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Point-Based Manifold Harmonics
Oct. 2012 (vol. 18 no. 10)
pp. 1693-1703
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.

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Index Terms:
Manifolds,Symmetric matrices,Eigenvalues and eigenfunctions,Harmonic analysis,Convergence,Laplace equations,Approximation methods,eigenfunction.,Point-sampled surface,Laplace-Beltrami operator
Yang Liu, Balakrishnan Prabhakaran, Xiaohu Guo, "Point-Based Manifold Harmonics," IEEE Transactions on Visualization and Computer Graphics, vol. 18, no. 10, pp. 1693-1703, Oct. 2012, doi:10.1109/TVCG.2011.152
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