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A local mesh method for solving PDEs on point clouds
1. | Department of mathematics, University of Southern California, Los Angeles, CA 90089-2532, United States |
2. | Department of mathematics, University of California, Irvine, Irvine, CA 92697-3875, United States |
3. | Department of Mathematics, University of California, Irvine, Irvine, CA 92697-3875 |
References:
[1] |
F. Camastra and A. Vinciarelli, Estimating the intrinsic dimension of data with a fractal-based method, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 24 (2002), 1404-1407.
doi: 10.1109/TPAMI.2002.1039212. |
[2] |
M. Belkin and P. Niyogi, Semi-supervised learning on riemannian manifolds, Machine Learning, 56 (2004), 209-239.
doi: 10.1023/B:MACH.0000033120.25363.1e. |
[3] |
G. Chen, A. V. Little, M. Maggioni and L. Rosasco, Some recent advances in multiscale geometric analysis of point clouds, Wavelets and Multiscale Analysis, 199-225.
doi: 10.1007/978-0-8176-8095-4_10. |
[4] |
M. Belkin, J. Sun and Y. Wang, Constructing laplace operator from point clouds in $\mathbbmathbb{R}^{d}$, In "Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms," 1031-1040, Philadelphia, PA, (2009). |
[5] |
C. Luo, J. Sun and Y. Wang, Integral estimation from point cloud in d-dimensional space: A geometric view, In "Symposium on Computational Geometry," (2009), 116-124.
doi: 10.1145/1542362.1542389. |
[6] |
J. Liang, R. Lai, T. W. Wong and H. Zhao, Geometric understanding of point clouds using Laplace-Beltrami operator, CVPR, (2012). |
[7] |
J. Liang and H. Zhao, Solving partial differential equations on point clouds, SIAM J. Sci. Comput., 35 (2013), A1461-A1486.
doi: 10.1137/120869730. |
[8] |
I. Jolliffe, "Principal Component Analysis," Wiley Online Library, 2005. |
[9] |
G. Taubin, Geometric signal processing on polygonal meshes, EUROGRAPHICS, (2000). |
[10] |
M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, 35-57, Math. Vis., Springer, Berlin, (2003). |
[11] |
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Discrete differential geometry operators in nd, In "Proc. VisMath'02 Berlin Germany," (2002). |
[12] |
Guoliang Xu, Convergent discrete Laplace-Beltrami operator over triangular surfaces, Proceedings of Geometric Modelling and Processing, (2004), 195-204. |
[13] |
R. Lai and T. F. Chan, A framework for intrinsic image processing on surfaces, UCLA CAM 10-25, submitted, 115 (2011), 1647-1661.
doi: 10.1016/j.cviu.2011.05.011. |
[14] |
J. W. Milnor and J. D. Stasherff, "Characteristic Classes," Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp. |
[15] |
J. M. Lee, "Riemannian Manifolds: An Introduction to Curvature," Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997, xvi+224 pp. |
[16] |
X. Li, W. Wang, R. R. Martin and A. Bowyer, Using low-discrepancy sequences and the crofton formula to compute surface areas of geometric models, Computer-Aided Design, 35 (2003), 771-782.
doi: 10.1016/S0010-4485(02)00100-8. |
[17] |
J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. U. S. A., 93 (1996), 1591-1595.
doi: 10.1073/pnas.93.4.1591. |
[18] |
R. Kimmel and J. A. Sethian, Computing geodesic paths on manifolds, Proc. Natl. Acad. Sci. USA, 95 (1998), 8431-8435.
doi: 10.1073/pnas.95.15.8431. |
[19] |
H. Zhao, A fast sweeping method for eikonal equations, Mathematics of Computation, 74 (2005), 603-627.
doi: 10.1090/S0025-5718-04-01678-3. |
[20] |
J. Qian, Y. T. Zhang and H. K. Zhao, Fast sweeping methods for eikonal equations on triangular meshes, SIAM Journal on Numerical Analysis, 45 (2007), 83-107.
doi: 10.1137/050627083. |
[21] |
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. of Math., 98 (1976), 1059-1078.
