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 pseu- dodistance 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.
Niccolò, C., Massimo, F., Claudia, L. (2015). Estimating multidimensional persistent homology through a finite sampling. INTERNATIONAL JOURNAL OF COMPUTATIONAL GEOMETRY & APPLICATIONS, 25, 187-205.
Estimating multidimensional persistent homology through a finite sampling
CAVAZZA, NICCOLO';FERRI, MASSIMO;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 pseu- dodistance 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.