Discrete Tomography through Distribution Theory

Abstract

Discrete tomography concerns with the problem of reconstruction of a function $f$ on ${\mathbf{Z}}^n$ from various sums $f_{\mathbf{t}+\mathbf{v}}={\Sigma }_{\mathbf{x}\in \mathbf{t}+\mathbf{v}}f(\mathbf{x})$, $\mathbf{v}\in {\mathbf{Z}}^n$ , where $\mathbf{t}$ is a fixed finite subset of ${\mathbf{Z}}^n$. In this paper we focus on the structure of the set of functions satisfying $f_{\mathbf{t}+\mathbf{v}}=0$ for any $\mathbf{v}$. 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.