Table Of ContentMathematical Problems
of Tomography
I. M. GELFAND
S. G. GINDIKIN, Editors
Volume 81
TRANSLATIONS OF
MATHEMATICAL MONOGRAPHS
American Mathematical Society
TRANSLATIONS OF MATHEMATICAL MONOGRAPHS
VOLUME 81
Mathematical Problems
of Tomography
I. M. GELFAND
S. G. GINDIKIN, Editors
American Mathematical Society * Providence • Rhode Island
Translated from the Russian by S. Gelfand
Translation edited by A. Sossinsky
1980 Mathematics Subject Classification (1985 Revision). Primary 44A05,
44A15; Secondary 46F12, 44-04, 90A15, 32A25.
Abstract. Papers in the book cover various mathematical problems arising from and related
to computerized tomography. The main idea unifying all the approaches in the book is that
these mathematical problems satisfy strong requirements imposed by practical applications
of computerized tomography: reconstruction of nonsmooth function is studied, pointwise
convergence is used, and discretization in computational algorithms is taken into account.
The mathematical areas discussed include integral geometry; theory of several complex
variables; theory of distributions; integral transformations; and applications to reconstruction
of biological objects and to mathematical economics.
Library of Congress Cataloging-in-Publication Data
Mathematical problems of tomography/edited by I. M. Gelfand and S. G. Gindikin; [trans
lated from the Russian by S. Gelfand].
p. cm.—(Translations of mathematical monographs; v. 81)
ISBN 0-8218-4534-9
1. Tomography—Mathematics. I. Gel'fand, I. M. (Izrail7 Moiseevich) II. Gindikin, S.
G. (Semen Grigor'evich) III. Gel'fand, S. I. (Sergei Izrailevich) IV. Series.
RC78.7.T6M38 1990 90-845
616.07' 5 7' 0151 —dc20 CIP
Copyright © 1990 by the American Mathematical Society. All rights reserved.
Translation authorized by the
All-Union Agency for Authors’ Rights, Moscow
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America
Information on Copying and Reprinting can be found at the back of this volume.
The paper used in this book is acid-free and falls within the guidelines
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This publication was typeset using -TeX,
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10 9 8 7 6 5 4 3 2 1 94 93 92 91 90
Contents
GELFAND, I. M. and GINDIKIN, S. G. Introduction. Integral ge
ometry and tomography.............................................................. 1
POPOV, D. A. On convergence of a class of algorithms for the in
version of the numerical Radon transform............................... 7
GONCHAROV, A. B. Three-dimensional reconstruction of arbitrar
ily arranged identical particles given their projections............. 67
GELFAND, M. S. and GONCHAROV, A. B. Spatial rotational align
ment of identical particles given their projections: theory and
practice....................................................................................... 97
PALAMODOV, V. P. Some singular problems in tomography....... 123
VVEDENSKAYA, N. D. and GINDIKIN, S. G. Discrete Radon trans
form and image reconstruction................................................. 141
HENKIN, G. M. and SHANANIN, A. A. Bernstein theorems and
the Radon transform. Application to the theory of production
functions..................................................................................... 189
BUCHSTABER, V. M. and MASLOV, V. K. Mathematical models
and algorithms of tomographic synthesis of wave fields and in
homogeneous media................................................................... 225
iii
TRANSLATIONS OF
MATHEMATICAL MONOGRAPHS
Volume 81, 1990
INTRODUCTION.
INTEGRAL GEOMETRY AND TOMOGRAPHY
As early as 1917 Radon derived an explicit formula for the reconstruc
tion of a function on the plane given its integrals over all lines. Even
earlier Minkovsky and Funk solved the essentially equivalent problem of
reconstructing a function on a sphere from its integrals over large circles.
In the late sixties, the first applications of the Radon formula appeared,
first in radio astronomy (R. N. Bracewell), and then in electron microgra
phy (A. Klug, B. K. Vainstein). At the same time, attempts to apply the
Radon transform to X-ray tomography (reconstruction of plane sections
of an object) were initiated. The study of tomograms was essential in ra
diography from the twenties at least, when the first analogue devices for
such studies had been developed. Presumably, it is not a pure coincidence
that the idea of using the Radon formula for constructing tomograms ap
peared simultaneously with the first computers, which allowed one to bring
this idea into practice. In 1970 the first computer tomograph that could
be used in clinical work was introduced. G. N. Hounsfield, its inven
tor, and A. M. Cormack, who developed the mathematical and computa
tional aspects of tomography, were jointly awarded the 1979 Nobel prize in
medicine. Today computerized tomography has become one of the most
important techniques in medicine. The 1982 Nobel prize in chemistry was
awarded to A. Klug for a series of papers in which tomographic methods
in electron micrography were essential.
Practical applications of the Radon transform, especially in medical to
mography, made it very popular among mathematicians. This is not only
because new possibilities for mathematicians in an applied research field
have appeared (and the number of spectacular applications of the Radon
transform constantly grows, ranging from the geological research carried
out to design the Washington subway system to the reconstruction of the
inner coma of Halley’s comet from observational data obtained by the
©1990 American Mathematical Society
0065-9282/90 $1.00 + $.25 per page
2 I. M. GELFAND AND S. G. GINDIKJN
“Vega-1” space station). Undoubtedly, what a mathematician is really at
tracted by is a chance to use really deep mathematics in applications, as
well as to tackle the new serious problems which immediately appear when
one starts doing practical applications.
Any mathematician entering the field of medical tomography meets with
a number of unusual features. First, he has to abandon the smooth func
tions he is so accustomed to. The reason is that the bones contained in
a human body cause discontinuities of optical density. Second, any con
vergence different from pointwise convergence is practically useless, and
obtaining estimates involving pointwise convergence is an extremely dif
ficult task. Third, in searching for an optimal algorithm, one has to take
discretization into account; otherwise it may happen that the computa
tional process becomes divergent as the discretization step tends to zero.
This list can be continued. Unfortunately, quite a lot of papers from the
stream of mathematical work somehow related to computerized tomog
raphy do not satisfy these strict practical requirements, and, at the same
time, are of no independent mathematical interest. On the other hand,
one can definitely affirm that medical requirements in tomography cannot
be met empirically and require serious theoretical analysis.
In this vein, D. A. Popov’s paper in this book appears to be extremely
important. In it the author considers the convergence problem for the
convolution and backprojection algorithm (which is most commonly used
in computerized tomography) in maximum generality. This is an example
of a rigorous mathematical paper in which, however, all practical require
ments are fully taken into consideration: the author studies discontinuous
functions and pointwise convergence, the discretization step plays the role
of a regularization parameter, etc. The reader can see by himself that
working in these extremal conditions requires very serious mathematical
techniques, and that the analytical difficulties one has to overcome are
indeed enormous.
The Radon transform is the simplest one in the class of integral trans
forms of geometrical nature which are studied by integral geometry - a
branch of modern functional analysis with very deep connections to vari
ous fields of mathematics and mathematical physics. It seems to us that a
specialist in tomography must become quite familiar with the present situ
ation in integral geometry. There are a number of examples where integral
geometry provides means to overcome certain difficulties in tomography
or to implement certain additional possibilities.
This is true even for the Radon transform itself. For instance, once
the projective nature of the Radon transform is understood, it is easy to
INTRODUCTION 3
grasp the equivalence of the Radon and Funk-Minkovsky transforms, or,
in the inversion formula, to pass from parallel beams to divergent beams
emanating from points on the scanner line. Some other examples are
provided by papers in this book.
