Particle-based anisotropic surface meshing

Figure 1: Anisotropic meshing results generated by our particle-based method. This paper introduces a particle-based approach for anisotropic sur-face meshing. Given an input polygonal mesh endowed with a Rie-mannian metric and a specified number of vertices, the method gen-erates a metric-adapted mesh. The main idea consists of mapping the anisotropic space into a higher dimensional isotropic one, called “embedding space”. The vertices of the mesh are generated by uni-formly sampling the surface in this higher dimensional embedding space, and the sampling is further regularized by optimizing an en-ergy function with a quasi-Newton algorithm. All the computations can be re-expressed in terms of the dot product in the embedding space, and the Jacobian matrices of the mappings that connect d-ifferent spaces. This transform makes it unnecessary to explicitly represent the coordinates in the embedding space, and also provides all necessary expressions of energy and forces for efficient compu-tations. Through energy optimization, it naturally leads to the de-sired anisotropic particle distributions in the original space. The tri-angles are then generated by computing the Restricted Anisotropic Voronoi Diagram and its dual Delaunay triangulation. We compare our results qualitatively and quantitatively with the state-of-the-art in anisotropic surface meshing on several examples, using the stan-dard measurement criteria