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 ) coincides with the distance induced by the "Finsler L∞ metric" defined in §2.2 of .
ASJC Scopus subject areas
- Surfaces and Interfaces