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
Previous Page Next Page