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

### From LRDE

- Authors
- Nicolas Boutry, Laurent Najman, Thierry Géraud
- Where
- Discrete Geometry for Computer Imagery – Proceedings of the 20th IAPR International Conference on Discrete Geometry for Computer Imagery (DGCI)
- Place
- Vienna, Austria
- Type
- inproceedings
- Publisher
- Springer
- Projects
- Olena
- Keywords
- Image
- Date
- 2017-06-01

## 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.} }