Unique reconstruction of bounded sets in discrete tomography

We consider the problem of recognizing arbitrary finite subsets of Z2 by X-rays corresponding to a small set of directions S. We show that if we fix any rectangle A in Z2 then there exists a so called valid set S of four directions (at least when A is not too "small") depending only on the size of A such that any two subsets of A can be distinguished, using these directions only. By our approach this result can easily be extended to any lattice, in any dimension.