A square root velocity framework for curves of bounded variation

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.

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