An exact computation of the persistent Betti numbers of a submanifold X of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of X is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of X from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.

Estimating multidimensional persistent homology through a finite sampling / Cavazza, Niccolò; Ferri, Massimo; Landi, Claudia. - In: INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS. - ISSN 0218-1959. - STAMPA. - 25:3(2015), pp. 187-205. [10.1142/S0218195915500119]

Estimating multidimensional persistent homology through a finite sampling

LANDI, Claudia
2015

Abstract

An exact computation of the persistent Betti numbers of a submanifold X of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of X is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of X from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.
13-ott-2015
25
3
187
205
Estimating multidimensional persistent homology through a finite sampling / Cavazza, Niccolò; Ferri, Massimo; Landi, Claudia. - In: INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS. - ISSN 0218-1959. - STAMPA. - 25:3(2015), pp. 187-205. [10.1142/S0218195915500119]
Cavazza, Niccolò; Ferri, Massimo; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/1075784
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