# Discrete set-valued continuity and interpolation

### From LRDE

- 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

## 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.} }