In the usual continuous model for tomography one attempts to reconstruct a function from a knowledge of its line integrals. All the reconstruction methods used in computerized tomography require a large number of projection images to obtain results of acceptable quality.

The field of discrete tomography focuses on the reconstruction of samples, that consist of only a few different materials, from a "small" number of projections. For instance, it can be applied to the reconstruction of nanocrystals at atomic resolution, where it is assumed that the crystal contains only a few types of atoms, and that the atoms lie on a regular grid, modeled by the integer lattice. The high energies required to produce the discrete X-rays of a crystal mean that just a few number of X-rays can be taken before the crystal is damaged, so that the conventional techniques of computerized tomography fail.

In general, this reconstruction task is a ill-posed inverse problem. In fact, for general data there need not exist a solution, if the data is consistent, the solution need not be uniquely determined and "small" changes in the data can lead to unique but disjoint solutions. Thus, one has to use a priori information, such as convexity or connectedness, about the sets that have to be reconstructed to satisfy existence, uniqueness and stability requirements.

By now there are many uniqueness results available for different classes of finite lattice sets, but just few stability results. In the present paper we introduce some new classes of lattice sets, and investigate the problem of their reconstruction by means of their X-rays taken in the directions belonging to given finite set D. The geometric structure of such sets enable us to prove results concerning additivity and uniqueness. When D is the set of coordinate directions, we give a sharp stability estimate which depends only on the data error, differently from all the known results, which also involve the sizes of the sets. Some of these results hold true in any dimension.