Uniqueness of models in persistent homology: the case of curves
Abstract
We consider generic curves in { {R}}^2 , i.e. generic C^{1} functions f:S^1\rightarrow { {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 o h, where h: S^{1} → S^{1} is a C^{1}diffeomorphism, if and only if the persistent homology groups of s o f and s o g coincide, for every s belonging to the group Σ_{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 maxnorm (up to reparameterizations) if and only if, for every s in Σ_{2}, the persistent Betti number functions of s o f and s o g are close to each other, with respect to a suitable distance.
 Publication:

Inverse Problems
 Pub Date:
 December 2011
 DOI:
 10.1088/02665611/27/12/124005
 arXiv:
 arXiv:1012.5783
 Bibcode:
 2011InvPr..27l4005F
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry;
 55N35;
 53A04;
 68U05
 EPrint:
 15 pages, 7 figures