Skip to main content
SpringerLink
Log in

Extension of half-edges for the representation of multiresolution subdivision surfaces

  • Original Article
  • Published:
The Visual Computer Aims and scope Submit manuscript

Abstract

We address in this paper the problem of the data structures used for the representation and the manipulation of multiresolution subdivision surfaces. The classically used data structures are based on quadtrees, straightforwardly derived from the nested hierarchy of faces generated by the subdivision schemes. Nevertheless, these structures have some drawbacks: specificity to the kind of mesh (triangle or quad); the time complexity of neighborhood queries is not optimal; topological cracks are created in the mesh in the adaptive subdivision case.

We present in this paper a new topological model for encoding multiresolution subdivision surfaces. This model is an extension to the well-known half-edge data structure. It allows instant and efficient navigation at any resolution level of the mesh. Its generality allows the support of many subdivision schemes including primal and dual schemes. Moreover, subdividing the mesh adaptively does not create topological cracks in the mesh. The extension proposed here is formalized in the combinatorial maps framework. This allows us to give a very general formulation of our extension.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bertram, M.: Biorthogonal loop-subdivision wavelets. Computing 72(1–2), 29–39 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biermann, H., Kristjansson, D., Zorin, D.: Approximate boolean operations on free-form solids. In: Proceedings of SIGGRAPH’01, pp. 185–194. ACM Press, Los Angeles (2001)

    Google Scholar 

  3. Biermann, H., Levin, A., Zorin, D.: Piecewise smooth subdivision surfaces with normal control. In: Proceedings of SIGGRAPH’00, pp. 113–120. ACM Press, New Orleans (2000)

    Google Scholar 

  4. Biermann, H., Martin, I., Bernardini, F., Zorin, D.: Cut-and-paste editing of multiresolution surfaces. ACM Trans. Graph. 21(3), 312–321 (2002)

    Article  Google Scholar 

  5. Biermann, H., Martin, I., Zorin, D., Bernardini, F.: Sharp features on multiresolution subdivision surfaces. Graph. Models 64(2), 61–77 (2002)

    Article  MATH  Google Scholar 

  6. Brun, L., Kropatsch, W.: Combinatorial pyramids. In: IEEE International Conference on Image Processing (ICIP), vol. 2, pp. 33–36. IEEE, Barcelona (2003)

    Google Scholar 

  7. Brun, L., Kropatsch, W.: Contains and inside relationships within combinatorial pyramids. Pattern Recogn. 39(4), 515–526 (2006)

    Article  MATH  Google Scholar 

  8. Catmull, E., Clark, J.: Recursively generated B-spline surfaces on arbitrary topological meshes. Comput. Aided Des. 10(6), 350–355 (1978)

    Article  Google Scholar 

  9. Cazier, D., Dufourd, J.: A formal specification of geometric refinements. Vis. Comput. 15, 279–301 (1999)

    Article  Google Scholar 

  10. DeFloriani, L., Kobbelt, L., Puppo, E.: A survey on data structures for level-of-detail models. In: Dodgsan, N.A., Floater, M.S., Sabin, M.A. (eds.) Advances in Multiresolution for Geometric Modelling, Series in Mathematics and Visualization, pp. 49–74. Springer (2004)

  11. Doo, D.: A subdivision algorithm for smoothing down irregularly shaped polygons. In: Proceedings of Interactive Techniques in Computer Aided Design, pp. 157–165. Bologna (1978)

  12. Doo, D., Sabin, M.: Analysis of the behaviour of recursive division surfaces near extraordinary points. Comput. Aided Des. 10(6), 356–360 (1978)

    Article  Google Scholar 

  13. Dufourd, J.F.: Algebras and formal specifications in geometric modeling. Vis. Comput. 13(3), 131–154 (1997)

    Article  Google Scholar 

  14. Dyn, N., Levin, D., Gregory, J.A.: A butterfly subdivision scheme for surface interpolation with tension control. ACM Trans. Graph. 9(2), 160–169 (1990)

    Article  MATH  Google Scholar 

  15. Edmonds, J.: A combinatorial representation for polyhedral surfaces. In: Notices of the American Mathematical Society, vol. 7 (1960)

  16. Grasset-Simon, C., Damiand, G., Lienhardt, P.: nD generalized map pyramids: definition, representations and basic operations. Pattern Recogn. 39(4), 527–538 (2006)

    Article  MATH  Google Scholar 

  17. Herzen, B.V., Barr, A.: Accurate triangulations of deformed, intersecting surfaces. Comput. Graph. 21(4), 103–110 (1987)

    Article  Google Scholar 

  18. Kobbelt, L.: \(\sqrt{3}\) subdivision. In: Proceedings of SIGGRAPH’00, pp. 103–112. ACM Press, New Orleans (2000)

    Google Scholar 

  19. Lee, A., Sweldens, W., Schröder, P., Cowsar, L., Dobkin, D.: MAPS: Multiresolution adaptive parameterization of surfaces. In: Proceedings of SIGGRAPH’98, pp. 95–104. ACM Press, Orlando (1998)