doi: 10.2307/2374041. |
[22] |
J. Sun, M. Ovsjanikov and L. Guibas, A concise and provabley informative multi-scale signature baded on heat diffusion, In " Eurographics Symposium on Geometry Processing," (2009), 1383-1392. |
[23] |
M. M. Bronstein and I. Kokkinos, Scale-invariant heat kernel signatures for non-rigid shape recognition, In "Proc. Computer Vision and Pattern Recognition (CVPR)", (2010), 1704-1711.
doi: 10.1109/CVPR.2010.5539838. |
[24] |
Y. Shi, R. Lai, R. Gill, D. Pelletier, D. Mohr, N. Sicotte and A. W. Toga, Conformal metric optimization on surface (CMOS) for deformation and mapping in Laplace-Beltrami embedding space, MICCAI, 14 (2011), 327-334. |
[25] |
R. Lai, Y. Shi, K. Scheibel, S. Fears, R. Woods, A. W. Toga and T. F. Chan, Metric-induced optimal embedding for intrinsic 3D shape analysis, Computer Vision and Pattern Recognition (CVPR), (2010), 2871-2878. |
[26] |
R. Lai, Y. Shi, N. Sicotte and A. W. Toga, Automated corpus callosum extraction via Laplace-Beltrami nodalâparcellation and intrinsic geodesic curvature flows on surfaces, ICCV, 11 (2011), 2034-2040. |
[27] |
I. Chavel, "Eigenvalues in Riemannian Geometry," Including a Chapter by Burton Randol. With an Appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. |
[28] |
J. Jost, "Riemannian Geometry And Geometric Analysis," Springer, 3rd edition, 2001. |
[29] |
M. Reuter, F. E. Wolter and N. Peinecke, Laplace-Beltrami spectra as shape-DNA of surfaces and solids, Computer-Aided Design, 38 (2006), 342-366.
doi: 10.1016/j.cad.2005.10.011. |
[30] |
J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces, Journal of Scientific Computing, 37 (2008), 282-315.
doi: 10.1007/s10915-008-9210-z. |
[31] |
W. Gao, R. Lai, Y. Shi, I. Dinov and A. W. Toga, A narrow-band approach for approximating the Laplace-Beltrami spectrum of 3D shapes, In "AIP Conference Proceedings," 1281 (2010), 1010-1013.
doi: 10.1063/1.3497791. |
[32] |
B. Levy, Laplace-Beltrami eigenfunctions: Towards an algorithm that understands geometry, IEEE International Conference on Shape Modeling and Applications, invited talk, (2006), 13pp.
doi: 10.1109/SMI.2006.21. |
[33] |
S. Dong, P.-T. Bremer, M. Garland, V. Pascucci and J. C. Hart, Spectral surface quadrangulation, ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH, (2006), 1057-1066. |
[34] |
R. M. Rustamov, Laplace-Beltrami eigenfunctions for deformation invariant shape representation, Eurographics Symposium on Geometry Processing, (2007), 225-233. |
[35] |
B. Vallet and B. Lévy, Spectral geometry processing with manifold harmonics, Computer Graphics Forum (Proceedings Eurographics), 27 (2008), 251-260.
doi: 10.1111/j.1467-8659.2008.01122.x. |
[36] |
M. Reuter, Hierarchical shape segmentation and registration via topological features of Laplace-Beltrami eigenfunctions, International Journal of Computer Vision, 89 (2009), 287-308.
doi: 10.1007/s11263-009-0278-1. |
[37] |
A. M. Bronstein, M. M. Bronstein, R. Kimmel, M. Mahmoudi and G. Sapiro, A gromov-hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching, Int. Jour. Comput. Vis., 89 (2010), 266-286.
