Discrete tomography determination of bounded lattice sets from four X-rays

We deal with the question of uniqueness, namely to decide when an unknown finite set of points in Z2Z2 is uniquely determined by its X-rays corresponding to a given set SS of lattice directions. In Hajdu (2005)  [11] proved that for any fixed rectangle AA in Z2Z2 there exists a non trivial set SS of four lattice directions, depending only on the size of AA, such that any two subsets of AA can be distinguished by means of their X-rays taken in the directions in SS. The proof was given by explicitly constructing a suitable set SS in any possible case. We improve this result by showing that in fact whole families of suitable sets of four directions can be found, for which we provide a complete characterization. This permits us to easily solve some related problems and the computational aspects.