# Euler Well-Composedness

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## 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 ${\displaystyle 1}$. A cell decomposition of a picture ${\displaystyle I}$ is a pair of regular cell complexes ${\displaystyle {\big (}K(I),K({\bar {I}}){\big )}}$ such that ${\displaystyle K(I)}$ (resp. ${\displaystyle K({\bar {I}})}$) is a topological and geometrical model representing ${\displaystyle I}$ (resp. its complementary, ${\displaystyle {\bar {I}}}$). Then, a cell decomposition of a picture ${\displaystyle I}$ is self-dual Euler well-composed if both ${\displaystyle K(I)}$ and ${\displaystyle K({\bar {I}})}$ are Euler well-composed. We prove in this paper that, firstself-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.

## Bibtex (lrde.bib)

@InProceedings{	  boutry.20.iwcia1,
author	= {Nicolas Boutry and Rocio Gonzalez-Diaz and Maria-Jose
Jimenez and Eduardo Paluzo-Hildago},
title		= {{E}uler 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
volume	= {12148},
series	= {Lecture Notes in Computer Science},
pages		= {3--19},
publisher	= {Springer},
doi		= {10.1007/978-3-030-51002-2_1},
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.}
}