Equivalence between DWCness and AWCness on n-D Cubical Grids

From LRDE

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 -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 setsand 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.

Documents

Bibtex (lrde.bib)

@Article{	  boutry.20.jmiv.2,
  author	= {Nicolas Boutry and Laurent Najman and Thierry G\'eraud},
  title		= {Equivalence between DWCness and AWCness on $n$-D Cubical
		  Grids},
  journal	= {Journal of Mathematical Imaging and Vision},
  volume	= {62},
  number	= {},
  pages		= {1285--1333},
  month		= sep,
  year		= {2020},
  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. }
}