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Transactions on Computational Science VI pp 342–361Cite as

A Discrete Approach to Multiresolution Curves and Surfaces

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Part of the book series: Lecture Notes in Computer Science ((TCOMPUTATSCIE,volume 5730))

Abstract

Subdivision surfaces have been widely adopted in modeling in part because they introduce a separation between the surface and the underlying basis functions. This separation allows for simple general-topology subdivision schemes. Multiresolution representations based on subdivision, however, incongruently return to continuous functional spaces in their construction and analysis. In this paper, we propose a discrete multiresolution framework applicable to many subdivision schemes and based only on the subdivision rules. Noting that a compact representation can only afford to store a subset of the detail information, our construction enforces a constraint between adjacent detail terms. In this way, all detail information is recoverable for reconstruction, and a decomposition approach is implied by the constraint. Our framework is demonstrated with case studies in Dyn-Levin-Gregory curves and Catmull-Clark surfaces, each of which our method produces results on par with earlier methods. It is further shown that our construction can be interpreted as biorthogonal wavelet systems.

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References

  1. Stollnitz, E., DeRose, T., Salesin, D.: Wavelets for Computer Graphics: Theory and Applications, pp. 152–159. Morgan Kaufmann Publishers, Inc., San Francisco (1996)

    Google Scholar 

  2. Warren, J., Weimer, H.: Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann, San Francisco (2001)

    Google Scholar 

  3. Finkelstein, A., Salesin, D.: Multiresolution curves. In: Proc. of SIGGRAPH 1994, pp. 261–268. ACM Press, New York (1994)

    Chapter  Google Scholar 

  4. Samavati, F., Mahdavi-Amiri, N., Bartels, R.: Multiresolution surfaces having arbitrary topologies by a reverse doo subdivision method. Computer Graphics Forum 21(2), 121–136 (2002)

    Article  Google Scholar 

  5. Bertram, M.: Biorthogonal Loop-Subdivision Wavelets. Computing 72(1-2), 29–39 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Li, D., Qin, K., Sun, H.: Unlifted loop subdivision wavelets. In: 12th Pacific Conference on Computer Graphics and Applications (2004)

    Google Scholar 

  7. Olsen, L., Samavati, F., Bartels, R.: Multiresolution for curves and surfaces based on constraining wavelets. Computers and Graphics (2007)

    Google Scholar 

  8. Bertram, M., Duchaineau, M., Hamann, B., Joy, K.: Generalized b-spline subdivision-surface wavelets for geometry compression. IEEE Transactions on Visualization and Computer Graphics 10, 326–338 (2004)

    Article  Google Scholar 

  9. Wang, H., Qin, K.H., Tang, K.: Efficient wavelet construction with catmull-clark subdivision. The Visual Computer 22, 874–884 (2006)

    Article  Google Scholar 

  10. Leibniz, G.: The Labyrinth of the Continuum: Writings on the Continuum Problem, pp. 1672–1686. Yale University Press, New Haven (2001)

    Google Scholar 

  11. Stollnitz, E., DeRose, T., Salesin, D.: Wavelets for computer graphics: A primer, part 1. IEEE Computer Graphics and Applications 15(3), 76–84 (1995)

    Article  Google Scholar 

  12. Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. In: Proc. of SIGGRAPH 1995, pp. 173–182 (1995)

    Google Scholar 

  13. Lounsbery, M., DeRose, T.D., Warren, J.: Multiresolution analysis for surfaces of arbitrary topological type. ACM Trans. Graph. 16(1), 34–73 (1994)

    Article  Google Scholar 

  14. Samavati, F., Bartels, R.: Multiresolution curve and surface representation by reversing subdivision rules. Computer Graphics Forum 18(2), 97–120 (1999)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Lee, A., Sweldens, W., Schröder, P., Cowsar, L., Dobkin, D.: Maps: multiresolution adaptive parameterization of surfaces. In: Proc. of SIGGRAPH 1998, pp. 95–104. ACM Press, New York (1998)

    Chapter  Google Scholar 

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

    Google Scholar 

  18. Boier-Martin, I., Rushmeier, H., Jin, J.: Parameterization of triangle meshes over quadrilateral domains. In: Proc. of ACM Symposium on Geometry Processing (SGP 2004), pp. 193–203 (2004)

