Anisotropic Geodesics for Perceptual Grouping and Domain Meshing

Abstract

Abstract. This paper shows how Voronoi diagrams and their dual Delaunay complexes, defined with geodesic distances over 2D Reimannian manifolds, can be used to solve two important problems encountered 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 noisy curves. From this latter curves, our grouping algorithm first designs an anisotropic tensor field that corresponds to a Reimannian metric. Then, according to this metric, the Delaunay graph is constructed and pruned in order to correctly link together salient features. The second problem studied is planar domain meshing, where one should build a good quality triangulation of a given domain. Our meshing algorithm is a geodesic Delaunay refinement method that exploits an anisotropic tensor field in order to locally impose the orientation and aspect ratio of the created triangles.

Image compression with anisotropic triangulations

“…We propose to define a truly Riemannian geodesic sampling, and use the anisotropic Delaunay triangulation for image approximation. This combines in a common framework several important features from previous works, including fast Delaunay insertions, anisotropic sampling, anisotropic triangulations [8,18,14,13,20,3,4]. We show that these properties are indeed crucial to improve over the state of the art for the compression of geometric images.…”

“…The main difference is we use a truly anisotropic geodesic distance, as in [4], meanwhile they use the modified metric (5). This allows to obtain better results, in particular for triangulations with few vertices, where the approximation with (5) can be important.…”

“…This allows to obtain a full rank tensor which performs a robust estimation in the presence of noise. From the diagonalization of , we propose the following modified structure tensor to represent the anisotropy: (4) where and are the eigenvalues and the eigenvectors of . The parameter controls the isotropic adaptivity of the triangulation (the variation of density, see [20]), while the parameter controls the anisotropic adaptivity.…”

Saliency Detection with VORONOI Diagram

“…Perceptual grouping and domain meshing are addressed in [24]. The article shows a solution for the problems by Voronoi diagrams and their dual Delaunay complexes, defined with geodesic distances over 2D Riemannian manifolds.…”

Adaptive Multi–Robot Coverage of Curved Surfaces

“…Discrete geodesic alternatives for covering a graph can be found in [9,10,11]. Furthermore, geodesic distances under the influence of anisotropy can be computed by Fast Marching methods [12]. The adaptive remeshing of triangular meshes [13], in contrast, applies similar techniques of approximating geodesics by Euclidean distances.…”