Calculus of Variations and Geometric Measure Theory

A. C. G. Mennucci - G. Sundaramoorthi - A. Yezzi

Properties of Sobolev--type metrics in the space of curves

created by mennucci on 20 Dec 2006
modified on 14 Nov 2012


Published Paper

Inserted: 20 dec 2006
Last Updated: 14 nov 2012

Volume: 10
Pages: 423-445
Year: 2008


We define a manifold $M$ where objects $c\in M$ are curves, which we parameterize as $c:S^1\to R^n$ ($n\ge 2$, $S^1$ is the circle). Given a curve $c$, we define the tangent space $T_cM$ of $M$ at $c$ including in it all deformations $h:S^1\to R^n$ of $c$.

We study geometries on the manifold of curves, provided by Sobolev--type Riemannian metrics $H^j$.

We initially present some mathematical examples to show how the metrics $H^j$ simplify or regularize gradient flows used in Computer Vision applications.

We then provide some basilar results of $H^j$ metrics; and, for the cases $j=1,2$, we characterize the completion of the space of smooth curves; we call this completion(s) ``$H^1$ and $H^2$ Sobolev--type Riemannian Manifolds of Curves''.

As a byproduct, we prove that the Fr├ęchet distance of curves coincides with the distance induced by the ``Finsler $L^\infty$ metric'' .

Keywords: space of curves