Gradient of the Objective Function for an Anisotropic Centroidal Voronoi Tessellation (CVT) - A revised, detailed derivation

In their recent article (2010), Levy and Liu introduced a generalization of Centroidal Voronoi Tessellation (CVT) namely the Lp-CVT that allows the computation of an anisotropic CVT over a sound mathematical framework. In this article a new objective function is defined, and both this function and its gradient are derived in closed-form for surfaces and volumes. This method opens a wide range of possibilities, also described in the paper, such as quad-dominant surface remeshing, hex-dominant volume meshing or fullyautomated capturing of sharp features. However, in the same paper, the derivations of the gradient and of the new objective function are only partially expanded, in the appendices, and some relevant requisites on the anisotropy field are left implicit. In order to better harness the possibilities described there, in this work the entire derivation process is made explicit. In the authors’ opinion, this also helps understanding the working conditions of the method and its possible applications. keywords: Centroidal Voronoi tessellation, anisotropic meshing, surface reconstruction, topology preservation, computational geometry and object modeling Introduction In their recent article (2010), Levy and Liu introduce a generalization of Centroidal Voronoi Tessellation namely the Lp-CVT that “minimizes a higher-order moment of the coordinates on the Voronoi cells”. Levy and Liu take as reference the standard CVT objective function, and extend its behavior injecting the Lp norm and an anisotropy term, in the form of a matrix obtained from the anisotropy field, in the function itself. This method opens a wide range of possible applications, also described in the article, such as quad-dominant surface remeshing, hex-dominant volume meshing or fully-automated capturing of sharp features. In particular, in the authors’ opinion, this method could also increase the resistance to noise in surface reconstruction and remeshing, which is relevant in the application of methods such as the one described in Piastra (2013). In the work by Levy and Liu (2010), however, the derivation of the gradient, as well as the definition of the new objective function, are only partially described, and some conditions on the anisotropy field are left implicit. Application of the same method under different conditions, e.g. a specific anisotropy field, or for different purposes, involves a complete comprehension of the mathematical frame on which the method is based. This work is intended to analyze thoroughly the derivation both of the objective function and of its gradient, in order to understand the functioning of the method and the conditions of applicability. For sake of clarity, the notation used here is slightly different from the one in the original work, due to of the different structure of this paper. ∗Corresponding author: [email protected]