Comparison of persistent homologies for vector functions: from continuous to discrete and back

Cavazza, Niccolò ; Ethier, Marc ; Frosini, Patrizio ; Kaczynski, Tomasz ; Landi, Claudia (2012) Comparison of persistent homologies for vector functions: from continuous to discrete and back.
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Abstract

The theory of multidimensional persistent homology was initially developed in the discrete setting, and involved the study of simplicial complexes filtered through an ordering of the simplices. Later, stability properties of multidimensional persistence have been proved to hold when topological spaces are filtered by continuous functions, i.e. for continuous data. This paper aims to provide a bridge between the continuous setting, where stability properties hold, and the discrete setting, where actual computations are carried out. More precisely, a stability preserving method is developed to compare rank invariants of vector functions obtained from discrete data. These advances confirm that multidimensional persistent homology is an appropriate tool for shape comparison in computer vision and computer graphics applications. The results are supported by numerical tests.

Abstract
Document type
Article
Creators
CreatorsAffiliationORCID
Cavazza, Niccolò
Ethier, Marc
Frosini, Patrizio
Kaczynski, Tomasz
Landi, Claudia
Keywords
Multidimensional persistent homology; axis-wise interpolation; filtration; matching distance; topological aliasing
Subjects
DOI
Deposit date
16 Jan 2012 16:47
Last modified
21 Feb 2012 10:20
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