Discrete set-valued continuity and interpolation. (English) Zbl 1382.68269
Hendriks, Cris L. Luengo (ed.) et al., Mathematical morphology and its applications to signal and image processing. 11th international symposium, ISMM 2013, Uppsala, Sweden, May 27–29, 2013. Proceedings. Berlin: Springer (ISBN 978-3-642-38293-2/pbk). Lecture Notes in Computer Science 7883, 37-48 (2013).
Summary: 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 [T. Géraud et al., Lect. Notes Comput. Sci. 7883, 98–110 (2013; Zbl 1382.68261)].
For the entire collection see [Zbl 1263.68019].
For the entire collection see [Zbl 1263.68019].
MSC:
| 68U05 | Computer graphics; computational geometry (digital and algorithmic aspects) |
| 68U10 | Computing methodologies for image processing |
Keywords:
tree of shapesCitations:
Zbl 1382.68261References:
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