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

## 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 ${\displaystyle n}$-D self-dual DWC interpolations to obtain a purely self-dual tree of shapes. However, it has been proved that we cannot have an ${\displaystyle n}$-D interpolation which is at the same time localself-dual, and well-composed. By removing the locality constraint, we have obtained an ${\displaystyle n}$-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.

## 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	= {},
number	= {},
pages		= {},
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. }
}```