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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Discrete Set-Valued Continuity and Interpolation
verfasst von : Laurent Najman, Thierry Géraud
Erschienen in: Mathematical Morphology and Its Applications to Signal and Image Processing
Verlag: Springer Berlin Heidelberg
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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 [10].