# Self-Duality and Digital Topology: Links Between the Morphological Tree of Shapes and Well-Composed Gray-Level Images

## Abstract

In digital topology, the use of a pair of connectivities is required to avoid topological paradoxes. In mathematical morphology, self-dual operators and methods also rely on such a pair of connectivities. There are several major issues: self-duality is impure, the image graph structure depends on the image values, it impacts the way small objects and texture are processed, and so on. A sub-class of images defined on the cubical grid, well-composed images, has been proposed, where all connectivities are equivalent, thus avoiding many topological problems. In this paper we unveil the link existing between the notion of well-composed images and the morphological tree of shapes. We prove that a well-composed image has a well-defined tree of shapes. We also prove that the only self-dual well-composed interpolation of a 2D image is obtained by the median operator. What follows from our results is that we can have a purely self-dual representation of images, and consequently, purely self-dual operators.

## Bibtex (lrde.bib)

@InProceedings{	  geraud.15.ismm,
author	= {Thierry G\'eraud and Edwin Carlinet and S\'ebastien Crozet},
title		= {Self-Duality and Digital Topology: {L}inks Between the
Morphological Tree of Shapes and Well-Composed Gray-Level
Images},
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		= {573--584},
abstract	= {In digital topology, the use of a pair of connectivities
is required to avoid topological paradoxes. In mathematical
morphology, self-dual operators and methods also rely on
such a pair of connectivities. There are several major
issues: self-duality is impure, the image graph structure
depends on the image values, it impacts the way small
objects and texture are processed, and so on. A sub-class
of images defined on the cubical grid, {\it well-composed}
images, has been proposed, where all connectivities are
equivalent, thus avoiding many topological problems. In
this paper we unveil the link existing between the notion
of well-composed images and the morphological tree of
shapes. We prove that a well-composed image has a
well-defined tree of shapes. We also prove that the only
self-dual well-composed interpolation of a 2D image is
obtained by the median operator. What follows from our
results is that we can have a purely self-dual
representation of images, and consequently, purely
self-dual operators.},
doi		= {10.1007/978-3-319-18720-4_48}
}