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Estimating the Laplace-Beltrami Operator by Restricting 3D Functions

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Executive Summary

The authors present a novel approach for computing and solving the Poisson equation over the surface of a mesh. As in previous approaches, the authors define the Laplace-Beltrami operator by considering the derivatives of functions defined on the mesh. However, in this work, they explore a choice of functions that is decoupled from the tessellation. Specifically, they use basis functions (second-order tensor-product B-splines) defined over 3D space, and then restrict them to the surface. They 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.

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