Uniqueness of models in persistent homology: the case of curves

作者：Frosini P^*; Landi C

来源：Inverse Problems, 2011, 27(12): 124005.

DOI：10.1088/0266-5611/27/12/124005

We consider generic curves in R(2), i.e. generic C(1) functions f : S(1) -> 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 re-parameterizations of S(1). We give a partially positive answer to this question. More precisely, we prove that f = g circle h, where h : S(1) -> S(1) is a C(1)-diffeomorphism, if and only if the persistent homology groups of s circle f and s circle g coincide, for every s belonging to the group Sigma(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 themax-norm (up to re-parameterizations) if and only if, for every s is an element of Sigma(2), the persistent Betti number functions of s circle f and s circle g are close to each other, with respect to a suitable distance.

• 出版日期2011-12