Properties of Sobolev-type metrics in the space of curves

Abstract

We define a manifold M where objects c ϵ M are curves, which we parameterize as c : S¹ → R ⁿ (n ≥2, S¹ is the circle). Given a curve c, we define the tangent space TcM of M at c including in it all deformations h : S¹ → R ⁿ of c. We study geometries on the manifold of curves, provided by Sobolev–type Riemannian metrics H[superscript j]. We initially present some mathematical examples to show how the metrics H[superscript j] simplify or regularize gradient flows used in Computer Vision applications. We then provide some basilar results of Hj metrics; and, for the cases j = 1, 2, we characterize the completion of the space of smooth curves; we call this completion(s) “H¹ and H² Sobolev–type Riemannian Manifolds of Curves”. As a byproduct, we prove that the Fréchet distance of curves (see [MM06b]) coincides with the distance induced by the “Finsler L H [superscript ∞] metric” defined in §2.2 in [YM04b].