Résumé en anglais

This paper shows how computational Riemannian manifold can be used to solve several problems in computer vision and graphics. Indeed, Voronoi segmentations and Delaunay graphs computed with geodesic distances are shaped according to the anisotropy of the metric. A careful design of a Riemannian manifold can thus help to solve some important difficulties in computer vision and graphics. The first contribution of this paper is thus a detailed exposition of Riemannian metrics as a tool for computer vision and graphics. The second contribution of this paper is the use of this new framework to solve two important problems in computer vision and graphics. The first problem studied is perceptual grouping which is a curve reconstruction problem where one should complete in a meaningful way a sparse set of curves. Our anisotropic grouping algorithm works over a Riemannian metric that propagates the direction of a sparse set of noisy incomplete curves over the whole domain. The proposed method prunes the Delaunay graph in order to correctly link together salient features in the image. The second problem studied is planar domain meshing, where one should build a good quality triangulation of a given domain. Our anisotropic meshing algorithm is a geodesic Delaunay refinement method that exploits a Riemannian metric in order to locally impose the orientation and aspect ratio of the created triangles.