Topological Properties of the First Non-Local Digitally Well-Composed Interpolation on n -D Cubical Grids
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Topological Properties of the First Non-Local Digitally Well-Composed Interpolation on n-D Cubical Grids Nicolas Boutry1
· Laurent Najman2 · Thierry Géraud1
Received: 30 May 2019 / Accepted: 21 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In discrete topology, we like digitally well-composed (shortly DWC) interpolations because they remove pinches in cubical images. Usual well-composed interpolations are local and sometimes self-dual (they treat in a same way dark and bright components in the image). In our case, we are particularly interested in n-D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However, it has been proved that we cannot have an n-D interpolation which is at the same time local, self-dual and well-composed. By removing the locality constraint, we have obtained an n-D interpolation with many properties in practice: it is self-dual, DWC, and in-between (this last property means that it preserves the contours). Since we did not publish the proofs of these results before, we propose to provide in a first time the proofs of the two last properties here (DWCness and in-betweeness) and a sketch of the proof of self-duality (the complete proof of self-duality requires more material and will come later). Some theoretical and practical results are given. Keywords Well-composed images · Critical configurations · Digital topology · Tree of shapes · Mathematical morphology
1 Introduction It is well-known that images coming from the digitization of the real world loose a part of their topological properties (their boundaries are no longer topological manifolds for example). In a discrete image, different possible connectivities exist (like c2n and c3n −1 ), to cite only the most famous, which means that depending on the chosen connectivity, some algorithms will work in a specific way; this can lead to topological paradoxes (it can happen that a simple, closed curve in the digital plane no longer separates this plane into an interior and an exterior like in the Euclidian case). Latecki introduced then well-composed images with topological properties sim-
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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
ilar to the ones of the objects in the real world; for example, in 2D and 3D, an image which is well-composed will not longer have pinches in its boundary. Note that a summary of the different flavors of well-composednesses can be found in [9]. The question which arises then is: since natural or synthetic discrete images are generally not well-composed, how can we compute a “good” DWC representation of a given image u? It has been shown in [6] that it is impossible in a local manner, and thereafter we proposed a way to compute a we
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