The Community for Technology Leaders
RSS Icon
Issue No.10 - Oct. (2012 vol.18)
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.
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 & Computer Graphics, vol.18, no. 10, pp. 1693-1703, Oct. 2012, doi:10.1109/TVCG.2011.152
[1] M. Belkin, J. Sun, and Y. Wang, "Constructing Laplace Operator from Point Clouds in ${\hbox{\rlap{I}\kern 2.0pt{\hbox{R}}}}^d$ ," Proc. 19th Ann. ACM-SIAM Symp. Discrete Algorithms, pp. 1031-1040, 2009.
[2] S. Rosenberg, The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds, ser. London Mathematical Society Student Texts. Cambridge Univ. Press, 1997.
[3] J. Jost, Riemannian Geometry and Geometric Analysis, ser. Universitext. Springer, 2002.
[4] M. Reuter, F.-E. Wolter, and N. Peinecke, "Laplace-Beltrami Spectra as "Shape-DNA" of Surfaces and Solids," Computer-Aided Design, vol. 38, no. 4, pp. 342-366, 2006.
[5] B. Vallet and B. Lévy, "Spectral Geometry Processing with Manifold Harmonics," Computer Graphics Forum, vol. 27, no. 2, pp. 251-260, 2008.
[6] J. Hu and J. Hua, "Salient Spectral Geometric Features for Shape Matching and Retrieval," The Visual Computer, vol. 25, nos. 5-7, pp. 667-675, 2009.
[7] H. Zhang, O. van Kaick, and R. Dyer, "Spectral Methods for Mesh Processing and Analysis," Proc. Eurographics State-of-the-Art Report, pp. 1-22, 2007.
[8] N. Peinecke, F.-E. Wolter, and M. Reuter, "Laplace Spectra as Fingerprints for Image Recognition," Computer Aided Design, vol. 39, pp. 460-476, http://portal.acm.orgcitation. cfm?id=1244485.1244880 , June 2007.
[9] K. Hildebrandt and K. Polthier, "On Approximation of the Laplace-Beltrami Operator and the Willmore Energy of Surfaces," Computer Graphics Forum, vol. 30, pp. 1513-1520, 2011.
[10] K. Huseyin, Vibration and Stability of Multiple Parameter Systems. Springer, 1978.
[11] B. Lévy, "Laplace-Beltrami Eigenfunctions Towards an Algorithm that 'Understands' Geometry," Proc. IEEE Int'l Conf. Shape Modeling and Applications, p. 13, 2006.
[12] L. Miao, J. Huang, X. Liu, H. Bao, Q. Peng, and B. Guo, "Computing Variation Modes for Point Set Surfaces," Proc. Eurographics/IEEE VGTC Symp. Point-Based Graphics, pp. 63-69, June 2005.
[13] G. Taubin, "A Signal Processing Approach to Fair Surface Design," Proc. ACM SIGGRAPH '95, pp. 351-358, 1995.
[14] R. Liu and H. Zhang, "Mesh Segmentation via Spectral Embedding and Contour Analysis," Computer Graphics Forum, vol. 26, pp. 385-394, 2007.
[15] Z. Karni and C. Gotsman, "Spectral Compression of Mesh Geometry," Proc. ACM SIGGRAPH '00, pp. 279-286, 2000.
[16] D. Cotting, T. Weyrich, M. Pauly, and M. Gross, "Robust Watermarking of Point-Sampled Geometry," Proc. the Shape Modeling Applications Int'l, pp. 233-242, 2004.
[17] J. Wu and L. Kobbelt, "Efficient Spectral Watermarking of Large Meshes with Orthogonal Basis Functions," The Visual Computer, vol. 21, nos. 8-10, pp. 848-857, 2005.
[18] S. Dong, P.-T. Bremer, M. Garland, V. Pascucci, and J.C. Hart, "Spectral Surface Quadrangulation," ACM Trans. Graphics, vol. 25, no. 3, pp. 1057-1066, 2006.
[19] J. Huang, M. Zhang, J. Ma, X. Liu, L. Kobbelt, and H. Bao, "Spectral Quadrangulation with Orientation and Alignment Control," ACM Trans. Graphics, vol. 27, no. 5, p. 147, 2008.