    Google Scholar 

  20. Lee, M., Samet, H.: Navigating through triangle meshes implemented as linear quadtrees. ACM Trans. Graph. 19(2), 79–121 (2000)

    Article  Google Scholar 

  21. Lienhardt, P.: Subdivision of n-dimensional spaces and n-dimensional generalized maps. In: 5th ACM Conference on Computational Geometry, pp. 228–236. ACM, Saarbrücken, Germany (1989)

    Google Scholar 

  22. Lienhardt, P.: Topological models for boundary representation: a comparison with n-dimensional generalized maps. Comput. Aided Des. 23(1), 59–82 (1991)

    Article  MATH  Google Scholar 

  23. Loop, C.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah (1987)

  24. Lounsberry, M., DeRose, T., Warren, J.: Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graph. 16(1), 34–73 (1997)

    Article  Google Scholar 

  25. Pajarola, R.: Large scale terrain visualization using the restricted quadtree triangulation. In: Proceedings of IEEE Visualization ’98, pp. 19–26. IEEE, Research Triangle Park, NC, USA (1998)

    Google Scholar 

  26. Pajarola, R., DeCoro, C.: Efficient implementation of real-time view-dependent multiresolution meshing. IEEE Trans. Vis. Comput. Graph. 10(3), 353–368 (2004)

    Article  Google Scholar 

  27. Peters, J., Reif, U.: The simplest subdivision scheme for smoothing polyhedra. ACM Trans. Graph. 16(4), 420–431 (1997)

    Article  Google Scholar 

  28. Peters, J., Shiue, L.J.: Combining 4- and 3-direction subdivision. ACM Trans. Graph. 23(4), 980–1003 (2004)

    Article  Google Scholar 

  29. Schrack, G.: Finding neighbors of equal size in linear quadtrees and octrees in constant time. CVGIP: Image Understanding 55(3), 221–230 (1992)

    Article  MATH  Google Scholar 

  30. Shiue, L.J., Peters, J.: A mesh refinement library based on generic design. In: ACM-SE 43: Proceedings of the 43rd Annual Southeast Regional Conference, pp. 104–108. Kennesaw, GA, USA (2005)

    Chapter  Google Scholar 

  31. Shiue, L.J., Peters, J.: A pattern-based data structure for manipulating meshes with regular regions. In: GI ’05: Proceedings of the 2005 Graphics Interface Conference. ACM International Conference Proceedings Series, vol. 112, pp. 153–160. Victoria, BC, Canada (2005)

  32. Stam, J., Loop, C.: Quad/triangle subdivision. Comput. Graph. Forum 22(1), 79–85 (2003)

    Article  Google Scholar 

  33. Taubin, G.: A signal processing approach to fair surface design. In: Proceedings of SIGGRAPH’95, pp. 351–358. ACM Press (1995)

  34. Taubin, G.: Detecting and reconstructing subdivision connectivity. Vis. Comput. 18(5–6), 357–367 (2002)

    Article  Google Scholar 

  35. Vince, A.: Combinatorial maps. J. Comb. Theory 34(1), 1–21 (1983)

    MATH  MathSciNet  Google Scholar 

  36. Weiler, K.: Edge-based data structures for modeling in curved-surface environments. IEEE Comput. Graph. Appl. 5(1), 21–40 (1985)

    Article  Google Scholar 

  37. Zorin, D.: Modeling with multiresolution subdivision surfaces. SIGGRAPH ’06 Course Notes. ACM Press, Boston (2006)

    Google Scholar 

  38. Zorin, D., Schröder, P., DeRose, T., Kobbelt, L., Levin, A., Sweldens, W.: Subdivision for modeling and animation. SIGGRAPH’00 Course Notes. ACM Press, New Orleans (2000)

    Google Scholar 

  39. Zorin, D., Schröder, P., Sweldens, W.: Interpolating subdivision for meshes with arbitrary topology. In: Proceedings of SIGGRAPH’96, pp. 189–192. ACM Press, New Orleans (1996)

    Google Scholar 

  40. Zorin, D., Schröder, P., Sweldens, W.: Interactive multiresolution mesh editing. In: Proceedings of SIGGRAPH’97, pp. 259–268. ACM Press, Los Angeles (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LSIIT, UMR CNRS 7005, Université Louis Pasteur, Strasbourg, France

    Pierre Kraemer, David Cazier & Dominique Bechmann

Authors
  1. Pierre Kraemer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David Cazier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dominique Bechmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierre Kraemer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kraemer, P., Cazier, D. & Bechmann, D. Extension of half-edges for the representation of multiresolution subdivision surfaces. Vis Comput 25, 149–163 (2009). https://doi.org/10.1007/s00371-008-0211-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00371-008-0211-6

Keywords

Search

Navigation