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

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 deformations 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 compl...

• Type: Journal
• Year: 2011
• Where: SIAMIS
• Authors: Ganesh Sundaramoorthi, Andrea Mennucci, Stefano Soatto, Anthony J. Yezzi
• Keywords: Metric, Riemannian Manifold, SIAMIS 2011, Smooth Riemannian Manifold, Software Engineering