Discrete Tomography through Distribution Theory

Abstract

Discrete tomography concerns with the problem of reconstruction of a function f on ℤ_n_ from various sums f_t+v = Σx∈t+v f(x), v ∈ ℤ_n , where t is a fixed finite subset of ℤ_n_. In this paper we focus on the structure of the set of functions satisfying _f_t+v = 0 for any v. Through the theory of distributions we deduce a dimension formula for the set of solutions. An intimate connection between the problem and certain types of PDE is revealed too, and it enables one to obtain an efficient algorithm, which constructs a solution from the corresponding PDE.