On Some Geometric Aspects of the Class of hv-Convex Switching Components

In the usual aim of discrete tomography, the reconstruction of an unknown discrete set is considered, by means of projection data collected along a set U of discrete directions. Possible ambiguous reconstructions can arise if and only if switching components occur, namely, if and only if non-empty images exist having null projections along all the directions in U. In order to lower the number of allowed reconstructions, one tries to incorporate possible extra geometric constraints in the tomographic problem. In particular, the class P of horizontally and vertically convex connected sets (briefly, hv-convex polyominoes) has been largely considered. In this paper we introduce the class of hv-convex switching components, and prove some preliminary results on their geometric structure. The class includes all switching components arising when the tomographic problem is considered in P, which highly motivates the investigation of such configurations. It turns out that the considered class can be partitioned in two disjointed subclasses of closed patterns, called windows and curls, respectively. It follows that all windows have a unique representation, while curls consist of interlaced sequences of sub-patterns, called Z-paths, which leads to the problem of understanding the combinatorial structure of such sequences. We provide explicit constructions of families of curls associated to some special sequences, and also give additional details on further allowed or forbidden configurations by means of a number of illustrative examples

Characterization of hv-Convex Sequences

Reconstructing a discrete object by means of X-rays along a finite set U of (discrete) directions represents one of the main task in discrete tomography. Indeed, it is an ill-posed inverse problem, since different structures exist having the same projections along all lines whose directions range in U. Characteristic of ambiguous reconstructions are special configurations, called switching components, whose understanding represents a main issue in discrete tomography, and an independent interesting geometric problem as well. The investigation of switching component usually bases on some kind of prior knowledge that is incorporated in the tomographic problem. In this paper, we focus on switching components under the constraint of convexity along the horizontal and the vertical directions imposed to the unknown object. Moving from their geometric characterization in windows and curls, we provide a numerical description, by encoding them as special sequences of integers. A detailed study of these sequences leads to the complete understanding of their combinatorial structure, and to a polynomial-time algorithm that explicitly reconstructs any of them from a pair of integers arbitrarily given