-free paper produced from ch1orine-free pulp. TCF ?
ISBN 978-3-0348-9629-0
987654321
To my family
Contents
Preface . Xl
Notation . ................ . xii
1 Distributions and Fourier Transform
1.1 Introduction. . . . . . . . . . . ..... 1
1.2 Distributions and generalized functions . 1
1.3 Tempered distributions. . . . . . 5
1.4 Homogeneous distributions ... 9
1.5 Manifolds and differential forms . 14
1.6 Push down and pull back .... 17
1.7 More on the Fourier transform 20
1.8 Bandlimited functions and interpolation 26
2 Radon Transform
2.1 Properties . . 29
2.2 Inversion formulae 31
2.3 Alternative formulae 34
2.4 Range conditions . . 37
2.5 Frequency analysis 38
2.6 Radon transform of differential forms . 41
3 The Funk Transform
3.1 Factorable mappings ....... 43
3.2 Spaces of constant curvature .. 47
3.3 Inversion of the Funk transform . 49
3.4 Radon's inversion via Funk's inversion 50
3.5 Unified form. . . . . . . . . . . . . . . 51
3.6 Funk-Radon transform and wave fronts 53
3.7 Integral transform of boundary discontinuities . 55
3.8 Nonlinear artifacts ........... 61
3.9 Pizetti formula for arbitrary signature ..... 62
viii Contents
4 Reconstruction from Line Integrals
4.1 Pencils of lines and John's equation. 65
4.2 Sources at infinity ........ . 68
4.3 Reduction to the Radon transform 71
4.4 Rays tangent to a surface . . . . . 73
4.5 Sources on a proper curve ..... 74
4.6 Reconstruction from plane integrals of sources. 77
4.7 Line integrals of differential forms . 78
4.8 Exponential ray transform . 83
4.9 Attenuated ray transform 86
4.10 Inversion formulae 87
4.11 Range conditions 89
5 Flat Integral Transform
5.1 Reconstruction problem 93
5.2 Odd-dimensional subspaces 94
5.3 Even dimension ...... . 98
5.4 Range of the fiat transform 99
5.5 Duality for the Funk transform 101
5.6 Duality in Euclidean space. 102
6 Incomplete Data Problems
6.1 Completeness condition ........ . 105
6.2 Radon transform of Gabor functions . . 106
6.3 Reconstruction from limited angle data 107
6.4 Exterior problem .... 108
6.5 The parametrix method ... 111
7 Spherical Transform and Inversion
7.1 Problems .......... . 115
7.2 Arc integrals in the plane .. 115
7.3 Hemispherical integrals in space. 119
7.4 Incomplete data ........ . 124
7.5 Spheres centred on a sphere .. 125
7.6 Spheres tangent to a manifold. 127
7.7 Characteristic Cauchy problem 130
7.8 Fundamental solution for the adjoint operator. 133
8 Algebraic Integral Transform
8.1 Problems .............. . 135
8.2 Special cases ............ . 136
8.3 Multiplicative differential equations. 139
8.4 Funk transform of Leray forms . . . 141
8.5 Differential equations for hypersurface integrals 142
Contents ix
8.6 Howard's equations ............ . 144
8.7 Range of differential operators. . . . . . . 146
8.8 Decreasing solutions of Maxwell's system 147
8.9 Symmetric differential forms ....... . 149
9 Notes
Notes to Chapter 1 153
Notes to Chapter 2 . 153
Notes to Chapter 3 . 153
Notes to Chapter 4 . 154
Notes to Chapter 5 . 155
Notes to Chapter 6 . 155
Notes to Chapter 7 . 156
Notes to Chapter 8 . 156
Bibliography 157
Index .... 163
Preface
One hundred years ago (1904) Hermann Minkowski [58] posed a problem: to re
construct an even function I on the sphere 82 from knowledge of the integrals
fc
MI (C) = Ids
over big circles C. Paul Funk found an explicit reconstruction formula for I from
data of big circle integrals. Johann Radon studied a similar problem for the Eu
clidean plane and space. The interest in reconstruction problems like Minkowski
Funk's and Radon's has grown tremendously in the last four decades, stimulated by
the spectrum of new modalities of image reconstruction. These are X-ray, MRI,
gamma and positron radiography, ultrasound, seismic tomography, electron mi
croscopy, synthetic radar imaging and others. The physical principles of these
methods are very different, however their mathematical models and solution meth
ods have very much in common. The umbrella name reconstructive integral geom
etryl is used to specify the variety of these problems and methods.
The objective of this book is to present in a uniform way the scope of well
known and recent results and methods in the reconstructive integral geometry.
We do not touch here the problems arising in adaptation of analytic methods to
numerical reconstruction algorithms. We refer to the books [61], [62] which are
focused on these problems.
Various aspects of interplay of integral geometry and differential equations
are discussed in Chapters 7 and 8. The results presented here are partially new.
The book is an extended version of a lecture course which was read for
students of Tel Aviv University. Chapter 1 contains a mini-course in distribution
theory, harmonic analysis and geometry. Not much of this knowledge is necessary
for reading and understanding Chapters 2, 3 and 6.
1 Blaschke's term "Integralgeometrie" looks somehow redundant; the word geometry itself has,
since ancient Greek times, meant calculation of lengths, areas, volumes, i.e., some integrals.
Notation
Z - the algebra of integers,
lR - the field of real numbers,
lR+ - the set of non-negative numbers
C - the field of complex numbers
A
j ~ 21fz, Z ~
1+ = max (j,O) , f- = max (-f,O)
r (A), A E C - Gamma-function
V - a vector space over lR of finite dimension
V* - the space dual to a vector space V
dx or dV - the volume form in a coordinate space V
E or En - a Euclidean space of dimension n < CXl
D (V) - the space of smooth functions of compact support (test functions) in the
vector space V
K (V) - the space of smooth densities of compact support (test densities) in V
S (V) - the space of smooth fast decreasing functions in V
F (j) = j - the Fourier transform of a function f or of the density f dx defined
in V; j (~) = Iv exp (-j~x) f (x) dx
Iv.
F* (j) = exp (j~x) f (~) d~ - the adjoint Fourier transform