On the algorithmic inversion of the discrete Radon transform

The present paper deals with the computational complexity of the discrete inverse problem of reconstructing finite point sets and more general functions with finite support that are accessible only through some of the values of their discrete Radon transform. It turns out that this task behaves quite differently from its well-studied companion problem involving 1-dimensional X-rays. Concentrating on the case of coordinate hyperplanes in ℝ^d and on functionals ψ: ℤ^d → D with D ε {{0,1,...,r}, ℕ₀} for some arbitrary but fixed r, we show in particular that the problem can be solved in polynomial time if information is available for m such hyperplanes when m ≤ d - 1 but is ℕℙ-hard for m = d and D = {0,1,...,r}. However, for D = ℕ₀, a case that is relevant in the context of contingency tables, the problem is still in ℙ. Similar results are given for the task of determining the uniqueness of a given solution and for a related counting problem.