# Euler Well-Composedness

### From LRDE

- Authors
- Nicolas Boutry, Rocio Gonzalez-Diaz, Maria-Jose Jimenez, Eduardo Paluzo-Hildago
- Where
- Combinatorial Image Analysis: Proceedings of the 20th International Workshop, IWCIA 2020, Novi Sad, Serbia, July 16–18, 2020
- Type
- inproceedings
- Publisher
- Springer
- Projects
- Olena
- Date
- 2020-07-21

## 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 . 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, ). Then, a cell decomposition of a picture is self-dual Euler well-composed if both and 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.

## Documents

## 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 N. Sladoje}, 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.} }