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

From LRDE

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.}
}