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

From LRDE

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 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 D. Contrasting with a previous result stating that it is not possible to obtain a discrete 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.

Documents

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},
  address	= {Reykjavik, Iceland},
  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.}
}