*Published Paper*

**Inserted:** 20 dec 2006

**Last Updated:** 14 nov 2012

**Journal:** INTERFACES AND FREE BOUNDARIES

**Volume:** 10

**Pages:** 423-445

**Year:** 2008

**Abstract:**

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 Fréchet distance of curves coincides with the distance induced by the ``Finsler $L^\infty$ metric'' .

**Keywords:**
space of curves

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