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Sparse approximation of 3D shapes via spectral graph wavelets

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Abstract

This paper investigates the compressive representation of 3D meshes and articulates a novel sparse approximation method for 3D shapes based on spectral graph wavelets. The originality of this paper is centering on the first attempt of exploiting spectral graph wavelets in the sparse representation for 3D shape geometry. Conventional spectral mesh compression employs the eigenfunctions of mesh Laplacian as shape bases. The Laplacian eigenbases, generalizing the Fourier bases from Euclidean domain to manifold, exhibit global support and are neither efficient nor precise in representing local geometry. To ameliorate, we advocate an innovative approach to 3D mesh compression using spectral graph wavelets as dictionary to encode mesh geometry. In contrast to Laplacian eigenbases, the spectral graph wavelets are locally defined at individual vertices and can better capture local shape information in a more accurate way. Nonetheless, the multiscale spectral graph wavelets form a redundant dictionary as shape bases, therefore we formulate the compression of 3D shape as a sparse approximation problem that can be readily handled by powerful algorithms such as orthogonal matching pursuit. Various experiments demonstrate that our method is superior to the existing spectral mesh compression methods.

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References

  1. Antoine, J.P., Roşca, D., Vandergheynst, P.: Wavelet transform on manifolds: old and new approaches. Appl. Comput. Harmon. Anal. 28(2), 189–202 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal. 21(1), 53–94 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  3. Donoho, D., Tsaig, Y., Drori, I., Starck, J.L.: Sparse solution of underdetermined systems of linear equations by stagewise orthogonal matching pursuit. IEEE Trans. Inf. Theory 58(2), 1094–1121 (2012)

    Article  MathSciNet  Google Scholar 

  4. Gavish, M., Nadler, B., Coifman, R.R.: Multiscale wavelets on trees, graphs and high dimensional data: theory and applications to semi supervised learning. In: Proceedings of ICML, pp. 367–374 (2010)

  5. Gumhold, S., Straßer, W.: Real time compression of triangle mesh connectivity. In: Proceedings of SIGGRAPH, pp. 133–140 (1998)

  6. Hammond, D.K., Vandergheynst, P., Gribonval, R.: Wavelets on graphs via spectral graph theory. Appl. Comput. Harmon. Anal. 30(2), 129–150 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Karni, Z., Gotsman, C.: Spectral compression of mesh geometry. In: Proceedings of SIGGRAPH, pp. 279–286 (2000)

  8. Karni, Z., Gotsman, C.: 3D mesh compression using fixed spectral bases. In: Proceedings of Graphics, Interface, pp. 1–8 (2001)

  9. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)

    Article  MathSciNet  Google Scholar 

  10. Li, C., Hamza, A.B.: A multiresolution descriptor for deformable 3D shape retrieval. Visual Comput. 29(6–8), 513–524 (2013)

    Article  Google Scholar 

  11. Mahadevan, S.: Adaptive mesh compression in 3D computer graphics using multiscale manifold learning. In: Proceedings of ICML, pp. 585–592 (2007)

  12. Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)

    Article  MATH  Google Scholar 

  13. Pati, Y.C., Rezaiifar, R., Krishnaprasad, P.: Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition. In: Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, pp. 40–44 (1993)

  14. Ram, I., Elad, M., Cohen, I.: Redundant wavelets on graphs and high dimensional data clouds. IEEE Signal Process. Lett. 19(5), 291–294 (2012)

    Article  Google Scholar 

  15. Rossignac, J.: Edgebreaker: connectivity compression for triangle meshes. IEEE Trans. Visual Comput. Graphics 5(1), 47–61 (1999)

    Google Scholar 

  16. Skodras, A., Christopoulos, C., Ebrahimi, T.: The JPEG 2000 still image compression standard. IEEE Signal Process. Mag. 18(5), 36–58 (2001)

    Article  Google Scholar 

  17. Tropp, J.A., Gilbert, A.C., Strauss, M.J.: Algorithms for simultaneous sparse approximation part I: Greedy pursuit. Signal Process. 86(3), 572–588 (2006)

    Article  MATH  Google Scholar 

  18. Vallet, B., Lévy, B.: Spectral geometry processing with manifold harmonics. Comput. Graphics Forum 27(2), 251–260 (2008)

    Google Scholar 

  19. Wallace, G.K.: The JPEG still picture compression standard. IEEE Trans. Consum. Electron. 38(1), 18–34 (1992)

    Google Scholar 

  20. Wardetzky, M., Mathur, S., Kälberer, F., Grinspun, E.: Discrete Laplace operators: no free lunch. In: Proceedings of 5th Eurographics Symposium on Geometry Processing, pp. 33–37 (2007)

  21. Zhang, H., Van Kaick, O., Dyer, R.: Spectral mesh processing. Comput. Graphics Forum 29(6), 1865–1894 (2010)

    Article  Google Scholar 

Download references

Acknowledgments

This research is supported in part by National Science Foundation of USA (Grant Nos. IIS-0949467, IIS-1047715, and IIS-1049448), and National Natural Science Foundation of China (Grant Nos. 61190120, 61190121, 61190125).

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  1. Department of Computer Science, Stony Brook University, Stony Brook, NY, 11794, USA

    Ming Zhong & Hong Qin

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  1. Ming Zhong
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  2. Hong Qin
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Correspondence to Ming Zhong.

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Zhong, M., Qin, H. Sparse approximation of 3D shapes via spectral graph wavelets. Vis Comput 30, 751–761 (2014). https://doi.org/10.1007/s00371-014-0971-0

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