On some geometric aspects of the class of hv-convex switching components

TL;DR

This paper explores the geometric structure of hv-convex switching components in discrete tomography, introducing a new class called hv-convex switching components and analyzing their properties, including their partition into windows and curls.

Contribution

It introduces the class of hv-convex switching components, analyzes their geometric structure, and classifies them into windows and curls, providing explicit constructions and examples.

Findings

Class of hv-convex switching components includes all switching components in the class of hv-convex sets.

Partition of switching components into two subclasses: windows with unique representation, and curls with complex Z-path sequences.

Explicit constructions of curls and analysis of configurations expand understanding of ambiguous reconstructions.

Abstract

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 $\mathbb{P}$ 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 $\mathbb{P}$, 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.