    Google Scholar 

  19. Valette, S., Prost, R.: Wavelet-based multiresolution analysis of irregular surface meshes. IEEE Transactions on Visualization and Computer Graphics 10(2), 113–122 (2004)

    Article  Google Scholar 

  20. Zorin, D., Schröder, P., Sweldens, W.: Interactive multiresolution mesh editing. In: Proc. of SIGGRAPH 1997, pp. 259–268. ACM Press, New York (1997)

    Chapter  Google Scholar 

  21. Gioia, P.: Reducing the number of wavelet coefficients by geometric partitioning. Computational Geometry: Theory and applications 14(1-3), 25–48 (1999)

    MATH  MathSciNet  Google Scholar 

  22. Botsch, M., Kobbelt, L.: Multiresolution surface representation based on displacement volumes. Computer Graphics Forum 22, 483–491 (2003)

    Article  Google Scholar 

  23. Sorkine, O., Cohen-Or, D., Lipman, Y., Alexa, M., Rössl, C., Seidel, H.P.: Laplacian surface editing. In: Proc. of ACM Symposium on Geometry Processing (SGP 2004), pp. 175–184 (2004)

    Google Scholar 

  24. Ju, T., Schaefer, S., Warren, J.: Mean value coordinates for closed triangular meshes. In: Proc. of SIGGRAPH 2005, pp. 561–566 (2005)

    Google Scholar 

  25. Botsch, M., Pauly, M., Gross, M., Kobbelt, L.: Primo: Coupled prisms for intuitive surface modeling. In: Proc. of ACM Symposium on Geometry Processing (SGP 2006), pp. 11–20 (2006)

    Google Scholar 

  26. Chaikin, G.: An algorithm for high speed curve generation. Computer Graphics and Image Processing 3(4), 346–349 (1974)

    Article  Google Scholar 

  27. Doo, D., Sabin, M.: Behaviour of recursive subdivision surfaces near extraordinary points. Computer-Aided Design 10(6), 356–260 (1978)

    Google Scholar 

  28. Dyn, N., Levine, D., Gregory, J.: A 4-point interpolatory subdivision scheme for curve design. CAGD 4, 257–268 (1987)

    MATH  Google Scholar 

  29. Bartels, R., Samavati, F.: Reversing subdivision rules: Local linear conditions and observations on inner products. Journal of Computational and Applied Mathematics 119, 29–67 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  30. National Geophysical Data Center: Coastline extractor (2007), http://rimmer.ngdc.noaa.gov/mgg/coast/getcoast.html

  31. Catmull, E., Clark, J.: Recursively generated b-spline surfaces on arbitrary topological surfaces. Computer-Aided Design 10(6), 350–355 (1978)

    Article  Google Scholar 

  32. Zorin, D., Schröder, P.: Subdivision for modeling and animation. In: SIGGRAPH 2000 Course Notes. ACM Press, New York (2000)

    Google Scholar 

  33. DeRose, T., Kass, M., Truong, T.: Subdivision surfaces in character animation. In: Proc. of SIGGRAPH 1998, pp. 85–94 (1998)

    Google Scholar 

  34. United States Geological Survey: Seamless data distribution system (2004), http://seamless.usgs.gov/website/seamless

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Author information

Authors and Affiliations

  1. University of Calgary,  

    Luke Olsen & Faramarz Samavati

Authors
  1. Luke Olsen
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  2. Faramarz Samavati
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Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Calgary, 2500 University Drive N.W, T2N 1N4, Calgary, AB, Canada

    Marina L. Gavrilova

  2. OptimaNumerics Ltd., Cathedral House, 23-31 Waring Street, BT1 2DX, Belfast, UK

    C. J. Kenneth Tan

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Olsen, L., Samavati, F. (2009). A Discrete Approach to Multiresolution Curves and Surfaces. In: Gavrilova, M.L., Tan, C.J.K. (eds) Transactions on Computational Science VI. Lecture Notes in Computer Science, vol 5730. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10649-1_20

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  • DOI: https://doi.org/10.1007/978-3-642-10649-1_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-10648-4

  • Online ISBN: 978-3-642-10649-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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