Anisotropic Simplicial Meshing Using Local Convex Functions

Figure 1: Anisotropic meshes generated by our method. Left: 2D meshing. The anisotropic metric is defined as the Hessian of an analytic function evincing a large range of anisotropy ratios (λ ∈ [1.9, 394.4]). Zoom in on the image to see meshing details. Middle: 3D surface meshing of the Happy Buddha from curvature tensors estimated from a high-resolution reference mesh (115474 vertices). Our relaxation produces a high quality result (63284 vertices) starting with an initial low-resolution mesh (5000 vertices). Right: volumetric meshing in a 3D cube. Anisotropy changes substantially (λ ∈ [1, 40]) and rapidly over the domain. The lower image shows a cross-section. We present a novel method to generate high-quality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approx-imation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and itera-tively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewise-linear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to state-of-the-art methods on a variety of domains and metrics