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.Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.
N., Cavazza, Ferri, M. e Claudia, Landi. "Estimating multidimensional persistent homology through a finite sampling" Working paper, AMS Acta, Università di Bologna, 2010.
Estimating multidimensional persistent homology through a finite sampling
LANDI, Claudia
2010
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.Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.
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