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, 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.

Yang
Liu, Balakrishnan Prabhakaran, Xiaohu Guo, "Point-Based
Manifold Harmonics", in IEEE Transactions on Visualization and Computer Graphics, Vol. 18, No. 10, pp. 1693-1703, 2012.