Discrete laplace operator on meshed surfaces

Abstract

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 [12,23,25] 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 [12].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 outperforms cotangent-based methods in providing accurate approximation of the Laplace operator for various meshes. *

Discrete Laplace–Beltrami operators for shape analysis and segmentation

“…It can be seen in Figure 3 (left) that the cubic FEM approach performs best, followed by the lumped cubic FEM, linear FEM, Meyer et al [36] and the lumped linear FEM [16]. The approach by Belkin et al [6] does not work for eigencomputations on meshes with boundary yet and gives large errors when one tries. It will therefore only be tested for the sphere below.…”

“…This leads to the system D −1/2 AD −1/2 y = λy with the same eigenvalues. The original eigenvectors can be retrieved by f = D −1/2 y. Belkin et al [5,6] describe a discretization of the LaplaceBeltrami operator on the k-nearest neighbor graph T of a point set {p i } n i=1 sampled on an underlying manifold and an extension to meshes by using the heat kernel to construct the weights. The mesh version [6] considers weights not only at the edges of the mesh, but in a larger neighborhood of a vertex (the heat kernel is cut off thus sparsity is maintained).…”

“…The original eigenvectors can be retrieved by f = D −1/2 y. Belkin et al [5,6] describe a discretization of the LaplaceBeltrami operator on the k-nearest neighbor graph T of a point set {p i } n i=1 sampled on an underlying manifold and an extension to meshes by using the heat kernel to construct the weights. The mesh version [6] considers weights not only at the edges of the mesh, but in a larger neighborhood of a vertex (the heat kernel is cut off thus sparsity is maintained). While the geometric operators in (3,4,5) are not convergent in general and cannot deal well with non-uniform meshes [60], this method exhibits convergence and does not depend much on the shape of the triangles, just on the density of the vertices.…”

Heat-mapping: A robust approach toward perceptually consistent mesh segmentation

“…The heat kernel quantitatively encodes the heat flow across a manifold M and is uniquely defined for any two vertices i, j on the manifold [12,3]. Suppose we apply a unit amount of heat at the node i, and allow the heat flow on the manifold across all of the edges.…”

Uniform Convergence of Discrete Curvatures from Nets of Curvature Lines

“…2 The three generic patterns of curvature lines near an umbilical point, called lemon, star, and monstar by Berry and Hannay [2] of (ii) and (iii) may be interpreted in this way: by decreasing the diameter of the domains over which discrete curvatures are integrated (measured), while simultaneously increasing the mesh refinement inside these domains at a sufficiently fast rate, one recovers classical pointwise notions of smooth curvatures in the limit. In a similar fashion, Belkin et al [1] proposed a discrete Laplace operator based on the heat kernel. This operator converges in a pointwise manner if the kernel is scaled down while the mesh resolution is increased sufficiently fast relative to the scaling of the kernel.…”