Skip to main content
SpringerLink
Log in

Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

The synthesizing information, achieving understanding, and deriving insight from increasingly massive, time-varying, noisy and possibly conflicting data sets are some of most challenging tasks in the present information age. Traditional technologies, such as Fourier transform and wavelet multi-resolution analysis, are inadequate to handle all of the above-mentioned tasks. The empirical model decomposition (EMD) has emerged as a new powerful tool for resolving many challenging problems in data processing and analysis. Recently, an iterative filtering decomposition (IFD) has been introduced to address the stability and efficiency problems of the EMD. Another data analysis technique is the local spectral evolution kernel (LSEK), which provides a near prefect low pass filter with desirable time-frequency localizations. The present work utilizes the LSEK to further stabilize the IFD, and offers an efficient, flexible and robust scheme for information extraction, complexity reduction, and signal and image understanding. The performance of the present LSEK based IFD is intensively validated over a wide range of data processing tasks, including mode decomposition, analysis of time-varying data, information extraction from nonlinear dynamic systems, etc. The utility, robustness and usefulness of the proposed LESK based IFD are demonstrated via a large number of applications, such as the analysis of stock market data, the decomposition of ocean wave magnitudes, the understanding of physiologic signals and information recovery from noisy images. The performance of the proposed method is compared with that of existing methods in the literature. Our results indicate that the LSEK based IFD improves both the efficiency and the stability of conventional EMD algorithms.

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. Aki, K., Richards, P.G.: Quantitative Seismology. Freeman, San Francisco (1980)

    Google Scholar 

  2. Anderson, J.G.: Strong motion seismology. Rev. Geophys. Suppl. 29, 700–720 (1991)

    Google Scholar 

  3. Archibald, R., Gelb, A., Yoon, Y.: Polynomial fitting for edge detection in irregularly sampled signals and images. SIAM J. Numer. Anal. 43, 259–279 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. Archibald, R., Gelb, A., Saxena, R., Xiu, D.B.: Discontinuity detection in multivariate space for stochastic simulations. J. Comput. Phys. 228, 2676–2689 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bao, W., Sun, F., Wei, G.W.: Numerical methods for the generalized Zakharov system. J. Comput. Phys. 190, 201–228 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bao, G., Wei, G.W., Zhao, S.: Local spectral time-domain method for electromagnetic wave propagation. Opt. Lett. 28, 513–515 (2003)

    Article  Google Scholar 

  7. Bao, G., Wei, G.W., Zhao, S.: Numerical solution of the Helmholtz equation with high wave numbers. Int. J. Numer. Methods Eng. 59, 389–408 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bendat, J.S., Piersol, A.G.: Random Data: Analysis and Measurement Procedures. Wiley, New York (1986)

    MATH  Google Scholar 

  9. Benjamin, T.B., Feir, J.E.: The disintegration of wavetrains on deep water. I. Theory. J. Fluid Mech. 27, 417–430 (1967)

    Article  MATH  Google Scholar 

  10. Bi, N., Sun, Q., Huang, D., Yang, Z., Huang, J.: Robust image watermarking based on multiband wavelets and empirical mode decomposition. IEEE Image Process. 16, 1956–1966 (2007)

    Article  MathSciNet  Google Scholar 

  11. Boashash, B.: Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals. Proc. IEEE 80, 520–538 (1992)

    Article  Google Scholar 

  12. Canny, J.: A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8, 679–698 (1986)

    Article  Google Scholar 

  13. Chan, Y.T.: Wavelet Basics. Springer, Berlin (1995)

    Book  Google Scholar 

  14. Chen, Z., Ivanov, P.C., Hu, K., Stanley, H.E.: Effects of nonstationarities on detrended fluctuation analysis. Phys. Rev. E 65, 041107 (2002)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Chen, K., Chen, X., Renaut, R., Alexander, G.E., Bandy, D., Guo, H., Reiman, E.M.: Characterization of the image-derived carotid artery input function using independent component analysis for the quantitation of [18 F] fluorodeoxyglucose positron emission tomography images. Phys. Med. Biol. 52, 7055–7071 (2007)

