Discrete Laplace Operator on Meshed Surfaces

In recent years a considerable amount of work in graphics and geometric optimization used tools based on the Laplace-Beltrami operator on a surface. The applications of the Laplacian include mesh editing, surface smoothing, and shape interpolations among others. However, it has been shown [13, 24, 26] that the popular cotangent approximation schemes do not provide convergent point-wise (or even L-2) estimates, while many applications rely on point-wise estimation. Existence of such schemes has been an open question [13]. In this paper we propose the first algorithm for approximating the Laplace operator of a surface from a mesh with point-wise convergence guarantees applicable to arbitrary meshed surfaces. We show that for a sufficiently fine mesh over an arbitrary surface, our mesh Laplacian is close to the Laplace-Beltrami operator on the surface at every point of the surface. Moreover, the proposed algorithm is simple and easily implementable. Experimental evidence shows that our algorithm exhibits convergence empirically and compares favorably with cotangent-based methods in providing accurate approximation of the Laplace operator for various meshes.