[20] P. Mullen, Y. Tong, P. Alliez, and M. Desbrun, "Spectral Conformal Parameterization," Computer Graphics Forum, vol. 27, pp. 1487-1494, 2008.
[21] M. Desbrun, M. Meyer, P. Schröder, and A.H. Barr, "Implicit Fairing of Irregular Meshes Using Diffusion and Curvature Flow," Proc. ACM SIGGRAPH '99, pp. 317-324, 1999.
[22] M. Meyer, M. Desbrun, P. Schröder, and A. Barr, "Discrete Differential-Geometry Operator for Triangulated 2-Manifolds," Proc. Visual Mathematics '02, 2002.
[23] G. Xu, "Discrete Laplace-Beltrami Operators and Their Convergence," Computer Aided Geometric Design, vol. 21, no. 8, pp. 767-784, 2004.
[24] U. Pinkall and K. Polthier, "Computing Discrete Minimal Surfaces and Their Conjugates," Experimental Math., vol. 2, no. 1, pp. 15-36, 1993.
[25] K. Hildebrandt, K. Polthier, and M. Wardetzky, "On the Convergence of Metric and Geometric Properties of Polyhedral Surfaces," Geometriae Dedicata, vol. 123, no. 1, pp. 89-112, 2006.
[26] M. Wardetzky, S. Mathur, F. Kälberer, and E. Grinspun, "Discrete Laplace Operators: No Free Lunch," Proc. Fifth Eurographics Symp. Geometry Processing, pp. 33-37, 2007.
[27] M. Reuter, S. Biasotti, D. Giorgi, G. Patanè, and M. Spagnuolo, "Discrete Laplace-Beltrami Operators for Shape Analysis and Segmentation," Computers & Graphics, vol. 33, no. 3, pp. 381-390, 2009.
[28] M. Belkin and P. Niyogi, "Towards a Theoretical Foundation for Laplacian-Based Manifold Methods," Proc. 18th Conf. Learning Theory (COLT), pp. 486-500, 2005.
[29] M. Belkin, J. Sun, and Y. Wang, "Discrete Laplace Operator on Meshed Surfaces," Proc. 24th Ann. Symp. Computational Geometry, pp. 278-287, 2008.
[30] T.K. Dey, P. Ranjan, and Y. Wang, "Convergence, Stability, and Discrete Approximation of Laplace Spectra," Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 650-663, 2010.
[31] R.R. Coifman and S. Lafon, "Diffusion Maps," Applied and Computational Harmonic Analysis, vol. 21, no. 1, pp. 5-30, 2006.
[32] H. Zhang, "Discrete Combinatorial Laplacian Operators for Digital Geometry Processing," Proc. SIAM Conf. Geometric Design and Computing, pp. 575-592, 2004.
[33] C. Luo, J. Sun, and Y. Wang, "Integral Estimation from Point Cloud in D-Dimensional Space: A Geometric View," SCG '09: Proc. 25th Ann. Symp. Computational Geometry, pp. 116-124, 2009.
[34] N. Amenta, M. Bern, and M. Kamvysselis, "A New Voronoi-Based Surface Reconstruction Algorithm," Proc. ACM SIGGRAPH '98, pp. 415-421, 1998.
[35] J. Giesen and U. Wagner, "Shape Dimension and Intrinsic Metric from Samples of Manifolds with High Co-Dimension," Proc. 19th Ann. Symp. Computational Geometry, pp. 329-337, 2003.
[36] N. Amenta and M. Bern, "Surface Reconstruction by Voronoi Filtering," Proc. 14th Ann. Symp. Computational Geometry, pp. 39-48, 1998.
[37] H. Blum, "A Transformation for Extracting New Descriptors of Shape," Models for the Perception of Speech and Visual Form, W.W. Dunn, ed., pp. 362-380, MIT Press, 1967.
[38] F.E. Wolter, Cut Locus & Medial Axis in Global Shape Interrogation & Representation. Sea Grant College Program, Massachusetts Inst. of Tech nology 1992.
[39] S. Har-Peled and K. Varadarajan, "Projective Clustering in High Dimensions Using Core-Sets," Proc. 18th Ann. Symp. Computational Geometry, pp. 312-318, 2002.
42 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool