A new intrinsic numerical method for PDE on surfaces

References

• Akima , H. 1984 . On estimating partial derivatives for bivariate interpolation of scattered data . Rocky Mountain J. , 14 : 41 – 52 .
   Google Scholar
• Brandman , J. 2008 . A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces . J. Sci. Comput , 37 ( 3 ) : 282 – 315 .
   Google Scholar
• Burger , M. 2008 . Finite element approximation of elliptic partial differential equations on implicit surfaces . Comput. Vis. Sci , 12 ( 3 ) : 87 – 100 .
   Google Scholar
• Bertalmío , M. , Cheng , L. T. and Osher , S. 2001 . Variational problems and partial differential equations on implicit surfaces . J. Comput. Phys , 174 ( 2 ) : 759 – 780 .
   Google Scholar
• Bertalmío , M. , Sapiro , G. , Cheng , L. T. and Osher , S. 2000 . “ A framework for solving surface partial differential equations for computer graphics applications ” . Mathematics Department, UCLA . CAM Report 00-43
   Google Scholar
• Brandman , J. 2008 . A level-set method for computing the eigenvalues of elliptic operators defined on compact hypersurfaces . J. Sci. Comput , 37 ( 3 ) : 282 – 315 .
   Google Scholar
• do Carmo , M. 1976 . “ Differential Geometry of Curves and Surfaces ” . Englewood Cliffs, NJ : Prentice-Hall .
   Google Scholar
• Chen , S.-G. and Wu , J.-Y. 2004 . Estimating normal vectors and curvatures by centroid weights . Comput. Aided Geom. Design , 21 : 447 – 458 .
   Google Scholar
• Chen , S.-G. , Chi , M.-H. and Wu , J.-Y. 2006 . Curvature estimation and curvature flow for digital curves . WSEAS Trans. Comput. , 5 : 804 – 809 .
   Google Scholar
• Chen , S.-G. , Chi , M.-H. and Wu , J.-Y. 2006 . Boundary and interior derivatives estimation for 2D scattered data points . WSEAS Trans. Comput. , 5 : 824 – 829 .
   Google Scholar
• Chen , S.-G. , Chi , M.-H. , Lin , Y.-J. and Wu , J.-Y. 2010 . A weighted curvature flow for Shape deformation . Appl. Math. Comput , 217 ( 5 ) : 2097 – 2117 .
   Google Scholar
• Clarenz , U. , Rumpf , M. and Telea , A. 2004 . Surface processing methods for point sets using finite elements . Comput. Graph , 28 ( 6 ) : 851 – 868 .
   Google Scholar
• Desbrun , M. , Hirani , A. N. and Marsden , J. E. Discrete exterior calculus for variational problems in computer vision and graphics . Proceedings of the 42nd IEEE Conference Decision and Control, Honolulu . pp. 4902 – 4907 .
   Google Scholar
• Dorsey , J. and Hanrahan , P. 2000 . Digital materials and virtual weathering . Sci. Amer , 282 ( 2 ) : 46 – 53 .
   Google Scholar
• Floater , M. S. and Hormann , K. 2005 . “ Surface parameterization: A tutorial and survey ” . In Advances in Multiresolution for Geometric Modelling , Edited by: Dodgson , N. A. , Floater , M. S. and Sabin , M. A. 157 – 186 . Heidelberg : Springer-Verlag .
   Google Scholar
• Greer , J. B. 2006 . An improvement of recent Eulerian method for solving PDEs on general geometries . J. Sci. Comput. , 29 : 321 – 352 .
   Google Scholar
• Jin , M. , Wang , Y. , Yau , S.-T. and Gu , X. “ Optimal global conformal surface parameterization ” . IEEE visualization, Austin, TX, October 2004, pp. 267–274. Available at http://www.math.ucla.edu/ ylwang/doc/vis04.pdf
   Google Scholar
• Hildebrandt , K. , Polthier , K. and Wardetzky , M. 2006 . On the convergence of metric and geometric properties of polyhedral surfaces . Geom. Dedicata , 123 : 89 – 112 .
   Web of Science ®Google Scholar
• Lai , R. , Shi , Y. , Dinov , I. , Chan , T. F. and Toga , A. W. 2009 . Laplace–Beltrami nodal counts: A new signature for 3D shape analysis . Proceedings of the Sixth IEEE international conference on Symposium on Biomedical Imaging: From Nano to Macro . 2009 , Boston, MA. pp. 694 – 697 . Available at http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=5193142
   Google Scholar
• Lai , R. and Chan , T. F. “ A framework for intrinsic image processing on surfaces ” . UCLA CAM Report, 2010
   Google Scholar
• Landsberg , C. and Voigt , A. 2010 . A multigrid finite element method for reaction–diffusion systems on surfaces . Comput. Vis. Sci , 13 ( 4 ) : 177 – 185 .
