Discrete set-valued continuity and interpolation

Authors

Laurent Najman, Thierry Géraud

Where

Mathematical Morphology and Its Application to Signal and Image Processing – Proceedings of the 11th International Symposium on Mathematical Morphology (ISMM)

Place

Uppsala, Sweden

Type

inproceedings

Publisher

Springer

Projects

Olena

Keywords

Image

Date

2013-03-14

Abstract

The main question of this paper is to retrieve some continuity properties on (discrete) T0-Alexandroff spaces. One possible application, which will guide us, is the construction of the so-called "tree of shapes" (intuitively, the tree of level lines). This tree, which should allow to process maxima and minima in the same wayfaces quite a number of theoretical difficulties that we propose to solve using set-valued analysis in a purely discrete setting. We also propose a way to interpret any function defined on a grid as a "continuous" function thanks to an interpolation scheme. The continuity properties are essential to obtain a quasi-linear algorithm for computing the tree of shapes in any dimension, which is exposed in a companion paper.

Documents

• Paper

Bibtex (lrde.bib)

@InProceedings{	  najman.13.ismm,
  author	= {Laurent Najman and Thierry G\'eraud},
  title		= {Discrete set-valued continuity and interpolation},
  booktitle	= {Mathematical Morphology and Its Application to Signal and
		  Image Processing -- Proceedings of the 11th International
		  Symposium on Mathematical Morphology (ISMM)},
  year		= 2013,
  editor	= {C.L. Luengo Hendriks and G. Borgefors and R. Strand},
  volume	= 7883,
  series	= {Lecture Notes in Computer Science Series},
  address	= {Uppsala, Sweden},
  publisher	= {Springer},
  pages		= {37--48},
  abstract	= {The main question of this paper is to retrieve some
		  continuity properties on (discrete) T0-Alexandroff spaces.
		  One possible application, which will guide us, is the
		  construction of the so-called "tree of shapes"
		  (intuitively, the tree of level lines). This tree, which
		  should allow to process maxima and minima in the same way,
		  faces quite a number of theoretical difficulties that we
		  propose to solve using set-valued analysis in a purely
		  discrete setting. We also propose a way to interpret any
		  function defined on a grid as a "continuous" function
		  thanks to an interpolation scheme. The continuity
		  properties are essential to obtain a quasi-linear algorithm
		  for computing the tree of shapes in any dimension, which is
		  exposed in a companion paper.}
}