Non-additive bounded sets of uniqueness in Zn

Abstract

A main problem in discrete tomography consists in looking for theoretical models which ensure uniqueness of reconstruction. To this, lattice sets of points, contained in a multidimensional grid A = [m1] × [m2] × ... × [mn] (where for p ∈ N, [p] = {0, 1,..., p - 1}), are investigated by means of X-rays in a given set S of lattice directions. Without introducing any noise effect, one aims in finding the minimal cardinality of S which guarantees solution to the uniqueness problem. In a previous work the matter has been completely settled in dimension two, and later extended to higher dimension. It turns out that d+1 represents the minimal number of directions one needs in Zn (n ≥ d ≥ 3), under the requirement that such directions span a d-dimensional subspace of Zn. Also, those sets of d + 1 directions have been explicitly characterized. However, in view of applications, it might be quite difficult to decide whether the uniqueness problem has a solution, when X-rays are taken in a set of more than two lattice directions. In order to get computational simpler approaches, some prior knowledge is usually required on the object to be reconstructed. A powerful information is provided by additivity, since additive sets are reconstructible in polynomial time by using linear programming. In this paper we compute the proportion of non-additive sets of uniqueness with respect to additive sets in a given grid A ⊂ Zn, in the important case when d coordinate directions are employed.