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

Abstract

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 A containing the object to be reconstructed. Instead of looking for uniqueness in the whole grid A, we want to address the problem from a local point of view. More precisely, given a limited number of directions, we aim in showing, first of all, which is the subregion of 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.

Algorithms for linear time reconstruction by discrete tomography II

“…In 2015 Dulio, Frosini and Pagani [6] showed that in the corners of A the function values are uniquely determined and can be computed in linear time if the number of directions d = 2. Later they proved conditional results for d = 3 [7,8]. Recently, Pagani and Tijdeman [19] generalized the result for any number of directions.…”

A linear time approach to three-dimensional reconstruction by discrete tomography

“…They also showed that the problem is NP-complete for d ≥ 2 when the function contains six or more different values. In 2015 Dulio, Frosini and Pagani proved that the corners of A can be uniquely determined in linear time for d = 2 [3], and gave conditional results for d = 3 [4,5]. This result was generalised by Pagani and Tijdeman for any number of directions [11].…”

“…The switching function is elementary. By (4) we have T = {(0, 2, 2), (1, 0, 3), (1, 2, 2), (1, 3, 0), (1,3,4), (2, 0, 3), (2, 1, 1), (2, 1, 5), (2, 3, 0), (2,3,4), (2, 4, 2), (3, 1, 1), (3, 1, 5), (3, 2, 3), (3, 4, 2), (4, 2, 3)} .…”

Further steps on the reconstruction of convex polyominoes from orthogonal projections

“…In general, the number of matrices sharing the same projections grows exponentially with their dimension, so in most practical applications some extra information are needed to achieve a solution as close as possible to a starting unknown object. So, researchers tackle the algorithmic challenges of limiting the class of possible solutions in different ways: increasing the number of projections (Gardner and Gritzmann 1997;Gardner et al 1999), fixing the wideness of the unknown object (Dulio et al 2015(Dulio et al , 2016(Dulio et al , 2017b or adding geometrical information (mainly connectedness and convexity). Concerning this last, among connected sets a dominant role deserved polyominoes, that are commonly intended as finite 4-connected sets of points of the integer lattice, considered up to translation.…”