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Biorthogonal wavelet construction for hybrid quad/triangle meshes

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Abstract

Ever since its introduction by Stam and Loop, the quad/triangle subdivision scheme, which is a generalization of the well-known Catmull–Clark subdivision and Loop subdivision, has attracted a great deal of interest due to its flexibility of allowing both quads and triangles in the same model. In this paper, we present a novel biorthogonal wavelet—constructed through the lifting scheme—that accommodates the quad/triangle subdivision. The introduced wavelet smoothly unifies the Catmull–Clark subdivision wavelet (for quadrilateral meshes) and the Loop subdivision wavelet (for triangular meshes) in a single framework. It can be used to flexibly and efficiently process any complicated semi-regular hybrid meshes containing both quadrilateral and triangular regions. Because the analysis and synthesis algorithms of the wavelet are composed of only local lifting operations allowing fully in-place calculations, they can be performed in linear time. The experiments demonstrate sufficient stability and fine fitting quality of the presented wavelet, which are similar to those of the Catmull–Clark subdivision wavelet and the Loop subdivision wavelet. The wavelet analysis can be used in various applications, such as shape approximation, progressive transmission, data compression and multiresolution edit of complex models.

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References

  1. Bajaj, C., Schaefer, S., Warren, J., Xu, G.: A subdivision scheme for hexahedral meshes. Vis. Comput. 18(5–6), 343–356 (2002)

    Article  Google Scholar 

  2. Ball, A., Storry, D.: A matrix approach to the analysis of recursively generated B-spline surfaces. Comput. Aided Des. 18, 437–442 (1986)

    Article  Google Scholar 

  3. Beets, K., Claes, J., Van Reeth, F.: Optimizing Mesh Construction for Quad/Triangle Schemes. Lecture Notes in Computer Science, vol. 4035, pp. 711–718. Springer, Berlin (2006)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Bertram, M., Duchaineau, M., Hamann, B., et al.: Bicubic subdivision-surface wavelets for large-scale isosurface representation and visualization. In: Proceedings Visualization’2000, pp. 389–396 (2000)

  6. Bertram, M., Duchaineau, M., Hamann, B., et al.: Generalized B-spline subdivision-surface wavelets for geometry compression. IEEE Trans. Vis. Comput. Graph. 10(3), 326–338 (2004)

    Article  Google Scholar 

  7. Bonneau, G.: Multiresolution analysis on irregular surface meshes. IEEE Trans. Vis. Comput. Graph. 4(4), 365–378 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  9. Doo, D., Sabin, M.: Behavior of recursive division surfaces near extraordinary points. Comput. Aided Des. 10, 356–360 (1978)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

  11. Eck, M., DeRose, T., Duchamp, T., et al.: Multiresolution analysis of arbitrary meshes. In: Proceedings of SIGGRAPH’1995, pp. 173–182 (1995)

  12. Khodakovsky, A., Schröder, P., Sweldens, W.: Progressive geometry compression. In: Proceedings of SIGGRAPH’2000, New Orleans, Louisiana, USA, pp. 271–278 (2000)

  13. Kobbelt, L.: Interpolatory subdivision on Open quadrilateral nets with arbitrary topology. Comput. Graph. Forum 15(3), 409–420 (1996) (Proceedings of Eurographics’96)

    Article  Google Scholar 

  14. Kobbelt, L.: Root 3-subdivision. In: Proceedings of SIGGRAPH’2000, New Orleans, Louisiana, USA, pp. 103–112 (2000)

  15. Kobbelt, L., Vorsatz, J., Labsik, U., et al.: A shrink wrapping approach to remeshing polygonal surfaces. Comput. Graph. Forum 18, C119–C130 (1999), Eurographics’99 issue

    Article  Google Scholar 

  16. Labsik, U., Greiner, G.: Interpolatory root 3-subdivision. Comput. Graph. Forum 19(3), 131–138 (2000)

    Article  Google Scholar 

  17. Lee, A., Sweldens, W., Schröder, P., et al.: MAPS: multiresolution adaptive parameterization of surfaces. In: Proceedings of SIGGRAPH’1998, Orlando, Florida, pp. 95–104 (1998)

