Local Tomography II

Abstract

Standard, or global, tomography involves the reconstruction of a function f from line integrals. Local tomography, in this paper, involves the reconstruction of a related function, $Lf = \alpha (\Lambda f +\mu\Lambda^{-1} f),$ where $\Lambda$ is the square root of the positive Laplacian, $-\Delta .$ 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 effects in local tomography, i.e., identification and reduction of the dependence of Lf(x) on the values of f at points at an intermediate distance from x; (3) an algorithm for computing approximate density jumps from $\Lambda f$ when f is a linear combination of characteristic functions and a smooth background. Several examples are given: some from real x-ray data, some from mathematical phantoms. They include three-dimensional 7-micron resolution reconstructions from microtomographic scans.