LOCAL TOMOGRAPHY .2.

Standard, or global, tomography involves the reconstruction of a funct ion f from line integrals. Local tomography, in this paper, involves t he reconstruction of a related function, Lf = alpha(Lambda f + mu Lamb da(-1)f), where is is the square root of the positive Laplacian, -Delt a. This article is a sequel to the article ''Local Tomography'' [SIAM J. Appl. Math., 52 (1992), pp. 459-484, 1193-1198] by Faridani, Ritman , and Smith. The principal new results are (1) good bounds for Lambda f and Lambda(-1)f outside the support of f, particularly when f has 0 moments up to some order; (2) identification and reduction of global e ffects in local tomography, i.e., identification and reduction of the dependence of Lf(x) on the values of f at points at an intermediate di stance from 2; (3) an algorithm for computing approximate density jump s from Lambda f when f is a linear combination of characteristic funct ions and a smooth background. Several examples are given: some from re al x-ray data, some from mathematical phantoms. They include three-dim ensional 7-micron resolution reconstructions from microtomographic sca ns.