Fast calculation of Laplace-Beltrami eigenproblems via subdivision linear subspace

Abstract

To solve the Laplace-Beltrami eigenproblem on 3D models, we develop an efficient and fast computation method based on surface fitting and linear subspace. First, our method generates a finite subdivision surface to approximate the original high-resolution model. Then, we restrict the eigenproblem by constructing a subdivision linear subspace, whose basis is generated during the surface fitting process. Finally, we obtain the required eigenpairs by solving the restricted eigenproblem, whose scale is much smaller than that of the original model. Experimental results demonstrate that our eigenvalues and eigenvectors effectively approximate the ground truth. Especially, in the low-frequency band, the spectrum of our method performs much better than those of comparisons. Moreover, the eigenvectors mapped to the original mesh keep the orthogonality, making themselves a set of filtering basis on the original mesh. Meanwhile, our method also shows good performance on time and memory consumption.