Persistent Homology as Stopping-Criterion for Voronoi Interpolation

In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fa...

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Published inCombinatorial Image Analysis Vol. 12148; pp. 29 - 44
Main Authors Melodia, Luciano, Lenz, Richard
Format Book Chapter
LanguageEnglish
Published Switzerland Springer International Publishing AG 2020
Springer International Publishing
SeriesLecture Notes in Computer Science
Subjects
Interpolation
Persistent homology
Voronoi triangulation
Online AccessGet full text
ISBN9783030510015
3030510018
ISSN0302-9743
1611-3349
DOI10.1007/978-3-030-51002-2_3

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Abstract In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.
AbstractList In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration. The technique is based on Voronoi tessellation, which induces a natural dual map to the Delaunay triangulation. Advantage is taken from this fact calculating the persistent homology on it after each iteration to capture the changing topology of the data. The boundary points are identified as critical. The Bottleneck and Wasserstein distance serve as a measure of quality between the original point set and the interpolation. If the norm of two distances exceeds a heuristically determined threshold, the algorithm terminates. We give the theoretical basis for this approach and justify its validity with numerical experiments.
Author Melodia, Luciano
Lenz, Richard
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Sladoje, Natasa
Barneva, Reneta P
Čomić, Lidija
Lukić, Tibor
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Snippet In this study the Voronoi interpolation is used to interpolate a set of points drawn from a topological space with higher homology groups on its filtration....
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StartPage 29
SubjectTerms Interpolation
Persistent homology
Voronoi triangulation
Title Persistent Homology as Stopping-Criterion for Voronoi Interpolation
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