# Uniqueness of models in persistent homology: the case of curves

@article{Frosini2010UniquenessOM, title={Uniqueness of models in persistent homology: the case of curves}, author={Patrizio Frosini and Claudia Landi}, journal={ArXiv}, year={2010}, volume={abs/1012.5783} }

We consider generic curves in , i.e. generic C1 functions . 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 o h, where h: S1 ? S1 is a C1-diffeomorphism, if and only if the persistent homology groups of s o f…

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## References

SHOWING 1-10 OF 21 REFERENCES

Natural pseudo-distances between closed curves

- Mathematics
- 2009

Abstract Let us consider two closed curves ℳ, of class C 1 and two functions of class C 1, called measuring functions. The natural pseudo-distance d between the pairs (ℳ, φ), (, ψ) is defined as the…

The theory of multidimensional persistence

- MathematicsSCG '07
- 2007

This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.

Riemannian Geometries on Spaces of Plane Curves

- Mathematics
- 2003

We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from the circle to the plane modulo the group of diffeomorphisms of the circle,…

Natural pseudodistances between closed manifolds

- Mathematics
- 2004

Let us consider two closed homeomorphic manifolds M;N of class C 1 and two functions j : M ! R, c : N ! R of class C . The natural pseudodistance d between the pairs ðM; jÞ; ðN;cÞ is defined as the…

Stability of persistence diagrams

- MathematicsSCG
- 2005

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram…

DIFFERENTIAL TOPOLOGY

- Mathematics
- 2002

This paper is to propose solutions to selected exercises in Differential Topology by Guillemin and Pollack, [1], and to comment on certain proofs in the book. Although the sections covered in this…

Natural Pseudo-Distance and Optimal Matching between Reduced Size Functions

- Computer ScienceArXiv
- 2008

The matching distance is shown to be resistant to perturbations, implying that it is always smaller than the natural pseudo-distance, and it is proved that the lower bound so obtained is sharp and cannot be improved by any other distance between size functions.

Size Functions from a Categorical Viewpoint

- Mathematics
- 2001

A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result…

Finiteness of rank invariants of multidimensional persistent homology groups

- MathematicsAppl. Math. Lett.
- 2011

Inequalities for the Curvature of Curves and Surfaces

- MathematicsFound. Comput. Math.
- 2007

The difference between the total mean curvatures of two closed surfaces in ${\Bbb R}^3$ in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose is bound.