Optimal reparametrizations in the square root velocity framework

TL;DR

This paper investigates the mathematical properties of the square root velocity framework in shape analysis, establishing conditions for optimal reparametrisations and analyzing the structure of the space of unparametrised curves.

Contribution

It provides new insights into the analytical and topological structure of the quotient space of unparametrised curves under the square root velocity framework, including existence results for optimal reparametrisations.

Findings

The square root velocity transform is a homeomorphism.

The action of the reparametrisation semigroup is continuous.

Existence of optimal reparametrisations for certain classes of curves.

Abstract

The square root velocity framework is a method in shape analysis to define a distance between curves and functional data. Identifying two curves if they differ by a reparametrisation leads to the quotient space of unparametrised curves. In this paper we study analytical and topological aspects of this construction for the class of absolutely continuous curves. We show that the square root velocity transform is a homeomorphism and that the action of the reparametrisation semigroup is continuous. We also show that given two $C^1$-curves, there exist optimal reparametrisations realising the minimal distance between the unparametrised curves represented by them. Furthermore we give an example of two Lipschitz curves, for which no pair of optimal reparametrisations exists.