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Abstract: In this paper we deal with uniqueness and reconstruction problems in Discrete Tomography. For a finite set of directions in , we introduce the class of -inscribable lattice sets, and give a detailed description of their geometric structure. This shows that such sets can be considered as the natural discrete counterpart of the same notion known in the continuous case, as well as a kind of generalization of the class of the so-called -convex polyominoes (or moon polyominoes). In view of reconstruction from projections along the directions in , two related questions of tomographic interest are investigated, namely uniqueness and additivity. We show that both properties are fulfilled by -inscribable lattice sets. Moreover, concerning the case , we provide an explicit reconstruction algorithm from the knowledge of directed horizontal and vertical X-rays, jointly with a few preliminary results towards a possible sharp stability inequality. [Copyright &y& Elsevier]

Abstract: A plane partition is a matrix , where and , with non-negative integer entries, and whose rows and columns are weakly decreasing. From a geometric point of view plane partitions are equivalent to pyramids, subsets of the integer lattice which play an important role in Discrete Tomography. As a consequence, some typical problems concerning the tomography of discrete lattice sets can be rephrased and considered via plane partitions. In this paper we focus on some of them. In particular, we get a necessary and sufficient condition for additivity, a canonical procedure for checking the existence of (weakly) bad configurations, and an algorithm which constructs minimal pyramids (with respect to the number of levels) with assigned projection of a bad configurations. [Copyright &y& Elsevier]

In this paper we investigate bounded additivity in Discrete Tomography. This notion has been previously introduced in [5] , as a generalization of the original one in [11] , which was given in terms of ridge functions . We exploit results from [6–8] to deal with bounded S non-additive sets of uniqueness, where S ⊂ Z n contains d coordinate directions { e 1 , … , e d } , | S | = d + 1 , and n ≥ d ≥ 3 . We prove that, when the union of two special subsets of { e 1 , … , e d } has cardinality k = n , then bounded S non-additive sets of uniqueness are confined in a grid A having a suitable fixed size in each coordinate direction e i , whereas, if k < n , the grid A can be arbitrarily large in each coordinate direction e i , where i > k . The subclass of pure bounded S non-additive sets plays a special role. We also compute explicitly the proportion of bounded S non-additive sets of uniqueness w.r.t. those additive, as well as w.r.t. the S -unique sets. This confirms a conjecture proposed by Fishburn et al. in [14] for the class of bounded sets. [ABSTRACT FROM AUTHOR]

This paper deals with the uniqueness problem for bounded sets in Discrete Tomography, namely to decide when an unknown finite subset of points in a multidimensional grid is uniquely determined by its X-rays corresponding to a given set of lattice directions. In Brunetti et al. (2013) the authors completely settled the problem in dimension two, by proving that whole families of suitable sets of four directions can be found, for which they provided a complete characterization. In this paper we examine the problem in higher dimensions. We show that d + 1 represents the minimal number of directions we need in Z n ( n ≥ d ≥ 3 ) to distinguish all the subsets of a finite grid by their X-rays in these directions, under the requirement that such directions span a d -dimensional subspace of Z n . It turns out that the situation differs from the planar case, where less than four directions are never sufficient for uniqueness. Moreover we characterize the minimal sets of directions of uniqueness by providing a necessary and sufficient condition. As a consequence, results in Brunetti et al. (2013) can be easily extended to the n -dimensional case. [ABSTRACT FROM AUTHOR]

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