# Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids

### From LRDE

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## Abstract

Among the different flavors of well-composednesses on cubical grids, two of them, called respectively Digital Well-Composedness (DWCness) and Well-Composedness in the sens of Alexandrov (AWCness), are known to be equivalent in 2D and in 3D. The former means that a cubical set does not contain critical configurations when the latter means that the boundary of a cubical set is made of a disjoint union of discrete surfaces. In this paper, we prove that this equivalence holds in ${\displaystyle n}$-D, which is of interest because today images are not only 2D or 3D but also 4D and beyond. The main benefit of this proof is that the topological properties available for AWC sets, mainly their separation properties, are also true for DWC sets, and the properties of DWC sets are also true for AWC sets: an Euler number locally computable, equivalent connectivities from a local or global point of view... This result is also true for gray-level images thanks to cross-section topology, which means that the sets of shapes of DWC gray-level images make a tree like the ones of AWC gray-level images.

## Bibtex (lrde.bib)

```@Article{	  boutry.20.jmiv.2,
author	= {Nicolas Boutry and Laurent Najman and Thierry G\'eraud},
title		= {Equivalence between Digital Well-Composedness and
Well-Composedness in the Sense of {A}lexandrov on {\$n\$-D}
Cubical Grids},
journal	= {Journal of Mathematical Imaging and Vision},
volume	= {62},
pages		= {1285--1333},
month		= sep,
year		= {2020},
doi		= {10.1007/s10851-020-00988-z},
abstract	= {Among the different flavors of well-composednesses on
cubical grids, two of them, called respectively Digital
Well-Composedness (DWCness) and Well-Composedness in the
sens of Alexandrov (AWCness), are known to be equivalent in
2D and in 3D. The former means that a cubical set does not
contain critical configurations when the latter means that
the boundary of a cubical set is made of a disjoint union
of discrete surfaces. In this paper, we prove that this
equivalence holds in \$n\$-D, which is of interest because
today images are not only 2D or 3D but also 4D and beyond.
The main benefit of this proof is that the topological
properties available for AWC sets, mainly their separation
properties, are also true for DWC sets, and the properties
of DWC sets are also true for AWC sets: an Euler number
locally computable, equivalent connectivities from a local
or global point of view... This result is also true for
gray-level images thanks to cross-section topology, which
means that the sets of shapes of DWC gray-level images make
a tree like the ones of AWC gray-level images. }
}```