Local Tomography

Tomography produces the reconstruction of a function f from a large number of line integrals of f. Conventional tomography is a global procedure in that the standard convolution formulas for reconstruction at a single point require the integrals over all lines within some plane containing the point. Local tomography, as introduced initially, produced the reconstruction of the related function $\Lambda f$, where $\Lambda $ is the square root of $ - \Delta $, the positive Laplace operator. The reconstruction of $\Lambda f$ is local in that reconstruction at a point requires integrals only over lines passing infinitesimally close to the point, and $\Lambda f$ has the same smooth regions and boundaries as f. However, $\Lambda f$ is cupped in regions where f is constant. $\Lambda ^{ - 1} f$, also amenable to local reconstruction, is smooth everywhere and contains a counter-cup. This article provides a detailed study of the actions of $\Lambda $ and $\Lambda ^{ - 1} $, and shows several examples of what can be achieved with a linear combination. It includes the results of x-ray experiments in which the line integrals are obtained from attenuation measurements on two-dimensional image intensifiers and fluorescent screens, instead of the usual linear detector arrays.