Properties of Sobolev-type metrics in the space of curves

  • A.C.G. Mennucci

    Scuola Normale Superiore, Pisa, Italy
  • A. Yezzi

    Georgia Institute of Technology, Atlanta, United States
  • G. Sundaramoorthi

    Georgia Institute of Technology, Atlanta, United States

Abstract

We define a manifold where objects are curves, which we parameterize as (, is the circle). We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics . 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 metrics; and, for the cases , we characterize the completion of the space of smooth curves. We call these completions \emph{`` and 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
}

Cite this article

A.C.G. Mennucci, A. Yezzi, G. Sundaramoorthi, Properties of Sobolev-type metrics in the space of curves. Interfaces Free Bound. 10 (2008), no. 4, pp. 423–445

DOI 10.4171/IFB/196