Preface
The strong relationship between deep pure mathematics and applications is an
exciting feature of integral geometry and its applied cousin tomography. This vol-
ume brings together fundamental research in these areas. It grew out of the special
session, "Tomography and Integral Geometry" that met at the 997th Meeting of
the American Mathematical Society at Rider University, April 17th and 18th, 2004.
Articles in these proceedings are written on spherical Radon transforms
[4,
18],
the k-plane transform
[25],
and transforms on Grassmannians and Stiefel
manifolds [15, 29]. Applications to partial differential equations are included in
the papers
[4, 18, 32],
and results on tomography may be found in
[7, 10, 20].
The
paper
[13]
is concerned with wavelets and exhibits the interplay between geometry
and analysis typical of integral geometry. The paper
[30]
contains several interesting
results including one on reconstructing functions from an averaging process.
Integral geometry divides into two major branches. Probabilistic integral geom-
etry is concerned with the application of probability theory to geometric problems
and is characterized by such results as Crofton's theorem and the Buffon needle
problem. The solution to the Buffon needle problem determines the probability
that a needle that is randomly dropped onto a plane with equally spaced parallel
lines will intersect one of these lines. The other branch of integral geometry may
be called Radon integral geometry. It investigates properties of functions that can
be determined by integration over families of submanifolds of a given manifold.
In more detail, this version of integral geometry is concerned with properties of
functions that can be determined by a pair of integral transforms called the Radon
transform and the dual Radon transform.
These transforms are intimately connected with the geometry of the ambient
manifold. If it is possible to specify an incidence relation between two families of
subsets
A
and
B
of the manifold, then one defines the Radon transform as the
integral operator which for each l E B integrates functions over sets a in A incident
to l. In all interesting cases one can define a dual incidence relation and consequently
a dual Radon transform. This yields a profound interaction between the geometry
and the analysis of the manifold which allows, in many cases, the determination
of a function by knowing its Radon transform. The classic example is the Radon
transform on lines in
JR2
.
In this case the Radon transform of a density function
represents the attenuation of X-rays travelling along lines through the object, and
the dual Radon transform corresponds to the operation of backprojection.
A practical application of this idea is X-ray computerized tomography in which
the structure of a two-dimensional object can be determined by its integrals over
vii
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