Properties of Sobolev-type metrics in the space of curves

A. C.G. Mennucci, A. Yezzi, G. Sundaramoorthi

Research output: Contribution to journalArticlepeer-review

28 Scopus citations

Abstract

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S1 → ℝn (n ≥ 2, S 1 is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics Hj. 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 on Hj 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 H2 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 metric" defined in §2.2 of [18].

Original languageEnglish (US)
Pages (from-to)423-445
Number of pages23
JournalInterfaces and Free Boundaries
Volume10
Issue number4
StatePublished - 2008
Externally publishedYes

ASJC Scopus subject areas

  • Surfaces and Interfaces

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