In Discrete Tomography, a main issue is to find conditions which ensure uniqueness of reconstruction in a given lattice grid A, by means of a finite set S of projections.
Several methods have been proposed in order to attack the problem, and theoretical results have been found, depending on the different employed discrete models.
In a previous paper, and in the context of binary tomography, we gave a necessary and sufficient uniqueness reconstruction condition, holding in the grid model for a set S consisting of four lattice directions. Starting from this result, we investigate two possible aspects which could be of interest in real applications.
On one hand we consider the tomographic reconstruction problem modeled as a linear system Wx=p, where x is the image to be reconstructed and p is the vector collecting the projections. We compute p for different sets S of four directions, pointing out how suitable choices allow perfect (noise-free) reconstructions. Further, in order to manage also nosy reconstructions, we propose to switch from the grid model to the strip model.
On the other hand, one could be prevented from using suitable sets of directions. Moreover, the tomographic problem is not necessary binary, and could involve gray-scale image reconstructions. In these cases a general uniqueness result is not known. However, it is often the case that one is not really interested in reconstructing the whole working grid A, but, rather, some portions of A which are of special interest for a given tomographic problem. To this, it is useful to know in advance the shape of the sub-regions of A where uniqueness can be quickly achieved.
A geometrical characterization of the uniqueness profile determined by a general set of two directions is presented, and examples are commented in view of possible applications.