The Visual Computer

, Volume 31, Issue 6–8, pp 1135–1145 | Cite as

Efficient EMD and Hilbert spectra computation for 3D geometry processing and analysis via space-filling curve

  • Xiaochao Wang
  • Jianping Hu
  • Dongbo Zhang
  • Hong Qin
Original Article

Abstract

Empirical Mode Decomposition (EMD) has proved to be an effective and powerful analytical tool for non-stationary time series and starts to exhibit its modeling potential for 3D geometry analysis. Yet, existing EMD-based geometry processing algorithms only concentrate on multi-scale data decomposition by way of computing intrinsic mode functions. More in-depth analytical properties, such as Hilbert spectra, are hard to study for 3D surface signals due to the lack of theoretical and algorithmic tools. This has hindered much more broader penetration of EMD-centric algorithms into various new applications on 3D surface. To tackle this challenge, in this paper we propose a novel and efficient EMD and Hilbert spectra computational scheme for 3D geometry processing and analysis. At the core of our scheme is the strategy of dimensionality reduction via space-filling curve. This strategy transforms the problem of 3D geometry analysis to 1D time series processing, leading to two major advantages. First, the envelope computation is carried out for 1D signal by cubic spline interpolation, which is much faster than existing envelope computation directly over 3D surface. Second, it enables us to calculate Hilbert spectra directly on 3D surface. We could take advantages of Hilbert spectra that contain a wealth of unexploited properties and utilize them as a viable indicator to guide our EMD-based 3D surface processing. Furthermore, to preserve sharp features, we develop a divide-and-conquer scheme of EMD by explicitly separating the feature signals from non-feature signals. Extensive experiments have been carried out to demonstrate that our new EMD and Hilbert spectra based method is both fast and powerful for 3D surface processing and analysis.

Keywords

Empirical mode decomposition  Hilbert spectra Space-filling curve Surface processing 

