Conformal Metrics and True "Gradient Flows" for Curves

We wish to endow the manifold M of smooth curves in lRn with a Riemannian metric that allows us to treat continuous morphs (homotopies) between two curves c0 and c1 as trajectories with computable lengths which are independent of the parameterization or representation of the two curves (and the curves making up the morph between them). We may then define the distance between the two curves using the trajectory of minimal length (geodesic) between them, assuming such a minimizing trajectory exists. At first we attempt to utilize the metric structure implied rather unanimously by the past twenty years or so of shape optimization literature in computer vision. This metric arises as the unique metric which validates the common references to a wide variety of contour evolution models in the literature as “gradient flows” to various formulated energy functionals. Surprisingly, this implied metric yields a pathological and useless notion of distance between curves. In this paper, we s...

• Type: Conference
• Year: 2005
• Where: ICCV
• Authors: Anthony J. Yezzi, Andrea Mennucci
• Keywords: Computer Vision, Gradient flows, ICCV 2005, Metric, Riemannian Metric