DISCRETE TOMOGRAPHY - DETERMINATION OF FINITE SETS BY X-RAYS

We study the determination of finite subsets of the integer lattice Z( n), n greater than or equal to 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on ea ch line parallel to u. For practical reasons, only X-rays in lattice d irections, that is, directions parallel to a nonzero vector in the lat tice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are Sour prescribed lattice direct ions such that convex subsets of Z(n) (i.e., finite subsets F with F = Z(n) boolean AND conv F) are determined, among all such sets, by thei r X-rays in these directions. We also show that three X-rays do not su ffice for this purpose. This answers a question of Larry Shepp, and yi elds a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice dir ections in Z(2) have the property that convex subsets of Z(2) are dete rmined, among all such sets, by their X-rays in these directions. We a lso consider the use of orthogonal projections in the interactive tech nique of successive determination, in which the information from previ ous projections can be used in deciding the direction for the next pro jection. We obtain results for finite subsets of the integer lattice a nd also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.