Geometrical Characterization of the Uniqueness Regions Under Special Sets of Three Directions in Discrete Tomography

The faithful reconstruction of an unknown object from projections, i.e., from measurements of its density function along a finite set of discrete directions, is a challenging task. Some theoretical results prevent, in general, both to perform the reconstruction sufficiently fast, and, even worse, to be sure to obtain, as output, the unknown starting object. In order to reduce the number of possible solutions, one tries to exploit some a priori knowledge. In the present paper we assume to know the size of a lattice grid $\mathcal{A}$ containing the object to be reconstructed. Instead of looking for uniqueness in the whole grid $\mathcal{A}$, we want to address the problem from a \textit{local} point of view. More precisely, given a limited number of directions, we aim in showing, first of all, which is the subregion of $\mathcal{A}$ where pixels are uniquely reconstructible, and then in finding where the reconstruction can be performed quickly (in linear time). In previous works we have characterized the shape of the region of uniqueness (ROU) for any pair of directions. In this paper we show that the results can be extended to special sets of three directions, by splitting them in three different pairs. Moreover, we show that such a procedure cannot be employed for general triples of directions. We also provide applications concerning the obtained characterization of the ROU, and further experiments which underline some regularities in the shape of the ROU corresponding to sets of three not-yet-considered directions.