Preface

One of the most exciting features of the fields of Radon Transforms and Tomog-

raphy is the strong relationship between high level pure mathematics and applica-

tions to areas such as medical imaging, remote sensing, and industrial nondestruc-

tive evaluation. These proceedings bring together fundamental research articles in

the major areas of tomography and Radon transforms.

The proceedings include six expository papers that we hope will be valuable to

beginners as well as advanced researchers. Local tomography is an extremely useful

new type of tomography in which local data are used to reconstruct the singularities

or shapes of the objects to be reconstructed. Fundamental articles are included on

local tomography and wavelets [Berenstein] as well as on Lambda tomography and

related methods [Faridani, Buglione, Huabsomboon, Iancu, McGrath]. Margaret

Cheney gives a mathematical tutorial of Synthetic Aperture RADAR and shows

how tomographic methods come up naturally in this setting. Ultrasound tomog-

raphy is based on an inverse problem for the Helmholz equation. Frank N atterer

discusses the general problem and linear approximations and he provides a survey

to iterative inversion methods. One article shows how microlocal analysis is used to

prove support theorems for the hyperplane transform and how such theorems for a

spherical transform characterize stationary sets for the wave equation [Quinto]. The

Pompeiu problem is to find sets that determine functions from integrals over rigid

motions of the given set. Larry Zalcman updates his comprehensive bibliogmphy

on this problem from 1992 and provides valuable commentary.

The major themes in Radon transforms and tomography are represented among

the research articles. Bailey and Eastwood use spectral sequences to formulate and

examine the X-ray and related transforms on Grassmmann manifolds, obtaining

useful results about null spaces and ranges. Vector tomography models integration

of vector (or tensor) fields over curves and surfaces. Support theorems and injectiv-

ity are proven for vectorial transforms on divergence free vector fields integrating

over lines in

Rn

in real analytic measures, and counterexamples are provided if the

measure is only

coo

[Boman]. Injectivity theorems are proven for the solenoidal

part of tensor fields from integrals over complex lines intersecting a given complex

curve in

en

[Vertgeim]. The paper by Ehrenpreis discusses three problems posed

during the conference. These involve the recovery of both test data and defining

measures from Radon transforms values, the uniqueness properties of a transform

that integrates over spheres tangent to a surface, and a variation on Morera's theo-

rem for detecting analyticity properties of functions from integrals. Gonzalez proves

a Paley-Wiener Theorem for the Fourier coefficients of central functions on compact

Lie groups as a consequence

of a corresponding Paley-Wiener Theorem for ordinary

ix