Skip to main content
SpringerLink
Log in

A PDE based and interpolation-free framework for modeling the sifting process in a continuous domain

  • Published:
Advances in Computational Mathematics Aims and scope Submit manuscript

Abstract

The empirical mode decomposition (EMD) is a powerful tool in signal processing. Despite its algorithmic origin making its theoretical analysis and formulation very difficult, a few recent works has contributed to its theoretical framework. Herein, the former local mean is formulated in a more convenient way by introducing operators to calculate local upper and lower envelopes. This enables the use of differential calculus and other classical calculations on the new local mean. Based on its more accurate formulation, a partial differential equation (PDE) consistency result is provided to approximate the sifting process iterations, without any envelope interpolation. In addition, a new stopping criterion based on the introduced local mean is proposed. This new criterion is a local measure and resolves the null integral conservative property of the previous derived PDE, which made any signal having a null integral be a PDE-based mode. Moreover, the δ inner model parameter is now linked to the signal intrinsic properties, providing to the latter a physical meaning and making the proposed model keep the auto-adaptive property of the EMD. New decomposition modes are now analytically and fully characterized, and also interpolation free. Finally, properties of the interpolation free PDE model are presented. Results obtained with our proposed approach by explicit computations thanks to the eigendecomposition of the Laplacian operator, and also by numerical resolution of the derived PDE, show noticeable improvements for both stationary and non stationary signals, in comparison to the former EMD algorithm.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 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. A 454, 903–995 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cexus, J., Boudraa, A.: Teager-Huang analysis applied to sonar target recognition. International Journal of Signal Process 1(1), 23–27 (2004)

    Google Scholar 

  3. Boudraa, A., Cexus, J., Salzenstein, F., Guillon, L.: IF estimation using empirical mode decomposition and nonlinear Teager energy operator. In: Proc. IEEE ISCCSP, pp. 45–48. Hammamet, Tunisia (2004)

  4. Cexus, J., Boudraa, A.: Nonstationary signals analysis by Teager-Huang Transform (THT). In: EUSIPCO, 5 p., Florence, Italy (2006)

  5. Bouchikhi, A., Boudraa, A., Ramdane, S.B., Diop, E.-H.: Empirical mode decomposition and some operators to estimate instantaneous frequency: a comparative study. In: Proc. IEEE ISCCSP, pp. 608–613. Malta (2008)

  6. Kazys, R., Pagodinas, D., Tumsys, O.: Application of the Hilbert-Huang signal processing to ultrasonic nondestructive testing of composite materials. Ultragarsas 50(1), 17–22 (2004)

    Google Scholar 

  7. Oonincx, P.J., Hermand, J.P.: Empirical mode decomposition of ocean acoustic data with constraint on the frequency range. In: Proceedings of the Seventh European Conference on Underwater Acoustics, ECUA 2004, Delft, The Netherlands (2004)

  8. Liang, H., Bressler, S.L., Buffalo, E.A., Desimone, R., Fries, P.: Empirical mode decomposition of field potentials from macaque V4 in visual spatial attention. Biol. Cybern. 92, 380–392 (2005)

    Article  MATH  Google Scholar 

  9. Cai, C., Liu, W., Fu, J.S., Lu, Y.: A new approach for ground moving target indication in foliage environments. Elsevier Signal Process. 86, 84–97 (2006)

    Article  MATH  Google Scholar 

  10. Benramdane, S., Cexus, J.C., Boudraa, A.O., Astolfi, J.A.: Time-frequency analysis of pressure fluctuations ona hydrofoil undergoing a transient pitching motion using Hilbert-Huang and Teager-Huang transforms. In: ASME PVP 2007/CREEP 8, San Antonio, Texas, USA (2007)

  11. Wu, Z., Huang, N.E.: A study of the characteristics of white noise using the empirical mode decomposition method. Proc. R. Soc. A 460, 1597–1611 (2004)

    Article  MATH  Google Scholar 

  12. Flandrin, P., Rilling, G.: Empirical Mode Decomposition as a Filter Bank. IEEE Signal Process. Lett. 11(2), 112–114 (2004)

    Article  Google Scholar 

  13. Mallat, S.: Une exploration des signaux en ondelettes. L. E. de l’École Polytechnique, Ed., Novembre (2000)

  14. Cohen, L.: Time-Frequency Analysis. E. C. Prentice Hall, Ed., NJ (1995)

  15. Daubechies, I., Lu, J., Wu, H.-T.: Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal. (2010). doi:10.1016/j.acha.2010.08.002

    Google Scholar 

  16. Delechelle, E., Lemoine, J., Niang, O.: Empirical mode decomposition: an analytical approach for sifting process. IEEE Signal Process. Lett. 12(11), 764–767 (2005)

    Article  Google Scholar 

  17. Sharpley, R.C., Vatchev, V.: Analysis of the intrinsic mode functions. Constr. Approx. 24, 17–47 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Meignen, S., Perrier, V.: A new formulation for empirical mode decomposition based on constrained optimization. IEEE Signal Process. Lett. 14(12), 932–935 (2007)

