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

## Abstract

An ${\displaystyle n}$-D pure regular cell complex ${\displaystyle K}$ is weakly well-composed (wWC) if, for each vertex ${\displaystyle v}$ of ${\displaystyle K}$, the set of ${\displaystyle n}$-cells incident to ${\displaystyle v}$ is face-connected. In previous work we proved that if an ${\displaystyle n}$-D picture ${\displaystyle I}$ is digitally well composed (DWC) then the cubical complex ${\displaystyle Q(I)}$ associated to ${\displaystyle I}$ is wWC. If ${\displaystyle I}$ is not DWC, we proposed a combinatorial algorithm to locally repair ${\displaystyle Q(I)}$ obtaining an ${\displaystyle n}$-D pure simplicial complex ${\displaystyle P_{S}(I)}$ homotopy equivalent to ${\displaystyle Q(I)}$ which is always wWC. In this paper we give a combinatorial procedure to compute a simplicial complex ${\displaystyle P_{S}({\bar {I}})}$ which decomposes the complement space of ${\displaystyle |P_{S}(I)|}$ and prove that ${\displaystyle P_{S}({\bar {I}})}$ is also wWC. This paper means one more step on the way to our ultimate goal: to prove that the ${\displaystyle n}$-D repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is an ${\displaystyle (n-1)}$-manifold.

## Bibtex (lrde.bib)

@InProceedings{	  boutry.19.dgci,
author	= {Nicolas Boutry and Rocio Gonzalez-Diaz and Maria-Jose
Jimenez},
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},
series	= {Lecture Notes in Computer Science},
volume	= {11414},
publisher	= {Springer},
editor	= {Michel Couprie and Jean Cousty and Yukiko Kenmochi and
Nabil Mustafa},
doi		= {doi.org/10.1007/978-3-030-14085-4_9},
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. }
}