Global Surface Remeshing Using Symmetric Delaunay Triangulation in Uniformization Spaces

Surface remeshing plays a fundamental role in digital geometry processing. In this paper, we present a novel framework for global surface remeshing based on symmetric Delaunay triangulations on the uniformization spaces. Surfaces with arbitrary topologies can be uniformized to one of three canonical uniformization spaces: the sphere, the Euclidean plane and the hyperbolic plane. The uniformization process induces a symmetry group. A triangulation on the original surface is converted to a triangulation on the uniformization space, which is invariant under the symmetry group, and vice versa. Furthermore, the uniformization process preserves angles, so a symmetric dense Delaunay triangulation on the uniformization space induces a good triangulation on the original surface. We construct the uniformization of surfaces with arbitrary topologies using Ricci flow, and apply orbit insertion technique to the Delaunay refinement algorithm which ensures the symmetry. Experimental results demonstrates that our method is global, intrinsic, general and efficient.