doi: 10.1007/s11263-009-0301-6. |
[38] |
A. Qiu, D. Bitouk and M. I. Miller, Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator, IEEE Trans. Med. Imag., 25 (2006), 1296-1306. |
[39] |
Y. Shi, R. Lai, S. Krishna, N. Sicotte, I. Dinov and A. W. Toga, Anisotropic Laplace-Beltrami eigenmaps: Bridging reeb graphs and skeletons, In "Proc. MMBIA," (2008). |
[40] |
Y. Shi, R. Lai, K. Kern, N. Sicotte, I. Dinov and A. W. Toga, Harmonic surface mapping with Laplace-Beltrami eigenmaps, In "Proc. MICCAI," (2008), 147-154.
doi: 10.1007/978-3-540-85990-1_18. |
[41] |
R. Lai, Y. Shi, I. Dinov, T. F. Chan and A. W. Toga, Laplace-Beltrami nodal counts: A new signature for 3D shape analysis, In "Proc. ISBI," (2009), 694-697.
doi: 10.1109/ISBI.2009.5193142. |
[42] |
Y. Shi, I. Dinov and A. W. Toga, Cortical shape analysis in the Laplace-Beltrami feature space, In "Proc. MICCAI," (2009), 208-215. |
[43] |
Y. Shi, R. Lai, J. Morra, I. Dinov, P. Thompson and A. W. Toga, Robust surface reconstruction via Laplace-Beltrami eigen-projection and boundary deformation, IEEE Trans. Medical Imaging, 29 (2010), 2009-2022. |
[44] |
Y. Shi, R. Lai and A. Toga, Unified geometry and topology correction for cortical surface reconstruction with intrinsic reeb analysis, MICCAI, (2012), 601-8. |
[45] |
F. Mémoli and G. Sapiro, Distance functions and geodesics on submanifolds of rd and point clouds, SIAM Journal on Applied Mathematics, (2005), 1227-1260. |
[46] |
M. De Berg, O. Cheong, M. Van Kreveld and M. Overmars, "Computational Geometry: Algorithms and Applications," Third edition. Springer-Verlag, Berlin, 2008. |
show all references
References:
[1] |
F. Camastra and A. Vinciarelli, Estimating the intrinsic dimension of data with a fractal-based method, Pattern Analysis and Machine Intelligence, IEEE Transactions on, 24 (2002), 1404-1407.
doi: 10.1109/TPAMI.2002.1039212. |
[2] |
M. Belkin and P. Niyogi, Semi-supervised learning on riemannian manifolds, Machine Learning, 56 (2004), 209-239.
doi: 10.1023/B:MACH.0000033120.25363.1e. |
[3] |
G. Chen, A. V. Little, M. Maggioni and L. Rosasco, Some recent advances in multiscale geometric analysis of point clouds, Wavelets and Multiscale Analysis, 199-225.
doi: 10.1007/978-0-8176-8095-4_10. |
[4] |
M. Belkin, J. Sun and Y. Wang, Constructing laplace operator from point clouds in $\mathbbmathbb{R}^{d}$, In "Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms," 1031-1040, Philadelphia, PA, (2009). |
[5] |
C. Luo, J. Sun and Y. Wang, Integral estimation from point cloud in d-dimensional space: A geometric view, In "Symposium on Computational Geometry," (2009), 116-124.
doi: 10.1145/1542362.1542389. |
[6] |
J. Liang, R. Lai, T. W. Wong and H. Zhao, Geometric understanding of point clouds using Laplace-Beltrami operator, CVPR, (2012). |
[7] |
J. Liang and H. Zhao, Solving partial differential equations on point clouds, SIAM J. Sci. Comput., 35 (2013), A1461-A1486.
doi: 10.1137/120869730. |
[8] |
I. Jolliffe, "Principal Component Analysis," Wiley Online Library, 2005. |
[9] |
G. Taubin, Geometric signal processing on polygonal meshes, EUROGRAPHICS, (2000). |
[10] |
M. Meyer, M. Desbrun, P. Schröder and A. H. Barr, Discrete differential-geometry operators for triangulated 2-manifolds, Visualization and Mathematics III, 35-57, Math. Vis., Springer, Berlin, (2003). |
[11] |
M. Desbrun, M. Meyer, P. Schröder and A. H. Barr, Discrete differential geometry operators in nd, In "Proc. VisMath'02 Berlin Germany," (2002). |
[12] |
Guoliang Xu, Convergent discrete Laplace-Beltrami operator over triangular surfaces, Proceedings of Geometric Modelling and Processing, (2004), 195-204. |
[13] |
R. Lai and T. F. Chan, A framework for intrinsic image processing on surfaces, UCLA CAM 10-25, submitted, 115 (2011), 1647-1661.
