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

Image"Image" is not in the list (Vaucanson, Spot, URBI, Olena, APMC, Tiger, Climb, Speaker ID, Transformers, Bison, ...) of allowed values for the "Related project" property.

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.

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.}
}