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

### From LRDE

- Authors
- Thierry Géraud, Edwin Carlinet, Sébastien Crozet
- Where
- Mathematical Morphology and Its Application to Signal and Image Processing – Proceedings of the 12th International Symposium on Mathematical Morphology (ISMM)
- Place
- Reykjavik, Iceland
- Type
- inproceedings
- Publisher
- Springer
- Projects
- Olena
- Keywords
- Image
- Date
- 2015-04-07

## 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.

## Documents

## Bibtex (lrde.bib)

@InProceedings{ geraud.15.ismm, author = {Thierry G\'eraud and Edwin Carlinet and S\'ebastien Crozet}, title = {Self-Duality and Digital Topology: Links 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}, address = {Reykjavik, Iceland}, 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.} }