# Weakly Well-Composed Cell Complexes over nD Pictures

## Abstract

In previous work we proposed a combinatorial algorithm to “locally repair” the cubical complex ${\displaystyle Q(I)}$ that is canonically associated with a given 3D picture I. The algorithm constructs a 3D polyhedral complex ${\displaystyle P(I)}$ which is homotopy equivalent to ${\displaystyle Q(I)}$ 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 ${\displaystyle n}$-dimensional picture the obtained cell complex is well-composed in a weaker sense but is still homotopy equivalent to the initial cubical complex.

## 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.}
}