# How to Make nD Functions Digitally Well-Composed in a Self-Dual Way

## Abstract

Latecki et al. introduced the notion of 2D and 3D well-composed images, i.e., a class of images free from the “connectivities paradox” of digital topology. Unfortunately natural and synthetic images are not a priori well-composed. In this paper we extend the notion of “digital well-composedness” to ${\displaystyle n}$D setsinteger-valued functions (gray-level images), and interval-valued maps. We also prove that the digital well-composedness implies the equivalence of connectivities of the level set components in ${\displaystyle n}$D. Contrasting with a previous result stating that it is not possible to obtain a discrete ${\displaystyle n}$D self-dual digitally well-composed function with a local interpolation, we then propose and prove a self-dual discrete (non-local) interpolation method whose result is always a digitally well-composed function. This method is based on a sub-part of a quasi-linear algorithm that computes the morphological tree of shapes.

## Bibtex (lrde.bib)

@InProceedings{	  boutry.15.ismm,
author	= {Nicolas Boutry and Thierry G\'eraud and Laurent Najman},
title		= {How to Make {$n$D} Functions Digitally Well-Composed in a
Self-Dual Way},
booktitle	= {Mathematical Morphology and Its Application to Signal and
Image Processing -- Proceedings of the 12th International
Symposium on Mathematical Morphology (ISMM)},
year		= {2015},
series	= {Lecture Notes in Computer Science Series},
volume	= {9082},
publisher	= {Springer},
editor	= {J.A. Benediktsson and J. Chanussot and L. Najman and H.
Talbot},
pages		= {561--572},
abstract	= {Latecki {\it et al.} introduced the notion of 2D and 3D
well-composed images, {\it i.e.}, a class of images free
from the connectivities paradox'' of digital topology.
Unfortunately natural and synthetic images are not {\it a
priori} well-composed. In this paper we extend the notion
of digital well-composedness'' to $n$D sets,
integer-valued functions (gray-level images), and
interval-valued maps. We also prove that the digital
well-composedness implies the equivalence of connectivities
of the level set components in $n$D. Contrasting with a
previous result stating that it is not possible to obtain a
discrete $n$D self-dual digitally well-composed function
with a local interpolation, we then propose and prove a
self-dual discrete (non-local) interpolation method whose
result is always a digitally well-composed function. This
method is based on a sub-part of a quasi-linear algorithm
that computes the morphological tree of shapes.}
}