Centroidal Voronoi tessellation (CVT) is a fundamental geometric structure that finds many applications in computational science and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. Our goal is to develop efficient methods for CVT computation, justify the fast convergence of these methods theoretically and demonstrate their superiority with experimental examples in various cases. Specifically, it is shown that the CVT energy function has C 2 smoothness in convex domains and in most other commonly encountered domains with smooth density, correcting the view in the literature that this function is non-smooth (that is, merely C 0 but not C 1 ). Due to its C 2 smoothness, it is therefore possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We apply quasi-Newton methods to computing CVT and demonstrate their faster convergence than Lloyd's method and their better robustness than the Lloyd-Newton method, a previous attempt at CVT computation acceleration. The application of these results to surface remeshing in computer graphics is also studied.

International audienceWe propose a new isotropic remeshing method, based on Centroidal Voronoi Tessellation (CVT). Constructing CVT requires to repeatedly compute Restricted Voronoi Diagram (RVD), defined as the intersection between a 3D Voronoi diagram and an input mesh surface. Existing methods use some approximations of RVD. In this paper, we introduce an efficient algorithm that computes RVD exactly and robustly. As a consequence, we achieve better remeshing quality than approximation-based approaches, without sacrificing efficiency. Our method for RVD computation uses a simple procedure and a kd-tree to quickly identify and compute the intersection of each triangle face with its incident Voronoi cells. Its time complexity is O(mlogn), where n is the number of seed points and m is the number of triangles of the input mesh. Fast convergence of CVT is achieved using a quasi-Newton method, which proved much faster than Lloyd's iteration. Examples are presented to demonstrate the better quality of remeshing results with our method than with the state-of-art approaches