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