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, such as the request for connectedness, or some reconstruction satisfying special convexity constraints. 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, also in view of possible uniqueness results for hv-convex polyominoes. It turns out that the considered class can be partitioned into two disjointed subclasses of closed patterns, called windows and curls, respectively, according to whether the pattern can be travelled by turning always clockwise (or always counterclockwise), or whether points with different turning directions exist. 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.

On Some Geometric Aspects of the Class of hv-Convex Switching Components / Dulio P.; Frosini A.. - STAMPA. - 12708:(2021), pp. 299-311. (Intervento presentato al convegno 1st International Joint Conference on Discrete Geometry and Mathematical Morphology, DGMM 2021 tenutosi a swe nel 2021) [10.1007/978-3-030-76657-3_21].

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

Frosini A.
2021

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, such as the request for connectedness, or some reconstruction satisfying special convexity constraints. 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, also in view of possible uniqueness results for hv-convex polyominoes. It turns out that the considered class can be partitioned into two disjointed subclasses of closed patterns, called windows and curls, respectively, according to whether the pattern can be travelled by turning always clockwise (or always counterclockwise), or whether points with different turning directions exist. 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.
2021
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
1st International Joint Conference on Discrete Geometry and Mathematical Morphology, DGMM 2021
swe
2021
Dulio P.; Frosini A.
File in questo prodotto:
File Dimensione Formato  
CHVSC_DGCI_Meeting.pdf

accesso aperto

Tipologia: Altro
Licenza: Tutti i diritti riservati
Dimensione 496.73 kB
Formato Adobe PDF
496.73 kB Adobe PDF

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/1242514
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? ND
social impact