Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n -D Cubical Grids
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Equivalence between Digital Well-Composedness and Well-Composedness in the Sense of Alexandrov on n-D Cubical Grids Nicolas Boutry1
· Laurent Najman2 · Thierry Géraud1
Received: 4 February 2019 / Accepted: 21 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 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 sense 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, while 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. Keywords Digital topology · Discrete surfaces · Well-composed images
1 Introduction In 1995, Latecki introduced in [23] the notion of well-composedness as an elegant manner to get rid of the connectivity paradoxes well known in digital topology. Roughly speaking, a set in Z2 is said to be well-composed if its connectivities are equivalent, that is, its set of connected components is the same whatever the chosen connectivity. This definition has then been extended to dimension 3 in [24] and in n-D in [10]; at the same moment, this definition of well-composedness has been renamed digital well-composedness (DWCness). Later, Najman and Géraud [28] introduced a new notion of well-composedness: a subset X in a Khalimsky grid, a
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Nicolas Boutry [email protected] Laurent Najman [email protected] Thierry Géraud [email protected]
1
EPITA Research and Development Laboratory (LRDE), 14-16 rue Voltaire, 94276 Le Kremlin-Bicêtre, France
2
Université Paris-Est, LIGM, Équipe A3SI, ESIEE, Cité Descartes 2 Boulevard Blaise Pascal, 93160 Noisy-le-Grand, France
topological analog of the cubical grid Zn but with combinatorial properties, is said to be well-composed in the sense of Alexandrov (AWC) if its boundary is made of a disjoint union of discrete (n − 1)-surfaces (the definition of a discrete surface is formally recalled later). This definition is used to be able to characterize gray-level images defined on a Khalimsky grid and whose the set of shapes makes a tree; then, we can call “tree of shapes” this last set. This hierarchical representation is known to be powerful to make image segmentation or image filtering thanks to shapings
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