Characterization of hv-Convex Sequences

Abstract

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.