We deal with the question of uniqueness, namely to decide when an unknown finite set of points in Z 2 is uniquely determined by its X-rays corresponding to a given set S of lattice directions. In Hajdu (2005) [11] proved that for any fixed rectangle A in Z 2 there exists a non trivial set S of four lattice directions, depending only on the size of A, such that any two subsets of A can be distinguished by means of their X-rays taken in the directions in S. The proof was given by explicitly constructing a suitable set S 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

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.

Abstract. In Discrete Tomography there is a wide literature concerning (weakly) bad configurations. These occur in dealing with several questions concerning the important issues of uniqueness and additivity. Discrete lattice sets which are additive with respect to a given set S of lattice directions are uniquely determined by X-rays in the direction of S. These sets are characterized by the absence of weakly bad configurations for S. On the other side, if a set has a bad configuration with respect to S, then it is not uniquely determined by the X-rays in the directions of S, and consequently it is also non-additive. Between these two opposite situations there are also the non-additive sets of uniqueness, which deserve interest in Discrete Tomography, since their unique reconstruction cannot be derived via the additivity property. In this paper we wish to investigate possible interplays among such notions in a given lattice grid A, under X-rays taken in directions belonging to a set S of four lattice directions.

Digital convex (DC) sets plays a prominent role in the framework of digital geometry providing a natural generalization to the concept of Euclidean convexity when we are dealing with polyominoes, i.e., finite and connected sets of points. A result by Brlek, Lachaud, Provençal and Reutenauer (see [4]) on this topic sets a bridge between digital convexity and combinatorics on words: the boundary word of a DC polyomino can be divided in four monotone paths, each of them having a Lyndon factorization that contains only Christoffel words. The intent of this paper is to provide some local properties that a boundary words has to fulfill in order to allow a single point modifications that preserves the convexity of the polyomino.