JournalsifbVol. 10 , No. 4DOI 10.4171/ifb/196

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
Properties of Sobolev-type metrics in the space of curves cover

Abstract

We define a manifold MM where objects cMc\in M are curves, which we parameterize as c:S1nc:S^1\to \real^n (n2n\ge 2, S1S^1 is the circle). We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics HjH^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 HjH^j metrics; and, for the cases j=1,2j=1,2, we characterize the completion of the space of smooth curves. We call these completions \emph{``H1H^1 and H2H^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\infinityL^\infinity metric'' defined in \S2.2 in \cite{YM
}