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

TL;DR

This paper provides a comprehensive, explicit derivation of the gradient and objective function for anisotropic Centroidal Voronoi Tessellations (CVT), enhancing understanding and enabling advanced applications like surface remeshing and feature capturing.

Contribution

It offers a complete, detailed derivation of the gradient and objective function for anisotropic CVT, clarifying previous partial derivations and implicit conditions.

Findings

Explicit formulas for the gradient and objective function on surfaces and volumes.

Enhanced understanding of the method's working conditions and applications.

Facilitation of advanced meshing techniques and feature capturing.

Abstract

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 fully-automated 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.