A. B. Goncharov considers the problem of electron micrography in which
one has to reconstruct relative projection angles from a given set of projec
tions when these angles are a priori unknown. The solution of this problem
stems from the fact that the Radon transform of a function is far from be
ing an arbitrary function on the set of lines; namely, it satisfies some rather
strong moment conditions (“Cavalieri conditions”). It turns out that these
conditions suffice to determine relative angles. Theoretical and practical
aspects of the applications of these methods to reconstruction of biological
objects (e.g., ribosomes) are discussed in the paper by M. S. Gelfand and
A. B. Goncharov.
It is known in integral geometry that the Radon transform of a com
pactly supported function also satisfies some other conditions of analyti
cal nature, which allow one to reconstruct this Radon transform from its
restriction to the set of lines whose direction vectors fill some solid angle.
V. P. Palamodov starts from this observation in dealing with the problem
of how to reconstruct a function from its incomplete Radon data. His pa
per shows that some important additional ideas are necessary in order to
transform this observation into a realistic computational algorithm. Some
other mathematical problems in tomography are also discussed in Palam-
odov’s paper. They include the reconstruction of nonsmooth functions
from their Radon transforms, and the study of a mathematical model for
nonlinear artifacts in practical tomography.
Another problem closely related to integral geometry is also discussed
in Palamodov’s paper. One of the first problems in integral geometry ap
peared in the paper by I. M. Gelfand and M. A. Naimark on the Plancherel
formula for the Lorentz group. The problem is to reconstruct a function
in three-dimensional complex space from its integrals over all lines inter
secting a fixed hyperbola. The point is that the family of all lines depends
on four parameters, and the natural problem would be to reconstruct a
function from its integrals over all lines from some three-parameter sub
family. The set of all lines intersecting a hyperbola is an example of such
a subfamily. In the above-mentioned paper an inversion formula was de
rived; as it turned out later, this formula can be automatically generalized
to an arbitrary algebraic curve.
Formulas for the complex case can be easily rewritten so as to become
applicable to the real case. It appears that the integrals over all lines inter
4 I. M. GELFAND AND S. G. GINDIKIN
secting a curve do not suffice to reconstruct a function. However, a func
tion can be reconstructed if, in addition, we know integrals over planes
that do not intersect the curve. Specialists in tomography recently became
interested in this fact in the following situation. Suppose the support of a
function satisfies the following condition: any plane that does not intersect
the curve does not intersect the support either. Then, to reconstruct the
function, it suffices to know only the integrals over lines intersecting the
curve.
The paper by N. D. Vvedenskaya and S. G. Gindikin also stems from a
fact of integral geometry. There exists a discrete Radon transform whose
relation to the standard Radon transform is similar to that of the Fourier
series to the Fourier integral. For this discrete Radon transform, there is
a simple inversion formula which yields a new inversion formula for the
Fourier transforms of compactly supported functions with fixed support.
Starting from this formula, a computational algorithm for the reconstruc
tion of the image from its projections was developed.
The paper by G. M. Henkin and A. A. Shananm contains yet another
application of the Radon transform, this time to economics, namely to
production function theory. This new application leads to a new natu
ral class of problems about the Radon transform. The main point is the
study of Radon transforms of positive functions supported on the posi
tive coordinate “octant” in «-dimensional space. In this study, interesting
connections with classical papers by S. N. Bernstein about integral trans
forms of positive measures and with current problems in multidimensional
complex analysis were clarified.
One must say that tomography has outgrown the scope of the Radon
transform long ago. For instance, the passage from A'-ray to other types
of radiation refraction and diffraction phenomena cannot be ignored. A
thorough analysis of problems in diffraction tomography is carried out
here by V. M. Buchstaber and V. K. Maslov. Their paper is oriented both
to exact mathematical problems and to realistic algorithms. New inverse
problems which appear to be quite promising are stated.
Very interesting mathematical problems also arise when one starts to
consider refraction phenomena in tomography. First of all, we have to pass
from integration over lines to integration over curves, and the resulting
“curved problem” of integral geometry is very appealing. For this problem
some uniqueness theorems are proved, but exact inversion formulas are
practically absent. The existence of local inversion formulas in the complex
analogue of this problem (when the value of a function at a point can be
reconstructed from its integrals over curves close to this point) has been
