Properties of Sobolev-type metrics in the space of curves

Abstract

We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to {\mathbb{R}}^n$ ($n\ge 2$, $S^1$ is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics $H^j$. These metrics have been shown to regularize gradient flows used in computer vision applications, see [13], [14], [16] and references therein.

We provide some basic results of $H^j$ metrics; and, for the cases $j=1,2$, we characterize the completion of the space of smooth curves. We call these completions$H^1$ and $H^2$ Sobolev-type Riemannian Manifolds of Curves.” This result is fundamental since it is a first step in proving the existence of geodesics with respect to these metrics. As a byproduct, we prove that the Fréchet distance of curves (see [7]) coincides with the distance induced by the “Finsler $L^{\infty }$ metric” defined in §2.2 of [18]