Issue No. 10 - Oct. (2012 vol. 18)
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TVCG.2011.152
Yang Liu , University of Texas at Dallas, Richardson
Balakrishnan Prabhakaran , University of Texas at Dallas, Richardson
Xiaohu Guo , University of Texas at Dallas, Richardson
This paper proposes an algorithm to build a set of orthogonal Point-Based Manifold Harmonic Bases (PB-MHB) for spectral analysis over point-sampled manifold surfaces. To ensure that PB-MHB are orthogonal to each other, it is necessary to have symmetrizable discrete Laplace-Beltrami Operator (LBO) over the surfaces. Existing converging discrete LBO for point clouds, as proposed by Belkin et al. [CHECK END OF SENTENCE], is not guaranteed to be symmetrizable. We build a new point-wisely discrete LBO over the point-sampled surface that is guaranteed to be symmetrizable, and prove its convergence. By solving the eigen problem related to the new operator, we define a set of orthogonal bases over the point cloud. Experiments show that the new operator is converging better than other symmetrizable discrete Laplacian operators (such as graph Laplacian) defined on point-sampled surfaces, and can provide orthogonal bases for further spectral geometric analysis and processing tasks.
Manifolds, Symmetric matrices, Eigenvalues and eigenfunctions, Harmonic analysis, Convergence, Laplace equations, Approximation methods, eigenfunction., Point-sampled surface, Laplace-Beltrami operator
B. Prabhakaran, X. Guo and Y. Liu, "Point-Based Manifold Harmonics," in IEEE Transactions on Visualization & Computer Graphics, vol. 18, no. , pp. 1693-1703, 2012.