We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 to R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to reparameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f = g ◦ h, where h : S^1 → S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of s ◦ f and s ◦g coincide, for every s belonging to the group S_2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ S_2, the persistent Betti number functions of s ◦ f and s ◦ g are close to each other, with respect to a suitable distance.

Uniqueness of models in persistent homology: the case of curves

FROSINI, PATRIZIO;LANDI, CLAUDIA
2011

Abstract

We consider generic curves in R^2, i.e. generic C^1 functions f from S^1 to R^2. We analyze these curves through the persistent homology groups of a filtration induced on S^1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to reparameterizations of S^1. We give a partially positive answer to this question. More precisely, we prove that f = g ◦ h, where h : S^1 → S^1 is a C^1-diffeomorphism, if and only if the persistent homology groups of s ◦ f and s ◦g coincide, for every s belonging to the group S_2 generated by reflections in the coordinate axes. Moreover, for a smaller set of generic functions, we show that f and g are close to each other in the max-norm (up to re-parameterizations) if and only if, for every s ∈ S_2, the persistent Betti number functions of s ◦ f and s ◦ g are close to each other, with respect to a suitable distance.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/151960
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