    Article  Google Scholar 

  17. Claasen, T.A.C.M., Mecklenbräuker, W.F.G.: The Wigner distributiona tool for time-frequency signal analysis. Part I: Continuous time signals. Philips J. Res. 35, 372–389 (1980)

    MathSciNet  MATH  Google Scholar 

  18. Cohen, L.: Time-Frequency Analysis. Prentice-Hall, Englewood Cliffs (1995)

    Google Scholar 

  19. Copson, E.T.: Asymptotic Expansions. Cambridge University Press, Cambridge (1967)

    Google Scholar 

  20. Drazin, P.G.: Nonlinear Systems. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  21. Echeverría, J.C., Crowe, J.A., Woolfson, M.S., Hayes-Gill, B.R.: Application of empirical mode decomposition to heart rate variability analysis. Med. Biol. Eng. Comput. 39, 471 (2001)

    Article  Google Scholar 

  22. Equis, S., Jacquot, P.: The empirical mode decomposition: a must-have tool in speckle interferometry? Opt. Express 17, 611–623 (2009)

    Article  Google Scholar 

  23. Farge, M.: Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech. 24, 395–457 (1992)

    Article  MathSciNet  Google Scholar 

  24. Guo, H., Renaut, R.A., Chen, K.: An input function estimation method for FDG-PET human brain studies. Nucl. Med. Biol. 34, 483–492 (2007)

    Article  Google Scholar 

  25. Guo, H., Renaut, R.A., Chen, K., Reiman, E.: FDG-PET parametric imaging by total variation minimization. Comput. Med. Imaging Graph. 33, 295–303 (2009)

    Article  Google Scholar 

  26. Hadley, P.K., Askar, A., Cakmak, A.S.: Subsoil geology and soil amplification in Mexico Valley. Soil Dyn. Earthq. Eng. 10, 101–109 (1991)

    Article  Google Scholar 

  27. Hildreth, E., Marr, D.: Theory of edge detection. Proc. R. Soc. Lond. B 207, 187–217 (1980)

    Article  Google Scholar 

  28. Hou, Z.J., Wei, G.W.: A new approach for edge detection. Pattern Recognit. 35, 1559–1570 (2002)

    Article  MATH  Google Scholar 

  29. Hu, K., Ivanov, P.C., Chen, Z., Carpena, P., Stanley, H.E.: Effects of trends on detrended fluctuation analysis. Phys. Rev. E 64, 011114 (2001)

    Article  Google Scholar 

  30. Huang, N.E., Wu, Z.: A review on Hilbert-Huang transform: method and its applications to geophysical studies. Rev. Geophys. 46, RG2006 (2008)

    Article  Google Scholar 

  31. Huang, N.E., Tung, C.C., Long, S.R.: Wave spectra. Sea 9, 197–237 (1990)

    Google Scholar 

  32. Huang, N.E., Long, S.R., Shen, Z.: The mechanism for frequency downshift in nonlinear wave evolution. Adv. Appl. Mech. 32, 59–111 (1996)

    Article  Google Scholar 

  33. 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, 903–995 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Huang, N.E., Shen, Z., Long, S.R.: A new view of nonlinear water waves: the Hilbert spectrum. Annu. Rev. Fluid Mech. 31, 417–457 (1999)

    Article  MathSciNet  Google Scholar 

  35. Hwang, W., Mallet, S.: Singularity detection and processing with wavelets. IEEE Trans. Inf. Theory 38, 617–643 (1992)

    Article  MATH  Google Scholar 

  36. Kevorkian, J.: Space Mathematics III. Lectures in Applied Mathematics, vol. 7, pp. 206–275. Am. Math. Soc., Providence (1966)