References

  1. 1.
    Akleman, E., Xing, Q., Garigipati, P., Taubin, G., Chen, J., Hu, S.: Hamiltonian cycle art: surface covering wire sculptures and duotone surfaces. Comput. Graph. 37(5), 316–332 (2013)CrossRefGoogle Scholar
  2. 2.
    Ali, H., Hariharan, M., Yaacob, S., Adom, A.H.: Facial emotion recognition using empirical mode decomposition. Expert Syst. Appl. 42(3), 1261–1277 (2015)CrossRefGoogle Scholar
  3. 3.
    Arkin, E., Held, M., Mitchell, J., Skiena, S.: Hamiltonian triangulations for fast rendering. Vis. Comput. 12(9), 429–444 (1996)CrossRefGoogle Scholar
  4. 4.
    Belyaev, A., Ohtake, Y.: A comparison of mesh smoothing methods. In: Israel-Korea bi-national conference on geometric modeling and computer graphics, pp. 83–87 (2003)Google Scholar
  5. 5.
    Coifman, R.R., Maggioni, M.: Diffusion wavelets. Appl. Comput. Harmon. Anal 21(1), 53–94 (2006)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Di, C., Yang, X., Wang, X.: A four-stage hybrid model for hydrological time series forecasting. PLoS ONE 9(8), e104–663 (2014)CrossRefGoogle Scholar
  7. 7.
    Fleishman, S., Drori, I., Cohen-Or, D.: Bilateral mesh denoising. ACM Trans. Graph. 22(3), 950–953 (2003)CrossRefGoogle Scholar
  8. 8.
    Gao, Y., Li, C.F., Ren, B., Hu, S.M.: View-dependent multiscale fluid simulation. IEEE Trans. Vis. Comput. Graph. 19(2), 178–188 (2013)CrossRefGoogle Scholar
  9. 9.
    Gurung, T., Luffel, M., Lindstrom, P., Rossignac, J.: Lr: compact connectivity representation for triangle meshes. ACM Trans. Graph. 30(4), 67:1–67:8 (2011)CrossRefGoogle Scholar
  10. 10.
    Guskov, I., Sweldens, W., Schröder, P.: Multiresolution signal processing for meshes. In: SIGGRAPH’99, pp. 325–334 (1999)Google Scholar
  11. 11.
    Hou, T., Qin, H.: Admissible diffusion wavelets and their applications in space-frequency processing. IEEE Trans. Vis. Comput. Graph. 19(1), 3–15 (2013)CrossRefGoogle Scholar
  12. 12.
    Hu, J., Wang, X., Qin, H.: Improved, feature-centric emd for 3d surface modeling and processing. Graph. Models 76(5), 340–354 (2014)CrossRefGoogle Scholar
  13. 13.
    Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. A 454(1971), 903–995 (1998)Google Scholar
  14. 14.
    Jones, T.R., Durand, F., Desbrun, M.: Non-iterative, feature-preserving mesh smoothing. ACM Trans. Graph. 22(3), 943–949 (2003)CrossRefGoogle Scholar
  15. 15.
    Kopsinis, Y., McLaughlin, S.: Development of emd-based denoising methods inspired by wavelet thresholding. IEEE Trans. Signal Process. 57(4), 1351–1362 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Mandic, D., Rehman, N., Wu, Z., Huang, N.: Empirical mode decomposition-based time-frequency analysis of multivariate signals: the power of adaptive data analysis. IEEE Signal Process. Mag. 30(6), 74–86 (2013)CrossRefGoogle Scholar
  17. 17.
    Peano, G.: Sur une courbe, qui remplit toute une aire plane. Mat. Ann. 36(1), 157–160 (1890)MATHMathSciNetCrossRefGoogle Scholar
  18. 18.
    Qin, X., Hong, C.X., Zhang, S.Q., Wang, W.H.: Emd based fairing algorithm for mesh surface. In: CAD/Graphics ’09., pp. 606–609 (2009)Google Scholar
  19. 19.
    Quinn, J., Langbein, F., Lai, Y.K., Martin, R.: Generalized anisotropic stratified surface sampling. IEEE Trans. Vis. Comput. Graph. 19(7), 1143–1157 (2013)CrossRefGoogle Scholar
  20. 20.
    Ren, B., Li, C.F., Lin, M., Kim, T., Hu, S.M.: Flow field modulation. IEEE Trans. Vis. Comput. Graph. 19(10), 1708–1719 (2013)CrossRefGoogle Scholar
  21. 21.
    Schröder, P., Sweldens, W.: Spherical wavelets: efficiently representing functions on the sphere. In: SIGGRAPH ’95, pp. 161–172 (1995)Google Scholar
  22. 22.
    Solomon, J., Crane, K., Butscher, A., Wojtan, C.: A general framework for bilateral and mean shift filtering. CoRR abs/1405.4734 (2014)Google Scholar
  23. 23.
    Subr, K., Soler, C., Durand, F.: Edge-preserving multiscale image decomposition based on local extrema. ACM Trans. Graph. 28(5), 147:1–147:9 (2009)CrossRefGoogle Scholar
  24. 24.
    Taubin, G.: A signal processing approach to fair surface design. In: SIGGRAPH, pp. 351–358 (1995)Google Scholar
  25. 25.
    Taubin, G.: Constructing hamiltonian triangle strips on quadrilateral meshes. In: Visualization and Mathematics III, Mathematics and Visualization, pp. 69–91 (2003)Google Scholar
  26. 26.
    Wang, H., Su, Z., Cao, J., Wang, Y., Zhang, H.: Empirical mode decomposition on surfaces. Graph. Models 74(4), 173–183 (2012)CrossRefGoogle Scholar
  27. 27.
    Wang, R., Yang, Z., Liu, L., Deng, J., Chen, F.: Decoupling noises and features via weighted l1-analysis compressed sensing. ACM Trans. Graph. 33(2), 18:1–18:12 (2014)CrossRefGoogle Scholar
  28. 28.
    Wang, S., Hou, T., Su, Z., Qin, H.: Multi-scale anisotropic heat diffusion based on normal-driven shape representation. Vis. Comput. 27(6–8), 429–439 (2011)CrossRefGoogle Scholar
  29. 29.
    Wang, X., Cao, J., Liu, X., Li, B., Shi, X., Sun, Y.: Feature detection of triangular meshes via neighbor supporting. J. Zhejiang Univ. Sci. C 13(6), 440–451 (2012)CrossRefGoogle Scholar
  30. 30.
    Wang, X., Liu, X., Lu, L., Li, B., Cao, J., Yin, B., Shi, X.: Automatic hole-filling of cad models with feature-preserving. Comput. Graph. 36(2), 101–110 (2012)CrossRefGoogle Scholar
  31. 31.
    Wu, Z., Huang, N.E., Chen, X.: The multi-dimensional ensemble empirical mode decomposition method. Adv. Adapt. Data Anal. 01(03), 339–372 (2009)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yoshizawa, S., Belyaev, A., peter Seidel, H.: Smoothing by example: mesh denoising by averaging with similarity based weights. In: Proceedings of the IEEE international conference on shape modeling and applications (2006), pp. 38–44. IEEE (2006)Google Scholar
  33. 33.
    Zang, Y., Huang, H., Zhang, L.: Efficient structure-aware image smoothing by local extrema on space-filling curve. IEEE Trans. Vis. Comput. Graph. 20(9), 1253–1265 (2014)CrossRefGoogle Scholar
  34. 34.
    Zhang, J., Zheng, C., Hu, X.: Triangle mesh compression along the hamiltonian cycle. Vis. Comput. 29(6–8), 717–727 (2013)CrossRefGoogle Scholar
  35. 35.
    Zhao, W., Gao, S., Lin, H.: A robust hole-filling algorithm for triangular mesh. Vis. Comput. 23(12), 987–997 (2007)CrossRefGoogle Scholar
  36. 36.
    Zhong, M., Qin, H.: Sparse approximation of 3d shapes via spectral graph wavelets. Vis. Comput. 30(6–8), 751–761 (2014)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Xiaochao Wang
    • 1
  • Jianping Hu
    • 2
  • Dongbo Zhang
    • 1
  • Hong Qin
    • 3
  1. 1.State Key Laboratory of Virtual Reality Technology and SystemsBeihang UniversityBeijingChina
  2. 2.College of SciencesNortheast Dianli UniversityJilinChina
  3. 3.Department of Computer ScienceStony Brook UniversityStony BrookUSA

Personalised recommendations