Properties of Sobolev-type metrics in the space of curves
A.C.G. Mennucci
Scuola Normale Superiore, Pisa, ItalyA. Yezzi
Georgia Institute of Technology, Atlanta, United StatesG. Sundaramoorthi
Georgia Institute of Technology, Atlanta, United States
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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 [13], [14], [16] 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 and 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 metric” defined in §2.2 of [18]
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