# One More Step Towards Well-Composedness of Cell Complexes over n-D Pictures

### From LRDE

- Authors
- Nicolas Boutry, Rocio Gonzalez-Diaz, JimenezMaria-Jose
- Where
- Proceedings of the 21st International Conference on Discrete Geometry for Computer Imagery (DGCI)
- Place
- Marne-la-Vallée, France
- Type
- inproceedings
- Publisher
- Springer
- Projects
- Olena
- Date
- 2019-06-18

## Abstract

An **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**
-D pure regular cell complex **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K}**
is weakly well-composed (wWC) if, for each vertex **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v}**
of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K}**
, the set of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**
-cells incident to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v}**
is face-connected. In previous work we proved that if an **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}**
-D picture **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I}**
is digitally well composed (DWC) then the cubical complex **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(I)}**
associated to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I}**
is wWC. If **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I}**
is not DWC, we proposed a combinatorial algorithm to locally repair **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(I)}**
obtaining an **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_S(I)}**
homotopy equivalent to **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(I)}**
which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_S(\bar{I})}**
which decomposes the complement space of **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |P_S(I)|}**
and prove that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_S(\bar{I})}**
is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n-1)}**
-manifold.

## Documents

## Bibtex (lrde.bib)

@InProceedings{ boutry.19.dgci, author = {Boutry, Nicolas and Gonzalez-Diaz, Rocio and Jimenez, Maria-Jose}, title = {One More Step Towards Well-Composedness of Cell Complexes over {$n$-D} Pictures}, booktitle = {Proceedings of the 21st International Conference on Discrete Geometry for Computer Imagery (DGCI)}, year = 2019, month = mar, pages = {101--114}, address = {Marne-la-Vall{\'e}e, France}, series = {Lecture Notes in Computer Science}, volume = {11414}, publisher = {Springer}, editor = {Michel Couprie and Jean Cousty and Yukiko Kenmochi and Nabil Mustafa}, abstract = {An {$n$-D} pure regular cell complex $K$ is weakly well-composed (wWC) if, for each vertex $v$ of $K$, the set of $n$-cells incident to $v$ is face-connected. In previous work we proved that if an {$n$-D} picture $I$ is digitally well composed (DWC) then the cubical complex $Q(I)$ associated to $I$ is wWC. If $I$ is not DWC, we proposed a combinatorial algorithm to locally repair $Q(I)$ obtaining an {$n$-D} pure simplicial complex $P_S(I)$ homotopy equivalent to $Q(I)$ which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex $P_S(\bar{I})$ which decomposes the complement space of $|P_S(I)|$ and prove that $P_S(\bar{I})$ is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the {$n$-D} repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an $(n-1)$-manifold. } }