Calculus of Variations and Geometric Measure Theory

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

A New Geometric Metric in the Space of Curves, and Applications to Tracking Deforming Objects by Prediction and Filtering

created by mennucci on 11 Jan 2010
modified by paolini on 07 Dec 2012


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


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 motionsdeformations. 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