Estimating the Laplace-Beltrami Operator by Restricting 3D Functions

We present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, we define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, we explore a choice of functions that is decoupled from the tessellation. Specifically, we use basis functions (second-order tensor-product B-splines) defined over 3D space, and then restrict them to the surface. We show that in addition to being invariant to mesh topology, this definition of the Laplace-Beltrami operator allows a natural multiresolution structure on the function space that is independent of the mesh structure, enabling the use of a simple multigrid implementation for solving the Poisson equation.

Resources:    Paper »

author = "Ming Chuang and Linjie Luo and Benedict J. Brown and Szymon Rusinkiewicz and
Michael Kazhdan",
title = "Estimating the {Laplace}-{Beltrami} Operator by Restricting {3D} Functions",
journal = "Symposium on Geometry Processing",
year = "2009",
month = jul