Discretizing Laplace–Beltrami Operator from Differential Quantities

Abstract

The Laplace–Beltrami operator (LBO) is the fundamental geometric object associated with manifold surfaces and has been widely used in various tasks in geometric processing. By understanding that the LBO can be computed by differential quantities, we propose an approach for discretizing the LBO on manifolds by estimating differential quantities. For a point on the manifold, we first fit a quadratic surface to this point and its neighborhood by minimizing the least-square energy function. Then we compute the first- and second-order differential quantities by the approximated quadratic surface. Finally the discrete LBO at this point is computed from the estimated differential quantities and thus the Laplacian matrix over the discrete manifold is constructed. Our approach has several advantages: it is simple and efficient and insensitive to noise and boundaries. Experimental results have shown that our approach performs better than most of the current approaches. We also propose a feature-aware scheme for modifying the Laplacian matrix. The modified Laplacian matrix can be used in other feature preserving geometric processing applications.