Euler Well-Composedness

From LRDE

Abstract

In this paper, we define a new flavour of well-composednesscalled Euler well-composedness, in the general setting of regular cell complexes: A regular cell complex is Euler well-composed if the Euler characteristic of the link of each boundary vertex is . A cell decomposition of a picture is a pair of regular cell complexes such that (resp. ) is a topological and geometrical model representing (resp. its complementary, ). Thena cell decomposition of a picture is self-dual Euler well-composed if both and are Euler well-composed. We prove in this paper that, first, self-dual Euler well-composedness is equivalent to digital well-composedness in dimension 2 and 3, and second, in dimension 4, self-dual Euler well-composedness implies digital well-composednessthough the converse is not true.

Documents

Bibtex (lrde.bib)

@InProceedings{	  boutry.20.iwcia1,
  author	= {Nicolas Boutry and Rocio Gonzalez-Diaz and Maria-Jose
		  Jimenez and Eduardo Paluzo-Hildago},
  title		= {Euler Well-Composedness},
  booktitle	= {Combinatorial Image Analysis: Proceedings of the 20th
		  International Workshop, IWCIA 2020, Novi Sad, Serbia, July
		  16--18, 2020},
  year		= 2020,
  editor	= {T. Lukic and R. P. Barneva and V. Brimkov and L. Comic and
		  N. Sladoje},
  volume	= {12148},
  series	= {Lecture Notes in Computer Science},
  pages		= {3--19},
  publisher	= {Springer},
  abstract	= {In this paper, we define a new flavour of
		  well-composedness, called Euler well-composedness, in the
		  general setting of regular cell complexes: A regular cell
		  complex is Euler well-composed if the Euler characteristic
		  of the link of each boundary vertex is $1$. A cell
		  decomposition of a picture $I$ is a pair of regular cell
		  complexes $\big(K(I),K(\bar{I})\big)$ such that $K(I)$
		  (resp. $K(\bar{I})$) is a topological and geometrical model
		  representing $I$ (resp. its complementary, $\bar{I}$).
		  Then, a cell decomposition of a picture $I$ is self-dual
		  Euler well-composed if both $K(I)$ and $K(\bar{I})$ are
		  Euler well-composed. We prove in this paper that, first,
		  self-dual Euler well-composedness is equivalent to digital
		  well-composedness in dimension 2 and 3, and second, in
		  dimension 4, self-dual Euler well-composedness implies
		  digital well-composedness, though the converse is not true.}
}