دانلود رایگان مقاله عملگر لاپلاس - بلترامی موضعی گسسته بر مش مثلثی

Abstract

The Laplace–Beltrami operator is the foundation of describing geometric partial differential equations, and it also plays an important role in the fields of computational geometry, computer graphics and image processing, such as surface parameterization, shape analysis, matching and interpolation. However, constructing the discretized Laplace–Beltrami operator with convergent property has been an open problem. In this paper we propose a new discretization scheme of the Laplace–Beltrami operator over triangulated surfaces. We prove that our discretization of the Laplace–Beltrami operator converges to the Laplace–Beltrami operator at every point of an arbitrary smooth surface as the size of the triangular mesh over the surface tends to zero. Numerical experiments are conducted, which support the theoretical analysis.

6. Conclusion

In this paper, we have developed an adaptive algorithm for approximating the Laplace–Beltrami operator on a triangular mesh with point-wise convergence guarantee. Such convergence is required in many applications, where quantities, such as mean curvature, need to be estimated at each node of the mesh. The convergence result does not require the aspect ratio of mesh elements to be bounded. Experimental results show that our algorithm indeed exhibits convergence empirically, and outperforms current popular methods in accuracy. Especially, our method provides quite good approximation of the LBO for surfaces with sharp features. In the future, we will investigate symmetrizable discrete LBO based on our method and then build a set of orthogonal bases for spectral analysis over triangulated manifold surfaces.