*Published Paper*

**Inserted:** 11 jan 2010

**Last Updated:** 7 dec 2012

**Journal:** SIAM Journal on Imaging Sciences

**Volume:** 4

**Number:** 1

**Year:** 2011

**Doi:** 10.1137/090781139

**Abstract:**

We define a novel metric on the space of closed planar curves which decomposes into three
intuitive components. According to this metric centroid translations, scale changes and defor-
mations are orthogonal, and the metric is also invariant with respect to reparameterizations
of the curve. While earlier related Sobolev metrics for curves exhibit some general similarities
to the novel metric proposed in this work, they lacked this important three-way orthogonal
decomposition which has particular relevance for tracking in computer vision. Another positive
property of this new metric is that the Riemannian structure that is induced on the space of
curves is a smooth Riemannian manifold, which is isometric to a classical well-known manifold.
As a consequence, geodesics and gradients of energies defined on the space can be computed
using fast closed-form formulas, and this has obvious benefits in numerical applications.
The obtained Riemannian manifold of curves is ideal to address complex problems in com-
puter vision; one such example is the tracking of highly deforming objects. Previous works have
assumed that the object deformation is smooth, which is realistic for the tracking problem, but
most have restricted the deformation to belong to a finite-dimensional group such as affine
motions or to finitely-parameterized models. This is too restrictive for highly deforming ob-
jects such as the contour of a beating heart. We adopt the smoothness assumption implicit in
previous work, but we lift the restriction to finite-dimensional motions*deformations. We define
a dynamical model in this Riemannian manifold of curves, and use it to perform filtering and
prediction to infer and extrapolate not just the pose (a finitely parameterized quantity) of an
object, but its deformation (an infinite-dimensional quantity) as well. We illustrate these ideas
using a simple first-order dynamical model, and show that it can be effective even on image
sequences where existing methods fail.*

**Keywords:**
space of curves, sobolev active contours

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