Reconstructing Multiresolution Mesh for Web Visualization Based on PDE Resampling

  • Ming-Yong Pang
  • Yun Sheng
  • Alexei Sourin
  • Gabriela González Castro
  • Hassan Ugail
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6670)


Various Partial Differential Equations (PDEs) have been used in computer graphics for approximating surfaces of geometric shapes by finding solutions to PDEs, subject to suitable boundary conditions. The PDE boundary conditions are defined as 3D curves on surfaces of the shapes. We propose how to automatically derive these curves from the surface of the original polygon mesh. Analytic solutions to the PDEs used throughout this work are fully determined by finding a set of coefficients associated with parametric functions according to the particular set of boundary conditions. When large polygon meshes are used, the PDE coefficients require an order of magnitude smaller space compared to the original polygon data and can be interactively rendered with different levels of detail. It allows for an efficient exchange of the PDE shapes in 3D Cyberworlds and their web visualization. In this paper we analyze and formulate the requirements for extracting suitable boundary conditions, describe the algorithm for the automatic deriving of the boundary curves, and present its implementation as a part of the function-based extension of VRML and X3D.


partial differential equations surface modeling surface approximation 3D reconstruction web visualization 


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  1. 1.
    Sheng, Y., Sourin, A., Gonzalez Castro, G., Ugail, H.: A PDE Method for Patchwise Approximation of Large Polygon Meshes. The Visual Computer 26(6-8), 975–984 (2010)CrossRefGoogle Scholar
  2. 2.
    Ugail, H., Sourin, A.: Partial Differential Equations for Function based Geometry Modelling within Visual Cyberworlds. In: 2008 Int Conf. on Cyberworlds, pp. 224–231. IEEE CS, Los Alamitos (2008)CrossRefGoogle Scholar
  3. 3.
    Ugail, H., González Castro, G., Sourin, A., Sourina, O.: Towards a Definition of Virtual Objects with Partial Differential Equations. In: 2009 Int. Conf. on Cyberworlds, Bradford, September 7-11, pp. 138–145 (2009)Google Scholar
  4. 4.
    Bloor, M., Wilson, M.: Generating blend surface using partial differential equations. Computer Aided Design 21(3), 165–171 (1989)CrossRefzbMATHGoogle Scholar
  5. 5.
    Schroeder, W., Zarge, J., Lorensen, W.: Decimation of triangle meshes. Computer Graphics 26(2), 65–70 (1992)CrossRefGoogle Scholar
  6. 6.
    Hoppe, H., DeRose, T., Duchamp, T., et al.: Mesh optimization. In: SIGGRAPH 1993, pp. 19–26 (1993)Google Scholar
  7. 7.
    Rossignac, J., Borrel, P.: Multi-resolution 3D approximations for rendering complex scenes. In: Falcidieno, B., Kunii, T. (eds.) Modeling in Computer Graphics: Methods Application, pp. 455–465 (1993)Google Scholar
  8. 8.
    Turk, G.: Re-tiling polygonal surfaces. In: SIGGRAPH 1992, pp. 55–64 (1992)Google Scholar
  9. 9.
    Lounsbery, M.: Multiresolution analysis for surfaces of arbitrary topological type [PhD dissertation], Department of Computer Science and Engineering, University of Washington (1994)Google Scholar
  10. 10.
    Garland, M., Heckbert, P.S.: Surface simplification using quadric error metrics. In: SIGGRAPH 1997, pp. 209–216 (1997)Google Scholar
  11. 11.
    Melia, S.: A simple, fast, effective polygon reduction algorithm. Game Developer 10, 44–49 (1998)Google Scholar
  12. 12.
    Levy, B.: Parameterization and deformation analysis on a manifold. Technical report, Alice (2007),
  13. 13.
    Floater, M.: Mean value coordinates. Computer Aided Geometric Design 20(1), 19–27 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Tutte, W.: How to draw a graph. Proc of the London Mathematical Society, 743–768 (1963)Google Scholar
  15. 15.
    Chen, G.L., Pang, M.Y., Wang, J.D.: Calculating shortest path on edge-based data structure of graph. In: 2nd International Workshop on Digital Media and its Application in Museum and Heritage, pp. 416–421 (2007)Google Scholar
  16. 16.
    Chen, J., Han, Y.: Shortest paths on a polyhedron, Part I: Computing shortest paths. International Journal of Computer Geometry Application 6, 127–144 (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Kaneva, B., O’Rourke, J.: An implementation of chen & han’s shortest paths algorithm. In: 21st Canadian Conference on Computer Geometry, pp. 139–146 (2000)Google Scholar
  18. 18.
    Kapoor, S.: Efficient computation of geodesic shortest paths. In: 31st ACM Symposium on Theory of Computing, pp. 770–779 (1999)Google Scholar
  19. 19.
    Lanthier, M., Maheshwari, A., Sack, J.R.: Approximating weighted shortest paths on polyhedral surfaces. In: 13th Annual Symposium on Computational Geometry, pp. 274–283 (1997)Google Scholar
  20. 20.
    Martínez, D., Velho, L., Carvalho, P.: Geodesic Paths on Triangular Meshes. In: SIBGRAPI/SIACG, pp. 210–217 (2004)Google Scholar
  21. 21.
    Mitchell, J.: Geometric shortest paths and network optimization. In: Sack, J., Urrutia, J. (eds.) Handbook of Computational Geometry, Elsevier Science, pp. 633–702 (2000)Google Scholar
  22. 22.
    Surazhsky, V., Surazhsky, T., Kirsanov, D., et al.: Fast exact and approximate geodesics on meshes. ACM Transactions on Graphics 24(3), 553–560 (2005)CrossRefGoogle Scholar
  23. 23.
    Sourin, A., Wei, L.: Visual Immersive Haptic Mathematics. Virtual Reality 13(4), 221–234 (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Ming-Yong Pang
    • 1
  • Yun Sheng
    • 2
  • Alexei Sourin
    • 2
  • Gabriela González Castro
    • 3
  • Hassan Ugail
    • 3
  1. 1.Department of Educational TechnologyNanjing Normal UniversityChina
  2. 2.School of Computer EngineeringNanyang Technological UniversitySingapore
  3. 3.Centre for Visual ComputingUniversity of BradfordUK

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