   Google Scholar
• Levy , H. and Lessman , F. 1992 . “ Finite Difference Equations ” . New York : Dover Publications .
   Google Scholar
• Lui , L. M. , Gu , X. , Chan , T. F. and Yau , S.-T. 2008 . Variational method on Riemann surfaces using conformal parameterization and its applications to image processing . J. Methods Appl. Anal , 15 ( 4 ) : 513 – 538 .
   Google Scholar
• Lui , L. M. , Thiruvenkadam , S. , Wang , Y. , Thompson , P. M. and Chan , T. F. 2010 . Optimized conformal surface registration with shape-based landmark matching . SIAM J. Imaging Sci , 3 ( 1 ) : 52 – 78 .
   Google Scholar
• Mémoli , F. , Sapiro , G. and Thompson , P. 2004 . Implicit brain imaging . NeuroImage , 23 ( S1 ) : 179 – 188 .
   Google Scholar
• Perez , L. L. , Deriche , R. and Sochen , N. A. The Beltrami flow over triangulated manifolds . Proceedings of the ECCV Workshops CVAMIA and MMBIA . pp. 135 – 144 .
   Google Scholar
• Pinkall , U. and Polthier , K. 1993 . Computing discrete minimal surfaces and their conjugates . Experiment. Math , 2 ( 1 ) : 15 – 36 .
   Google Scholar
• Ratz , A. and Voigt , A. 2006 . PDE's on surfaces – a diffuse interface approach . Comm. Math. Sci , 4 ( 3 ) : 575 – 590 .
   Google Scholar
• Shi , Y. , Lai , R. , Krishna , S. , Sicotte , N. , Dinov , I. and Toga , A. W. 2008 . Anisotropic Laplace–Beltrami eigenmaps: Bridging Reeb graphs and skeletons . Computer Vision and Pattern Recognition Workshops, 2008 (CVPRW’08), Anchorage, AK, IEEE Computer Society Conference . June 23–28 2008 . pp. 1 – 7 .
   Google Scholar
• Stam , J. Stable fluids . Proceedings of the 26th Annual conference on Computer graphics and interactive techniques (SIGGRAPH’99) . New York, NY, USA. pp. 121 – 128 .
   Google Scholar
• Stam , J. 2003 . Flows on surfaces of arbitrary topology . ACM Trans. Graph. (TOG) (Proceedings of SIGGRAPH 2003) , 22 ( 3 ) : 724 – 731 .
   Google Scholar
• Taubin , G. A signal processing approach to fair surface design . Proceeding of the 22nd annual conference on computer graphics and interactive techniques (SIGGRAPH’ 95) . New York, NY, USA. pp. 351 – 358 .
   Google Scholar
• Taubin , G. Estimating the tensor of curvature of a surface from a polyhedral approximation . Proceedings of the Fifth International Conference on Computer Vision . Washington, DC, USA. pp. 902 – 907 .
   Google Scholar
• Turing , A. 1952 . The chemical basis of morphogenesis . Phil. Trans. R. Stat. Soc. B , 237 : 37 – 72 .
   Web of Science ®Google Scholar
• Turk , G. 1991 . Generating textures on arbitrary surfaces using reaction–diffusion . Comput. Graph. (SIGGRAPH) , 25 ( 4 ) : 289 – 298 .
   Google Scholar
• Witkin , A. and Kass , M. 1991 . Reaction–diffusion textures . Comput. Graph. (SIGGRAPH) , 25 ( 4 ) : 299 – 308 .
   Google Scholar
• Wu , J.-Y. , Chi , M.-H. and Chen , S.-G. 2010 . A local tangential lifting differential method for triangular meshes . Math. Comput. Simulat. , 80 : 2386 – 2402 .
   Google Scholar
• Xu , G. 2004 . Discrete Laplace–Beltrami operators and their convergence . Comput. Aided Geom. Design , 21 : 767 – 784 .
   Google Scholar
• Xu , G. , Pan , Q. and Bajaj , C. 2003 . “ Discrete surface modeling using geometric flows ” . TICAM Report 03–38, August 2003, Institute for Computational Engineering and Sciences, The University of Texas at Austin
   Google Scholar
• Zhang , H. Discrete combinatorial Laplacian operators for digital geometry processing . Proceedings of the SIAM Conference on Geometric Design and Computing . Seattle. pp. 575 – 592 .
   Google Scholar