Modeling Wavelet Coefficients for Wavelet Subdivision Transforms of 3D Meshes

  • Shahid M. Satti
  • Leon Denis
  • Adrian Munteanu
  • Jan Cornelis
  • Peter Schelkens
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6474)


In this paper, a Laplacian Mixture (LM) model is proposed to accurately approximate the observed histogram of the wavelet coefficients produced by lifting-based subdivision wavelet transforms. On average, the proposed mixture model gives better histogram fitting for both normal and non-normal meshes compared to the traditionally used Generalized Gaussian (GG) distributions. Exact closed-form expressions for the rate and the distortion of the LM probability density function quantized using generic embedded deadzone scalar quantizer (EDSQ) are derived, without making high-rate assumptions. Experimental evaluations carried out on a set of 3D meshes reveals that, on average, the D-R function for the LM model closely follows and gives a better indication of the experimental D-R compared to the D-R curve of the competing GG model. Optimal embedded quantization for the proposed LM model is experimentally determined. In this sense, it is concluded that the classical Successive Approximation Quantization (SAQ) is an acceptable, but in general, not an optimal embedded quantization solution in wavelet-based scalable coding of 3D meshes.


Wavelet Coefficient Subdivision Scheme Standard Deviation Ratio Wavelet Subbands Mesh Code 
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  1. 1.
    Rossignac, J.: Edgebreaker: Connectivity compression for triangle meshes. IEEE Transaction on Visualization and Computer Graphics 5(1), 47–61Google Scholar
  2. 2.
    Touma, C., Gotsman, C.: Triangular mesh compression. In: Proceedings of Graphics Interface, pp. 26–34 (1998)Google Scholar
  3. 3.
    Taubin, G., Rossignac, J.: Geometric compression through topological surgery. ACM Transactions on Graphics 17(2), 84–115 (1998)CrossRefGoogle Scholar
  4. 4.
    Pajarola, R., Rossignac, J.: Compressed progressive meshes. IEEE Transactions on Visualization and Computer Graphics 6(1-3), 79–93 (2000)CrossRefGoogle Scholar
  5. 5.
    Taubin, G., Guéziec, A., Horn, W., Lazarus, F.: Progressive forest-split compression. In: Proceedings of SIGGRAPH, pp. 123–132 (1998)Google Scholar
  6. 6.
    Li, J., Kuo, C.-C.J.: Progressive coding of 3-D graphic models. Proceedings of the IEEE 86(6), 1052–1063 (1998)CrossRefGoogle Scholar
  7. 7.
    Eck, M., Hoppe, H.: Automatic reconstruction of B-spline surfaces of arbitrary topological type. In: Proceedings of SIGGRAPH, pp. 325–334 (1996)Google Scholar
  8. 8.
    Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multiresolution analysis of arbitrary meshes. In: Proceedings of SIGGRAPH, pp. 173–182 (1995)Google Scholar
  9. 9.
    Lee, A.W.F., Sweldens, W., Schröder, P., Cowsar, L., Dobkin, D.: Multiresolution adaptive parameterization of surfaces. In: Proceedings of SIGGRAPH, pp. 95–104 (1998)Google Scholar
  10. 10.
    Khodakovsky, A., Schroder, P., Sweldens, W.: Progressive geometry compression. In: Proceedings of SIGGRAPH, pp. 271–278 (2000)Google Scholar
  11. 11.
    Shapiro, J.M.: Embedded image coding using zerotrees of wavelet coefficients. IEEE Transactions on Signal Processing 41(12), 3445–3462 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Satti, S.M., Denis, L., Munteanu, A., Schelkens, P., Cornelis, J.: Estimation of interband and intraband statistical dependencies in wavelet-based decomposition of meshes. In: Proceedings of IS&T/SPIE, San Jose, California, pp. 72480A–10A (2009)Google Scholar
  13. 13.
    Denis, L., Satti, S.M., Munteanu, A., Schelkens, P., Cornelis, J.: Fully scalable intraband coding of wavelet-decomposed 3D meshes. In: Proceedings of Digital Signal Processing (DSP), Santorini (2009)Google Scholar
  14. 14.
    Denis, L., Satti, S.M., Munteanu, A., Schelkens, P., Cornelis, J.: Context-conditioned composite coding of 3D meshes based on wavelets on surfaces. In: Proceedings of IEEE International Conference on Image Processing (ICIP), Cairo, pp. 3509–3512 (2009)Google Scholar
  15. 15.
    Payan, F., Antonini, M.: An efficient bit allocation for compressing normal meshes with an error-driven quantization. Computer Aided Geometric Design 22(5), 466–486 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Mallat, S.: A theory for multiresolution signal decomposition: the wavelet representation. IEEE Transactions on Pattern Analysis and Machine Intelligence 11(7), 674–693 (1989)CrossRefzbMATHGoogle Scholar
  17. 17.
    Antonini, M., Barlaud, M., Mathieu, P., Daubechies, I.: Image coding using wavelet transform. IEEE Transactions on Image Processing 1, 205–220 (1992)CrossRefGoogle Scholar
  18. 18.
    Fraysse, A., Pesquet-Popescu, B., Pesquet, J.-C.: Rate-distortion results for generalized Gaussian distributions. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Las Vegas, pp. 3753–3756Google Scholar
  19. 19.
    Fraysse, A., Pesquet-Popescu, B., Pesquet, J.-C.: On the uniform quantization of a class of sparse sources. IEEE Transactions on Information Theory 55(7), 3243–3263 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Taubman, D., Marcelin, M.W.: JPEG2000: Image Compression Fundamentals, Standards, and Practice. Kluwer Academic Publishers, Norwell (2002)CrossRefGoogle Scholar
  21. 21.
    Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: Proceedings of SIGGRAPH, pp. 161–172 (1995)Google Scholar
  22. 22.
    Dyn, N., Levin, D., Gregory, J.: A butterfly subdivision scheme for surface interpolation with tension control. Transaction on Graphics 9(2), 160–169 (1990)CrossRefzbMATHGoogle Scholar
  23. 23.
    Loop, C.: Smooth subdivision surfaces based on triangles. Department of Mathematics, Master’s Thesis, University of Utah (1987)Google Scholar
  24. 24.
    Dempster, A.P., Laird, N.M., Rubin, D.B.: Maximum likelihood from incomplete data via the EM algorithm. The Royal Statistical Society, Series B 39(1), 1–38 (1977)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Cover, T.M., Thomas, J.A.: Elements of Information Theory. Wiley Series in Telecommunications. Wiley, New York (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Shahid M. Satti
    • 1
  • Leon Denis
    • 1
  • Adrian Munteanu
    • 1
  • Jan Cornelis
    • 1
  • Peter Schelkens
    • 1
  1. 1.Department of Electronics and Informatics – Interdisciplinary Institute for Broadband TechnologyVrije Universiteit BrusselBrusselsBelgium

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