Uniqueness of models in persistent homology: the case of curves

We consider generic curves in R2 , i.e. generic C1 functions f : S1 R2 . We analyze these curves through the persistent homology groups of a filtration induced on S1 by f. In particular, we consider the question whether these persistent homology groups uniquely characterize f, at least up to re-parameterizations of S1 . We give a partially positive answer to this question. More precisely, we prove that f = g h, where h : S1 S1 is a C1 -diffeomorphism, if and only if the persistent homology groups of s f and s g coincide, for every s belonging to the group 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 2, the persistent Betti numbers functions of sf and sg are close to each other, with respect to a suitable distance. AMS classification scheme numbers: 55N35, 53A04, 68U05

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Patrizio Frosini, Claudia Landi
• Keywords: CORR 2010, Education, Generic C1 Functions, Persistent Betti Numbers, Persistent Homology Groups