Topological Properties of the First Non-Local Digitally Well-Composed Interpolation on n-D Cubical Grids

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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 -D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However, it has been proved that we cannot have an -D interpolation which is at the same time local, self-dualand well-composed. By removing the locality constraint, we have obtained an -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 published 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.

Documents

Bibtex (lrde.bib)

@Article{	  boutry.20.jmiv.1,
  author	= {Nicolas Boutry and Laurent Najman and Thierry G\'eraud},
  title		= {Topological Properties of the First Non-Local Digitally
		  Well-Composed Interpolation on {$n$-D} Cubical Grids},
  journal	= {Journal of Mathematical Imaging and Vision},
  volume	= {62},
  pages		= {1256--1284},
  month		= sep,
  year		= {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 published 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. }
}