In Discrete Tomography, objects are reconstructed by means of their projections along certain directions. It is known that, for any given lattice grid, special sets of four valid projections exist that ensure uniqueness of reconstruction in the whole grid. However, in real applications, some physical or mechanical constraints could prevent the use of such theoretical uniqueness results, and one must employ projections fitting some further constraints. It turns out that global uniqueness cannot be guaranteed, even if, in some special areas included in the grid, uniqueness might be still preserved. In this paper we address such a question of local uniqueness. In particular, we wish to focus on the problem of characterizing, in a sufficiently large lattice rectangular grid, the sub-region which is uniquely determined under a set S of generic projections. It turns out that the regions of local uniqueness consist of some curious twisting of rectangular areas. This deserves a special interest even from the pure combinatorial point of view, and can be explained by means of numerical relations among the entries of the employed directions.

In the reconstruction problem of Discrete Tomography, projections are considered from a finite set S of lattice directions. Employing a limited number of projections implies that the injectivity of the Radon transform is lost, and, in general, images consistent with a given set of projections form a huge class. In order to lower the number of allowed solutions, one usually tries to include in the problem some a priori information. This suggests that modeling the tomographic reconstruction problem as a linear system of equations is preferable.In this paper we propose to restrict the usual notion of uniqueness, related to the solutions of the linear system, and to provide, for each set S, a geometrical characterization of the shape of a lattice subset, say region of uniqueness (ROU), forming a partial, fast reconstructible, solution.Any selected set S intrinsically determines its ROU inside an arbitrary lattice grid. For instance, trivially, if |S| = 1, the ROU is represented by two rectangles having sizes equal to the absolute values of the entries of the unique direction in S, and placed at two opposite corners of the chosen grid. Surprisingly, if |S| = 2, the problem becomes much more complicated.Our purpose is to provide a geometrical characterization of the ROU. This is based on a double Euclidean division algorithm (DEDA), which runs in polynomial time. It turns out that the ROU is delimited by a zigzag profile obtained by means of numerical relations among the entries of the employed directions. According to different inputs in DEDA, the shape of the ROU can change consistently, as it can be easily observed from the provided examples. Moreover, after selecting a region of interest (ROI) from a given phantom, we exploit DEDA to reconstruct the part of the ROI which falls in the ROU and, with a few further a priori knowledge, even parts of the ROI which are outside the ROU.

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In discrete tomographic image reconstruction, projections are taken along a finite set S of valid directions for a working grid A. In general, uniqueness cannot be achieved in the whole grid A. Usually, some information on the object to be reconstructed is introduced, that, sometimes, allows possible ambiguities to be removed. From a different perspective, one aims in finding subregions of A where uniqueness can be guaranteed, and obtained in linear time, only from the knowledge of S. When S consists of two lattice directions, the shape of any such region of uniqueness, say ROU, have been completely characterized in previous works by means of a double Euclidean division algorithm called DEDA. Results have been later extended to special triples of directions, under a suitable assumption on their entries. In this paper we remove the previous assumption, so providing a complete characterization of the shape of the ROU for such kind of triples. We also show that the employed strategy can be even applied to more general sets of three directions, where the corresponding ROU can be characterized as well. Independently of the combinatorial interest of the problem, the result can be exploited to define in advance, namely before using any kind of radiation, suitable sets of directions that allow regions of interest to be included in the corresponding ROU. Results have been proved in all details, and several experiments are considered, in order to support the theoretical steps and to clarify possible applications