Optimal Reparametrizations in the Square Root Velocity Framework

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 reparametrization leads to the quotient space of unparametrized 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 reparametrization semigroup is continuous. We also show that given two C¹-curves, there exist optimal reparametrizations realizing the minimal distance between the unparametrized curves represented by them. Furthermore we give an example of two Lipschitz curves, for which no pair of optimal reparametrizations exists.