    Google Scholar 

  37. Kopsinis, Y., McLaughlin, S.: Development of EMD-based denoising methods inspired by wavelet thresholding. IEEE Trans. Signal Process. 57, 1351–1362 (2009)

    Article  MathSciNet  Google Scholar 

  38. Lake, B.M., Yuan, H.C.: A new model for nonlinear gravity waves. I. Physical model and experimental evidence. J. Fluid Mech. 88, 33–62 (1978)

    Article  MATH  Google Scholar 

  39. Li, S.Z.: Markov Random Field Modeling in Image Analysis. Springer, London (2009)

    MATH  Google Scholar 

  40. Liang, H., Lin, Q.-H., Chen, J.D.Z.: Application of the empirical mode decomposition to the analysis of esophageal manometric data in gastroesophageal reflux disease. IEEE Trans. Biomed. Eng. 52, 1692–1701 (2005)

    Article  Google Scholar 

  41. Lin, L., Wang, Y., Zhou, H.: Iterative filtering as an alternative algorithm for empirical mode decomposition. Adv. Adapt. Data Anal. 1, 543–560 (2009)

    Article  MathSciNet  Google Scholar 

  42. Liu, B., Riemenschneidera, S., Xu, Y.: Gearbox fault diagnosis using empirical mode decomposition and Hilbert spectrum. Mech. Syst. Signal Process. 20, 718–734 (2006)

    Article  Google Scholar 

  43. Long, S.R., Huang, N.E., Tung, C.C., Wu, M.L., Lin, R.Q., Mollo-Christensen, E., Yuan, Y.: The Hilbert techniques: an alternate approach for non-steady time series analysis. IEEE Geosci. Remote Sens. Soc. Lett. 3, 6–11 (1995)

    Google Scholar 

  44. Lu, Z., Liu, Y.: Analysis of daily river flow fluctuations using empirical mode decomposition and arbitrary order Hilbert spectral analysis. J. Hydrol. 373, 103–111 (2009)

    Article  MathSciNet  Google Scholar 

  45. Mao, D., Rockmore, D.N., Wang, Y., Wu, Q.: EMD Analysis for Visual Stylometry. Preprint

  46. Mao, D., Wang, Y., Wu, Q.: A new approach for analyzing physiological time series. Preprint

  47. Miller, L., Cheney, R.E.: Large-scale meridional transport in the tropic Pacific Ocean during the 1986–1987 El Nino from GEOSAT. J. Geophys. Res. 95, 17905–17919 (1990)

    Article  Google Scholar 

  48. Miller, L., Cheney, R.E., Douglas, B.C.: GEOSAT altimeter observation of Kelvin waves and the 1986–1987 El Niño. Science 239, 52–54 (1988)

    Article  Google Scholar 

  49. Newmark, N.M., Rosenblueth, E.: Fundamentals of Earthquake Engineering. Prentice-Hall, Englewood Cliffs (1971)

    Google Scholar 

  50. Oppenheim, A.V., Schafer, R.W.: Digital Signal Processing. Prentice-Hall, Englewood Cliffs (1989)

    Google Scholar 

  51. Pines, D., Salvino, L.: Health monitoring of one dimensional structures using empirical mode decomposition and the Hilbert-Huang transform. Proc. SPIE 4701, 127–143 (2002)

    Article  Google Scholar 

  52. Radke, R.J., Andra, S., Al-Kofahi, O., Roysam, B.: Image change detection algorithms: a systematic survey. IEEE Trans. Image Process. 14, 294–307 (2005)

    Article  MathSciNet  Google Scholar 

  53. Ramamonjiarisoa, A., Mollo-Christensen, E.: Modulation characteristics of sea surface waves. J. Geophys. Res. 84, 7769–7775 (1979)

    Article  Google Scholar 

  54. Rezaei, D., Taheri, F.: Experimental validation of a novel structural damage detection method based on empirical mode decomposition. Smart Mater. Struct. 18, 045004 (2009)

