A square root velocity framework for curves of bounded variation

TL;DR

This paper extends the square root velocity framework to include curves of bounded variation, providing explicit formulas and analyzing the shape distance and reparametrisations for discontinuous curves.

Contribution

It generalizes the square root velocity distance to curves of bounded variation, including explicit formulas and analysis of reparametrisations for discontinuous curves.

Findings

Explicit formula for the extended distance on curves of bounded variation

Analysis of the existence of optimal reparametrisations

Extension of shape distance to discontinuous curves

Abstract

The square root velocity transform is a powerful tool for the efficient computation of distances between curves. Also, after factoring out reparametrisations, it defines a distance between shapes that only depends on their intrinsic geometry but not the concrete parametrisation. Though originally formulated for smooth curves, the square root velocity transform and the resulting shape distance have been thoroughly analysed for the setting of absolutely continuous curves using a relaxed notion of reparametrisations. In this paper, we will generalise the square root velocity distance even further to a class of discontinuous curves. We will provide an explicit formula for the natural extension of this distance to curves of bounded variation and analyse the resulting quotient distance on the space of unparametrised curves. In particular, we will discuss the existence of optimal reparametrisations for which the minimal distance on the quotient space is realised.