doi: 10.1016/j.cviu.2011.05.011. |
[14] |
J. W. Milnor and J. D. Stasherff, "Characteristic Classes," Annals of Mathematics Studies, No. 76. Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. vii+331 pp. |
[15] |
J. M. Lee, "Riemannian Manifolds: An Introduction to Curvature," Graduate Texts in Mathematics, 176. Springer-Verlag, New York, 1997, xvi+224 pp. |
[16] |
X. Li, W. Wang, R. R. Martin and A. Bowyer, Using low-discrepancy sequences and the crofton formula to compute surface areas of geometric models, Computer-Aided Design, 35 (2003), 771-782.
doi: 10.1016/S0010-4485(02)00100-8. |
[17] |
J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. U. S. A., 93 (1996), 1591-1595.
doi: 10.1073/pnas.93.4.1591. |
[18] |
R. Kimmel and J. A. Sethian, Computing geodesic paths on manifolds, Proc. Natl. Acad. Sci. USA, 95 (1998), 8431-8435.
doi: 10.1073/pnas.95.15.8431. |
[19] |
H. Zhao, A fast sweeping method for eikonal equations, Mathematics of Computation, 74 (2005), 603-627.
doi: 10.1090/S0025-5718-04-01678-3. |
[20] |
J. Qian, Y. T. Zhang and H. K. Zhao, Fast sweeping methods for eikonal equations on triangular meshes, SIAM Journal on Numerical Analysis, 45 (2007), 83-107.
doi: 10.1137/050627083. |
[21] |
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. of Math., 98 (1976), 1059-1078.
doi: 10.2307/2374041. |
[22] |
J. Sun, M. Ovsjanikov and L. Guibas, A concise and provabley informative multi-scale signature baded on heat diffusion, In " Eurographics Symposium on Geometry Processing," (2009), 1383-1392. |
[23] |
M. M. Bronstein and I. Kokkinos, Scale-invariant heat kernel signatures for non-rigid shape recognition, In "Proc. Computer Vision and Pattern Recognition (CVPR)", (2010), 1704-1711.
doi: 10.1109/CVPR.2010.5539838. |
[24] |
Y. Shi, R. Lai, R. Gill, D. Pelletier, D. Mohr, N. Sicotte and A. W. Toga, Conformal metric optimization on surface (CMOS) for deformation and mapping in Laplace-Beltrami embedding space, MICCAI, 14 (2011), 327-334. |
[25] |
R. Lai, Y. Shi, K. Scheibel, S. Fears, R. Woods, A. W. Toga and T. F. Chan, Metric-induced optimal embedding for intrinsic 3D shape analysis, Computer Vision and Pattern Recognition (CVPR), (2010), 2871-2878. |
[26] |
R. Lai, Y. Shi, N. Sicotte and A. W. Toga, Automated corpus callosum extraction via Laplace-Beltrami nodalâparcellation and intrinsic geodesic curvature flows on surfaces, ICCV, 11 (2011), 2034-2040. |
[27] |
I. Chavel, "Eigenvalues in Riemannian Geometry," Including a Chapter by Burton Randol. With an Appendix by Jozef Dodziuk. Pure and Applied Mathematics, 115. Academic Press, Inc., Orlando, FL, 1984. xiv+362 pp. |
[28] |
J. Jost, "Riemannian Geometry And Geometric Analysis," Springer, 3rd edition, 2001. |
[29] |
M. Reuter, F. E. Wolter and N. Peinecke, Laplace-Beltrami spectra as shape-DNA of surfaces and solids, Computer-Aided Design, 38 (2006), 342-366.
doi: 10.1016/j.cad.2005.10.011. |
[30] |
J. Brandman, A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces, Journal of Scientific Computing, 37 (2008), 282-315.
