Learning Shape Metrics based on Deformations and Transport

Shape evolutions, as well as shape matchings or image segmentation with shape prior, involve the preliminary choice of a suitable metric in the space of shapes. Instead of choosing a particular one, we propose a framework to learn shape metrics from a set of examples of shapes, designed to be able to handle sparse sets of highly varying shapes, since typical shape datasets, like human silhouettes, are intrinsically high-dimensional and non-dense. We formulate the task of finding the optimal metrics on an empirical manifold of shapes as a classical minimization problem ensuring smoothness, and compute its global optimum fast. First, we design a criterion to compute point-to-point matching between shapes which deals with topological changes. Then, given a training set of shapes, we use these matchings to transport deformations observed on any shape to any other one. Finally, we estimate the metric in the tangent space of any shape, based on transported deformations, weighted by their reliability. Experiments on difficult sets are shown, and applications are proposed.