## Abstract

We define a manifold M where objects c ∈ M are curves, which we parameterize as c : S^{1} → ℝ^{n} (n ≥ 2, S ^{1} is the circle). We study geometries on the manifold of curves, provided by Sobolev-type Riemannian metrics H^{j}. 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 H^{j} 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 H^{2} 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 language | English (US) |
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Pages (from-to) | 423-445 |

Number of pages | 23 |

Journal | Interfaces and Free Boundaries |

Volume | 10 |

Issue number | 4 |

DOIs | |

State | Published - 2008 |

Externally published | Yes |

## ASJC Scopus subject areas

- Applied Mathematics