    Article  Google Scholar 

  55. Rilling, G., Flandrin, P., Gonalves, P., Lilly, J.M.: Bivariate empirical mode decomposition. IEEE Signal Process. Lett. 14, 936–939 (2007)

    Article  Google Scholar 

  56. Robinson, A.R., Huang, N.E., Leitao, C.D., Parra, C.G.: A study of the variability of ocean currents in the Northwestern Atlantic using satellite altimetry. J. Phys. Oceanogr. 13, 565–585 (1983)

    Article  Google Scholar 

  57. Saxena, R., Gelb, A., Mittelmann, H.: A high order method for determining the edges in the gradient of a function. Commun. Comput. Phys. 5, 694–711 (2009)

    MathSciNet  Google Scholar 

  58. Shao, Z.H., Wei, G.W., Zhao, S.: DSC time-domain solution of Maxwell equations. J. Comput. Phys. 189, 427–453 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  59. Siddiqi, K., Kimia, B.B., Shu, C.-W.: Geometric shock capturing ENO-schemes for subpixel interpolation, computation and curve evolution. Graph. Models Image Process. 59, 278–302 (1997)

    Article  Google Scholar 

  60. Spedding, G.R., Browand, F.K., Huang, N.E., Long, S.R.: A 2D complex wavelet analysis of an unsteady wind-generated surface wavelet analysis of an unsteady wind-generated surface wave field. Dyn. Atmos. Oceans 20, 55–77 (1993)

    Article  Google Scholar 

  61. Tanaka, T., Mandic, D.P.: Complex empirical mode decomposition. Signal Processing Letters, IEEE 14(101–104) (2007)

  62. Tang, Y.-W., Tai, C.-C., Su, C.-C., Chen, C.-Y., Chen, J.-F.: A correlated empirical mode decomposition method for partial discharge signal denoising. Meas. Sci. Technol. 21, 085106 (2010)

    Article  Google Scholar 

  63. Titchmarsh, E.C.: Introduction to the Theory of Fourier Integrals. Oxford University Press, London (1948)

    Google Scholar 

  64. Vasudevan, K., Cook, F.A.: Empirical mode skeletonization of deep crustal seismic data: theory and applications. J. Geophys. Res. 105, 7845–7856 (2000)

    Article  Google Scholar 

  65. Wan, D.C., Patnaik, B.S.V., Wei, G.W.: Discrete singular convolution-finite subdomain method for the solution of incompressible viscous flows. J. Comput. Phys. 180, 229–255 (2002)

    Article  MATH  Google Scholar 

  66. Wang, Y., Zhou, Z.F.: On the convergence of iterative filtering empirical mode decomposition. Preprint

  67. Wei, G.W.: Discrete singular convolution for the Fokker-Planck equation. J. Chem. Phys. 110, 8930–8942 (1999)

    Article  Google Scholar 

  68. Wei, G.W.: A unified approach for the solution of the Fokker-Planck equation. J. Phys. A, Math. Gen. 33, 4935–4953 (2000)

    Article  MATH  Google Scholar 

  69. Wei, G.W.: Discrete singular convolution for the sine-Gordon equation. Physica D 137, 247–259 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  70. Wei, G.W.: A new algorithm for solving some mechanical problems. Comput. Methods Appl. Mech. Eng. 190, 2017–2030 (2001)

    Article  MATH  Google Scholar 

  71. Wei, G.W., Jia, Y.Q.: Synchronization-based image edge detection. Europhys. Lett. 59, 814–819 (2002)

    Article  Google Scholar 

  72. Wei, G.W., Zhao, Y.B., Xiang, Y.: Discrete singular convolution and its application to the analysis of plates with internal supports. I. Theory and algorithm. Int. J. Numer. Methods Eng. 55, 913–946 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  73. Whitham, G.B.: Linear and Nonlinear Waves. Wiley, New York (1975)

