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

From LRDE

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 n} -D pure 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(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} -D repaired complex is continuously well-composed (CWC), that is, the boundary of its continuous analog is 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-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. }
}