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

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## Abstract

In digital topology, it is well-known that, in 2D and in 3D, a digital set ${\displaystyle X\subseteq Z^{n}}$ is emphdigitally well-composed (DWC), i.e., does not contain any critical configuration, if its immersion in the Khalimsky grids ${\displaystyle H^{n}}$ is emphwell-composed in the sense of Alexandrov (AWC), i.e., its boundary is a disjoint union of discrete ${\displaystyle (n-1)}$-surfaces. We show that this is still true in ${\displaystyle n}$-D, ${\displaystyle n\geq 2}$, which is of prime importance since today 4D signals are more and more frequent.

## 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,
3D, a digital set $X \subseteq Z^n$ is \emph{digitally
grids $H^n$ is \emph{well-composed in the sense of
union of discrete $(n-1)$-surfaces. We show that this is
still true in $n$-D, $n \geq 2$, which is of prime
}