  18. Li, D., Qin, K., Sun, H.: Unlifted Loop subdivision wavelets. In: Proceedings of Pacific Graphics’2004, Seoul, Korea, pp. 25–33 (2004)

  19. Li, G., Ma, W., Bao, H.: Root 2-subdivision for quadrilateral meshes. Vis. Comput. 20(2–3), 180–198 (2004)

    Article  Google Scholar 

  20. Litke, N., Levin, A., Schröder, P.: Fitting subdivision surfaces. In: Proceedings of IEEE Visualization’2001, pp. 319–324 (2001)

  21. Loop, C.: Smooth subdivision surfaces based on triangles. M.S. Thesis, Department of Mathematics, University of Utah (1987)

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  24. Samavati, F., Bartels, R.: Multiresolution curve and surface editing: reversing subdivision rules by least-squares data fitting. Comput. Graph. Forum 18(2), 97–119 (1999)

    Article  Google Scholar 

  25. Samavati, F., Mahdavi-Amiri, N., Bartels, R.: Multiresolution surfaces having arbitrary topologies by a reverse Doo subdivision method. Comput. Graph. Forum 21(2), 121–136 (2002)

    Article  Google Scholar 

  26. Schaefer, S., Warren, J.: On C2 triangle/quad subdivision. ACM Trans. Graph. 24(1), 28–36 (2005)

    Article  Google Scholar 

  27. Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. Comput. Graph. 161–172 (1995) (Proceedings of SIGGRAPH’95)

  28. Sederberg, T., Zheng, J., Sewell, D., et al.: Non-uniform recursive subdivision surfaces. Comput. Graph. 387–394 (1998) (Proceedings of SIGGRAPH’98)

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

    Article  Google Scholar 

  30. Sweldens, W.: The lifting scheme: a custom-design construction of biorthogonal wavelets. Appl. Comput. Harmon. Anal. 3(2), 186–200 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sweldens, W.: The lifting scheme: a construction of second generation wavelets. SIAM J. Math. Anal. 29(2), 511–546 (1997)

    Article  MathSciNet  Google Scholar 

  32. Valette, S., Prost, R.: Wavelet-based multiresolution analysis of irregular surface meshes. IEEE Trans. Vis. Comput. Graph. 10(2), 113–122 (2004)

    Article  Google Scholar 

  33. Valette, S., Prost, R.: Wavelet-based progressive compression scheme for triangle meshes: Wavemesh. IEEE Trans. Vis. Comput. Graph. 10(2), 123–129 (2004)

    Article  Google Scholar 

  34. Velho, L., Zorin, D.: 4–8 subdivision. Comput. Aided Geom. Des. 18(5), 397–427 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Wang, H., Qin, K., Tang, K.: Efficient wavelet construction with Catmull–Clark subdivision. Vis. Comput. 22(9–11), 874–884 (2006)

    Article  Google Scholar 

  36. Wang, H., Qin, K., Sun, H.: Root 3-subdivision-based biorthogonal wavelets. IEEE Trans. Vis. Comput. Graph. 13(5), 914–925 (2007)

    Article  Google Scholar 

  37. Wang, H., Tang, K., Qin, K.: Biorthogonal wavelets based on gradual subdivision of quadrilateral meshes. Comput. Aided Geom. Des. (2007), doi:10.1016/j.cagd.2007.11.002

    Google Scholar 

  38. Wu, J., Amaratunga, K.: Wavelet triangulated irregular networks. Int. J. Geograph. Inf. Sci. 17(3), 273–289 (2003)

    Article  Google Scholar 

  39. Zorin, D., Schröder, P.: A unified framework for primal/dual quadrilateral subdivision schemes. Comput. Aided Geom. Des. 18(5), 429–454 (2001)

    Article  MATH  Google Scholar 

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

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Authors and Affiliations

  1. Dept. of Mechanical Engineering, Hong Kong University of Science & Technology, Hong Kong, China

    Huawei Wang & Kai Tang

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  1. Huawei Wang
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  2. Kai Tang
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Correspondence to Kai Tang.

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Wang, H., Tang, K. Biorthogonal wavelet construction for hybrid quad/triangle meshes. Vis Comput 25, 349–366 (2009). https://doi.org/10.1007/s00371-008-0300-6

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  • DOI: https://doi.org/10.1007/s00371-008-0300-6

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