    Google Scholar 

  74. Wu, Z., Huang, N.: Ensemble empirical mode decomposition: a noise-assisted data analysis method. Adv. Adapt. Data Anal. 1(1), 1–41 (2009)

    Article  Google Scholar 

  75. Yang, W., Tavner, P.: Empirical mode decomposition, an adaptive approach for interpreting shaft vibratory signals of large rotating machinery. J. Sound Vib. 321, 1144–1170 (2009)

    Article  Google Scholar 

  76. Yang, S.Y., Zhou, Y.C., Wei, G.W.: Comparison of the discrete singular convolution algorithm and the Fourier pseudospectral method for solving partial differential equations. Comput. Phys. Commun. 143, 113–135 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  77. Yang, Z., Yang, L., Qi, D., Suen, C.: An EMD-based recognition method for Chinese fonts and styles. Pattern Recognit. Lett. 27, 1692–1701 (2006)

    Article  Google Scholar 

  78. Yang, P., Wang, G., Bian, J., Zhou, X.: The prediction of non-stationary climate series based on empirical mode decomposition. Adv. Atmos. Sci. 27, 845–854 (2010)

    Article  Google Scholar 

  79. Yeh, J.-R., Fan, S.-Z., Shieh, J.-S.: Human heart beat analysis using a modified algorithm of detrended fluctuation analysis based on empirical mode decomposition. Med. Eng. Phys. 31, 92–100 (2009)

    Article  Google Scholar 

  80. Yu, S., Zhao, S., Wei, G.W.: Local spectral time splitting method for first- and second-order partial differential equations. J. Comput. Phys. 206(2), 727–780 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  81. Yu, S., Zhou, Y., Wei, G.W.: Matched interface and boundary (MIB) method for elliptic problems with sharp-edged interfaces. J. Comput. Phys. 224(2), 729–756 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  82. Yu, Z.-G., Anh, V., Wang, Y., Mao, D.: Modeling and simulation of the horizontal component of the magnetic field by fractional stochastic differential equations in conjunction with empirical mode decomposition. J. Geophys. Res. 115, A10219 (2010), 11 pp. doi:10.1029/2009JA015206

    Article  Google Scholar 

  83. Zhao, S., Wei, G.W.: Comparison of the discrete singular convolution and three other numerical schemes for Fishers equation. SIAM J. Sci. Comput. 25, 127–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  84. Zheng, Q., Yan, X.H., Ho, C.R., Tai, C.K.: The effects of shear flow on propagation of Rossby waves in the equatorial oceans. J. Phys. Oceanogr. 24, 1680–1686 (1994)

    Article  Google Scholar 

  85. Zheng, Q., Yan, X.H., Ho, C.R., Tai, C.K.: Observation of equatorially trapped waves in the Pacific using GEOSAT altimeter data. Deep Sea Res. 42, 797–817 (1995)

    Article  Google Scholar 

  86. Zhou, Y.C., Patnaik, B.S.V., Wan, D.C., Wei, G.W.: DSC solution for flow in a staggered double lid driven cavity. Int. J. Numer. Methods Eng. 57, 211–234 (2003)

    Article  MATH  Google Scholar 

  87. Zhou, Y.C., Wei, G.W.: High-resolution conjugate filters for the simulation of flows. J. Comput. Phys. 189, 150–179 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Yang Wang, Guo-Wei Wei & Siyang Yang

  2. Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI, 48824, USA

    Guo-Wei Wei

Authors
  1. Yang Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guo-Wei Wei
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Siyang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guo-Wei Wei.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wang, Y., Wei, GW. & Yang, S. Iterative Filtering Decomposition Based on Local Spectral Evolution Kernel. J Sci Comput 50, 629–664 (2012). https://doi.org/10.1007/s10915-011-9496-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-011-9496-0

Keywords

Search

Navigation