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 $c:S^1\to \real^n$ ($n\ge 2$, $S^1$ 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 \cite{ganesh

, ganesh_sobol_activ_contour08} 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 \emph{``$H^1$ and $H^2$ Sobolev--type Riemannian Manifolds of Curves''}. \gsinsert{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 Fr\'echet distance of curves (see \cite{Michor-Mumford}) coincides with the distance induced by the ``Finsler $L^\infinity$ metric'' defined in \S2.2 in \cite{YM}