    Article  Google Scholar 

  19. Vatchev, V., Sharpley, R.: Decomposition of Functions into Pairs of Intrinsic Mode Functions. Proc. R. Soc. A 464, 2265–2280 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Diop, E.H.S., Alexandre, R., Boudraa, A.O.: Analysis of intrinsic mode functions: a pde approach. IEEE Signal Process. Lett. 17(4), 398–401 (2010)

    Article  Google Scholar 

  21. Diop, E.H.S., Alexandre, R., Boudraa, A.O.: A PDE model for 2D intrinsic mode functions. In: IEEE ICIP, pp. 3961–3964. Cairo, Egypt (2009)

  22. Diop, E.H.S., Alexandre, R., Moisan, L.: Intrinsic nonlinear multiscale image decomposition: a 2D empirical mode decomposition-like tool. Comput. Vis. Image Underst. 116(1), 102–119 (2012)

    Article  Google Scholar 

  23. Nunes, J., Bouaoune, Y., Delechelle, E., Niang, O., Bunel, P.: Image analysis by bidimensional empirical mode decomposition. Elsevier Image Vis. Comput. 21, 1019–1026 (2003)

    Article  Google Scholar 

  24. Damerval, C., Meignen, S., Perrier, V.: A fast algorithm for bidimensional EMD. IEEE Signal Process. Lett. 12(10), 701–704 (2005)

    Article  Google Scholar 

  25. Xu, Y., Liu, B., Liu, J., Riemenschneider, S.: Two-dimensional empirical mode decomposition by finite elements. Proc. R. Soc. A 462(2074), 3081–3096, (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Frei, M.G., Osorio, I.: Intrinsic time-scale decomposition: time-frequency-energy analysis and real-time filtering of non-stationary signal. Proc. R. Soc. A 463, 321–342 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Flandrin, P., Goncalves, P.: Empirical mode decompositions as data-driven wavelet-like expansions. IJWMIP 2(4), 1–20 (2004)

    MathSciNet  Google Scholar 

  28. Boashash, B.: Time Frequency Signal Analysis and Processing: A Comprehensive Reference. Elsevier, The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK (2003)

  29. Rice, S.O.: Mathematical analysis of random noise. Bell Syst. Tech. J. 23, 282–332 (1944)

    MathSciNet  MATH  Google Scholar 

  30. Cexus, J.C.: Analyse des signaux non-stationnaires par transformation de huang, opérateur de Teager-Kaiser, et transformation de Huang-Teager. Ph.D. dissertation, Université de Rennes 1 (2005)

  31. Rilling, G.: Décompositions Modales Eempiriques. Contributions à la théorie, l’algorithmie et l’analyse de performances. Ph.D. dissertation, École Normale Supérieure de Lyon (2007)

  32. Hou, T.Y., Yan, M.P.: A variant of the EMD method for multi-scale data. AADA 1(4), 483–516 (2009)

    MathSciNet  Google Scholar 

  33. Chen, Q., Huang, N., Riemenschneider, S., Xu, Y.: A b-spline approach for empirical mode decompositions. Adv. Comput. Math. 24(1), 171–195 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. DiBenedetto, E.: Partial Differential Equations, ch. 4, pp. 208–209. Birkhäuser, Boston, USA (1995)

    Google Scholar 

  35. Diop, E.H.S.: A PDE-based approach for the 1D and 2D empirical mode decomposition, and AM-FM models for image analysis. Ph.D. dissertation, Université de Rennes 1, France (2009)

  36. Pinchover, Y., Rubinstein, J.: An Introduction to Partial Differential Equations. Cambridge University Press (2005)

  37. Allaire, G.: Numerical Analysis and Optimization. Oxford University Press Inc., New York, USA (2007)

    MATH  Google Scholar 

  38. Flandrin, P.: Available Online. http://perso.ens-lyon.fr/patrick.flandrin/emd.html (2007). Accessed Sept 2011

  39. Wu, Z., Huang, N.E.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. AADA 1(1), 1–41 (2009)

    Google Scholar 

  40. Boudraa, A.O., Cexus, J.C., Saidi, Z.: EMD-based signal noise reduction. International Journal of Signal Processing 1(1), 33–37 (2004)

    Google Scholar 

  41. Savitzky, A., Golay, M.J.E.: Smoothing and differentiation of data by simplified least squares procedures. Anal. Chem. 36(8), 1627–1639 (1964)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Center of Mathematical Morphology, Department of Mathematics and Systems, MINES ParisTech, 35 rue Saint-Honoré, 77305, Fontainebleau Cedex, France

    El Hadji S. Diop

  2. Department of Mathematics, Shanghai Jia Tong University, Shanghai, 200240, People’s Republic of China

    Radjesvarane Alexandre

  3. Laboratoire Jean-Kunztmann, Tour IRMA, BP 53, 38041, Grenoble Cedex 9, France

    Valérie Perrier

Authors
  1. El Hadji S. Diop
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Radjesvarane Alexandre
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Valérie Perrier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to El Hadji S. Diop.

Additional information

Communicated by Yuesheng Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diop, E.H.S., Alexandre, R. & Perrier, V. A PDE based and interpolation-free framework for modeling the sifting process in a continuous domain. Adv Comput Math 38, 801–835 (2013). https://doi.org/10.1007/s10444-011-9260-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10444-011-9260-x

Keywords

Mathematics Subject Classifications (2010)

Search

Navigation