Triangle Mesh Duality: Reconstruction and Smoothing

  • Giuseppe Patanè
  • Michela Spagnuolo
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2768)


Current scan technologies provide huge data sets which have to be processed considering several application constraints. The different steps required to achieve this purpose use a structured approach where fundamental tasks, e.g. surface reconstruction, multi-resolution simplification, smoothing and editing, interact using both the input mesh geometry and topology. This paper is twofold; firstly, we focus our attention on duality considering basic relationships between a 2-manifold triangle mesh \({\mathcal M}\) and its dual representation \({\mathcal M}'\). The achieved combinatorial properties represent the starting point for the reconstruction algorithm which maps \({\mathcal M}'\) into its primal representation \({\mathcal M}\), thus defining their geometric and topological identification. This correspondence is further analyzed in order to study the influence of the information in \({\mathcal M}\) and \({\mathcal M}'\) for the reconstruction process. The second goal of the paper is the definition of the “dual Laplacian smoothing”, which combines the application to the dual mesh \({\mathcal M}'\) of well-known smoothing algorithms with an inverse transformation for reconstructing the regularized triangle mesh. The use of \({\mathcal M}'\) instead of \({\mathcal M}\) exploits a topological mask different from the 1-neighborhood one, related to Laplacian-based algorithms, guaranteeing good results and optimizing storage and computational requirements.


Computational geometry mesh duality Laplacian smoothing filtering 


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  1. 1.
    Aurenhammer, F.: Voronoi diagrams: A survey of a fundamental geometric data structure. ACM Computing Surveys 23(3), 345–405 (1991)CrossRefGoogle Scholar
  2. 2.
    Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. To appear in Israel-Korea Bi-National Conference on Geometric Modeling and Computer Graphics, Tel-Aviv, Fabr. 12-14 (2003) Google Scholar
  3. 3.
    Bern, M.W., Eppstein, D.: Mesh generation and optimal triangulation. In: Hwang, F.K., Du, D.-Z. (eds.) Computing in Euclidean Geometry, pp. 23–90. World Scientific, Singapore (1992)Google Scholar
  4. 4.
    Cignoni, P., Rocchini, C., Scopigno, R.: Metro: Measuring error on simplified surfaces. Computer Graphics Forum 17(2), 167–174 (1998)CrossRefGoogle Scholar
  5. 5.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Berlin (1997)zbMATHGoogle Scholar
  6. 6.
    DeFloriani, L., Magillo, P.: Multiresolution mesh representation: Models and data structures. In: Tutorials on Multiresolution in Geometric Modeling, Munich, pp. 363–417. Springer, Heidelberg (2002)Google Scholar
  7. 7.
    Desbrun, M., Meyer, M., Schröder, P., Barr, A.H.: Implicit fairing of irregular meshes using diffusion and curvature flow. In: Rockwood, A. (ed.) Siggraph 1999. Annual Conference Series, ACM Siggraph, pp. 317–324. Addison Wesley Longman, Los Angeles (1999)CrossRefGoogle Scholar
  8. 8.
    Edelsbrunner, H.: Algorithms in Combinatorial Geometry. EATCS Monographs on Theoretical Computer Science, vol. 10. Springer, Heidelberg (1987)zbMATHGoogle Scholar
  9. 9.
    Golub, G., VanLoan, G.: Matrix Computations, 2nd edn. John Hopkins University Press, Baltimore (1989)zbMATHGoogle Scholar
  10. 10.
    Guibas, L.J., Stolfi, J.: Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. In: Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston, Massachusetts, April 25-27, pp. 221–234 (1983)Google Scholar
  11. 11.
    Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., Stuetzle, W.: Surface reconstruction from unorganized points. Computer Graphics 26(2), 71–78 (1992)CrossRefGoogle Scholar
  12. 12.
    Isenburg, M., Gumhold, S., Gotsman, C.: Connectivity shapes. In: Visualization 2001 Conference Proceedings, pp. 135–142 (2001)Google Scholar
  13. 13.
    Li, J., Kuo, C.: A dual graph approach to 3D triangular mesh compression. In: IEEE 1998 International Conference on Image Processing, Chicago, October 4-7, pp. 891–894 (1998)Google Scholar
  14. 14.
    Mäntylä, M.: An Introduction to Solid Modeling. Computer Science Press, Rockville (1987)Google Scholar
  15. 15.
    Mohar, B.: The laplacian spectrum of graphs. Graph Theory, Combinatorics and Applications, 871–898 (1991)Google Scholar
  16. 16.
    Preparata, F.P., Shamos, M.: Computational Geometry. Springer, New York (1985)Google Scholar
  17. 17.
    Taubin, G.: A signal processing approach to fair surface design. In: Cook, R., (ed.): SIGGRAPH 1995 Conference Proceedings. Annual Conference Series, ACM SIGGRAPH, Addison Wesley, held in Los Angeles, California, August 06-11, pp. 351–358 (1995) Google Scholar
  18. 18.
    Taubin, G.: Dual mesh resampling. In: Proceedings of Pacific Graphics 2001, Tokyo, Japan, pp. 180–188 (October 2001)Google Scholar
  19. 19.
    Taubin, G.: Geometric signal processing on polygonal meshes. In: Eurographics 2000, State of the Art Report (August 2000) Google Scholar
  20. 20.
    Taubin, G., Zhang, T., Golub, G.: Optimal surface smoothing as filter design. In: Buxton, B., Cipolla, R. (eds.) ECCV 1996. LNCS, vol. 1064-1065, pp. 283–292. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  21. 21.
    Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Computer Aided Geometric Design, Special issue on Subdivision Surfaces, 18 (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Giuseppe Patanè
    • 1
  • Michela Spagnuolo
    • 1
  1. 1.Istituto di Matematica Applicata e Tecnologie Informatiche Consiglio Nazionale delle RicercheGenovaItalia

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