doi: 10.1007/s10915-008-9210-z. |
[31] |
W. Gao, R. Lai, Y. Shi, I. Dinov and A. W. Toga, A narrow-band approach for approximating the Laplace-Beltrami spectrum of 3D shapes, In "AIP Conference Proceedings," 1281 (2010), 1010-1013.
doi: 10.1063/1.3497791. |
[32] |
B. Levy, Laplace-Beltrami eigenfunctions: Towards an algorithm that understands geometry, IEEE International Conference on Shape Modeling and Applications, invited talk, (2006), 13pp.
doi: 10.1109/SMI.2006.21. |
[33] |
S. Dong, P.-T. Bremer, M. Garland, V. Pascucci and J. C. Hart, Spectral surface quadrangulation, ACM Transactions on Graphics (TOG) - Proceedings of ACM SIGGRAPH, (2006), 1057-1066. |
[34] |
R. M. Rustamov, Laplace-Beltrami eigenfunctions for deformation invariant shape representation, Eurographics Symposium on Geometry Processing, (2007), 225-233. |
[35] |
B. Vallet and B. Lévy, Spectral geometry processing with manifold harmonics, Computer Graphics Forum (Proceedings Eurographics), 27 (2008), 251-260.
doi: 10.1111/j.1467-8659.2008.01122.x. |
[36] |
M. Reuter, Hierarchical shape segmentation and registration via topological features of Laplace-Beltrami eigenfunctions, International Journal of Computer Vision, 89 (2009), 287-308.
doi: 10.1007/s11263-009-0278-1. |
[37] |
A. M. Bronstein, M. M. Bronstein, R. Kimmel, M. Mahmoudi and G. Sapiro, A gromov-hausdorff framework with diffusion geometry for topologically-robust non-rigid shape matching, Int. Jour. Comput. Vis., 89 (2010), 266-286.
doi: 10.1007/s11263-009-0301-6. |
[38] |
A. Qiu, D. Bitouk and M. I. Miller, Smooth functional and structural maps on the neocortex via orthonormal bases of the Laplace-Beltrami operator, IEEE Trans. Med. Imag., 25 (2006), 1296-1306. |
[39] |
Y. Shi, R. Lai, S. Krishna, N. Sicotte, I. Dinov and A. W. Toga, Anisotropic Laplace-Beltrami eigenmaps: Bridging reeb graphs and skeletons, In "Proc. MMBIA," (2008). |
[40] |
Y. Shi, R. Lai, K. Kern, N. Sicotte, I. Dinov and A. W. Toga, Harmonic surface mapping with Laplace-Beltrami eigenmaps, In "Proc. MICCAI," (2008), 147-154.
doi: 10.1007/978-3-540-85990-1_18. |
[41] |
R. Lai, Y. Shi, I. Dinov, T. F. Chan and A. W. Toga, Laplace-Beltrami nodal counts: A new signature for 3D shape analysis, In "Proc. ISBI," (2009), 694-697.
doi: 10.1109/ISBI.2009.5193142. |
[42] |
Y. Shi, I. Dinov and A. W. Toga, Cortical shape analysis in the Laplace-Beltrami feature space, In "Proc. MICCAI," (2009), 208-215. |
[43] |
Y. Shi, R. Lai, J. Morra, I. Dinov, P. Thompson and A. W. Toga, Robust surface reconstruction via Laplace-Beltrami eigen-projection and boundary deformation, IEEE Trans. Medical Imaging, 29 (2010), 2009-2022. |
[44] |
Y. Shi, R. Lai and A. Toga, Unified geometry and topology correction for cortical surface reconstruction with intrinsic reeb analysis, MICCAI, (2012), 601-8. |
[45] |
F. Mémoli and G. Sapiro, Distance functions and geodesics on submanifolds of rd and point clouds, SIAM Journal on Applied Mathematics, (2005), 1227-1260. |
[46] |
M. De Berg, O. Cheong, M. Van Kreveld and M. Overmars, "Computational Geometry: Algorithms and Applications," Third edition. Springer-Verlag, Berlin, 2008. |
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