Local reconstructions in exponential tomography

Abstract

CIC-3, MS K-990, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 In this paper, pseudolocal and local approaches to the tomographic reconstruction of discontinuities of an unknown function f from its exponential Radon transform data g(θ, p) are developed. A function f is supposed to be piecewise-continuous and compactly supported. A pseudolocal tomography function fd(x) is introduced and it is proved that the difference f - fd is continuous. Therefore, locations and values of jumps of f can be recovered from fd, the computation of which is pseudolocal: for the reconstruction of fd at a point x one needs to know g(θ, p) only for (θ, p) satisfying |Θ · x - p| ≤ d. Investigation of the properties of fd as d → 0 is given. Also a local exponential tomography function f(μ)∧ is proposed and it is proved that f(μ)∧ is the result of action on f of an elliptic pseudo-differential operator with the principal symbol |ξ|. Thus sing supp(f(μ)∧) = sing supp(f) and, moreover, the asymptotics of f(μ)∧(x) as x → S, the discontinuity curve of f, are established. These asymptotics allow one to find values of jumps of f across S using local exponential tomography. © 1996 Academic Press, Inc.