JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.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,where A is the square root of the positive Laplacian, -A. This article is a sequel to the article "] by Faridani, Ritman, and Smith. The principal new results are (1) good bounds for Af and A-lf 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 Af 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.
We consider Shannon sampling theory for sampling sets which are unions of shifted lattices. These sets are not necessarily periodic. A function f can be reconstructed from its samples provided the sampling set and the support of the Fourier transform of f satisfy certain compatibility conditions. An explicit reconstruction formula is given for sampling sets which are unions of two shifted lattices. While explicit formulas for unions of more than two lattices are possible, it is more convenient to use a recursive algorithm. The analysis is presented in the general framework of locally compact abelian groups, but several specific examples are given, including a numerical example implemented in MATLAB. Our methods also provide a new tool for designing sampling sets of minimal density.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.