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
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