# Weakly Well-Composed Cell Complexes over nD Pictures

### From LRDE

- Authors
- Nicolas Boutry, Rocio Gonzalez-Diaz, Maria-Jose Jimenez
- Journal
- Information Sciences
- Type
- article
- Projects
- Olena
- Keywords
- Image
- Date
- 2018-07-04

## Abstract

In previous work we proposed a combinatorial algorithm to “locally repair” 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)}**
that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral 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(I)}**
which is 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)}**
and whose boundary surface is a 2D manifold. A polyhedral complex satisfying these properties is called well-composed. In the present paper we extend these results to higher dimensions. We prove that for a given **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}**
-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.

## Documents

## Bibtex (lrde.bib)

@Article{ boutry.18.is, author = {Nicolas Boutry and Rocio Gonzalez-Diaz and Maria-Jose Jimenez}, title = {Weakly Well-Composed Cell Complexes over {$n$D} Pictures}, journal = {Information Sciences}, volume = {499}, pages = {62--83}, month = oct, year = {2019}, abstract = {In previous work we proposed a combinatorial algorithm to ``locally repair'' the cubical complex $Q(I)$ that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral complex $P(I)$ which is homotopy equivalent to $Q(I)$ and whose boundary surface is a 2D manifold. A polyhedral complex satisfying these properties is called {\it well-composed}. In the present paper we extend these results to higher dimensions. We prove that for a given $n$-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.} }