Well-Composedness in Alexandrov spaces implies Digital Well-Composedness in Z^n

From LRDE

Abstract

In digital topology, it is well-known that, in 2D and in 3D, a digital set 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 X \subseteq Z^n} is emphdigitally well-composed (DWC), i.e., does not contain any critical configuration, if its immersion in the Khalimsky grids 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 H^n} is emphwell-composed in the sense of Alexandrov (AWC), i.e., its boundary is a disjoint union of discrete 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)} -surfaces. We show that this is still true in 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, 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 \geq 2} , which is of prime importance since today 4D signals are more and more frequent.

Documents

Bibtex (lrde.bib)

@InProceedings{	  boutry.17.dgci,
  author	= {Nicolas Boutry and Laurent Najman and Thierry G\'eraud},
  title		= {Well-Composedness in {A}lexandrov spaces implies Digital
		  Well-Composedness in $Z^n$},
  booktitle	= {Discrete Geometry for Computer Imagery -- Proceedings of
		  the 20th IAPR International Conference on Discrete Geometry
		  for Computer Imagery (DGCI)},
  year		= {2017},
  series	= {Lecture Notes in Computer Science},
  volume	= {10502},
  publisher	= {Springer},
  editor	= {W.G. Kropatsch and N.M. Artner and I. Janusch},
  pages		= {225--237},
  month		= sep,
  address	= {Vienna, Austria},
  abstract	= {In digital topology, it is well-known that, in 2D and in
		  3D, a digital set $X \subseteq Z^n$ is \emph{digitally
		  well-composed (DWC)}, {\it i.e.}, does not contain any
		  critical configuration, if its immersion in the Khalimsky
		  grids $H^n$ is \emph{well-composed in the sense of
		  Alexandrov (AWC)}, {\it i.e.}, its boundary is a disjoint
		  union of discrete $(n-1)$-surfaces. We show that this is
		  still true in $n$-D, $n \geq 2$, which is of prime
		  importance since today 4D signals are more and more frequent.}
}