Discrete tomography: Determination of finite sets by X-rays

Abstract

Abstract. We study the determination of finite subsets of the integer lattice Z n , n ≥ 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 each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are four prescribed lattice directions such that convex subsets of Z n (i.e., finite subsets F with F = Z n ∩ conv F ) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in Z 2 have the property that convex subsets of Z 2 are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.

13. On the reconstruction of static and dynamic discrete structures

“…Theorem 6 ( [96]). There are S 1 , S 2 , S 3 , S 4 ∈ L d such that every finite convex lattice set F is uniquely determined by X S 1 F, .…”

Ghosts in Discrete Tomography

“…A complete characterization of bad configurations (weakly or not weakly) has been obtained in [25] with a new algebraic approach employed then in several papers (see for instance [27][28][29][30]). S-bad configurations, with the extra condition of convexity, are known as S-polygons, and reveal to be useful both in geometric tomography and in discrete tomography (see for instance [31,32], and [33] for an algorithmic approach), as well as very interesting also from a purely geometric point of view (see for instance [34][35][36]). In this paper we focus on "minimal" bad configurations.…”

Discrete tomography of icosahedral model sets

“…For S & R d , we denote by L S u the subset of L d u consisting of lines in L d u which pass through at least one point of S. Lemma 2.2. [Lemmas 5.1 and 5.4 of Gardner & Gritzmann (1997).] Let d 2 N and let u 2 S dÀ1 be a direction.…”

“…Here, a Ã-direction is parallel to a nonzero interpoint vector of Ã. Since this reconstruction problem can possess rather different solutions, we also study the uniqueness problem of finding a small number of suitably prescribed Ã-directions that eliminate these non-uniqueness phenomena [see Gardner & Gritzmann (1997, Fishburn et al (1991) and Fishburn & Shepp (1999)]. More precisely, a subset E of the set of all finite subsets of a fixed icosahedral model set à is said to be determined by the X-rays in a finite set U of directions if different sets in E cannot have the same X-rays in the directions of U.…”