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Licensed Unlicensed Requires Authentication Published online by De Gruyter February 23, 2023

Two-dimensional EMD with shape-preserving spline interpolation

  • Wesley G. Brown and Maria D. van der Walt EMAIL logo


Empirical mode decomposition (EMD) is a popular, user-friendly, data-driven algorithm to decompose a given (non-stationary) signal into its constituting components, utilizing spline interpolation. This algorithm was first proposed in 1998 in the one-dimensional setting, and it employed standard cubic spline interpolation. Since then, different two-dimensional extensions of EMD have been proposed. In this paper, we consider one of these two-dimensional extensions and adapt it to use a shape-preserving interpolation scheme based on quadratic B-splines, ensuring that monotonicity and concavity in the input data are preserved. Using multiple numerical experiments, we show that this new scheme outperforms the original EMD, both qualitatively and quantitatively.

MSC 2010: 41A15


[1] C.-S. Chen and Y. Jeng, Two-dimensional nonlinear geophysical data filtering using the multidimensional EEMD method, J. Appl. Geophys. 111 (2014), 256–270. 10.1016/j.jappgeo.2014.10.015Search in Google Scholar

[2] W.-K. Chen, J.-C. Lee, W.-Y. Han, C.-K. Shih and K.-C. Chang, Iris recognition based on bidimensional empirical mode decomposition and fractal dimension, Inform. Sci. 221 (2013), 439–451. 10.1016/j.ins.2012.09.021Search in Google Scholar

[3] C. K. Chui and M. D. van der Walt, Signal analysis via instantaneous frequency estimation of signal components, Int. J. Geomath. 6 (2015), 1–42. 10.1007/s13137-015-0070-zSearch in Google Scholar

[4] Z.-P. Fan and G.-L. Zhang, The research of improved envelope algorithm of EMD, Comput. Simul. 27 (2010), no. 6, 126–129. Search in Google Scholar

[5] N. E. Huang, M.-L. C. Wu, S. R. Long, S. S. P. Shen, W. g Qu, P. Gloersen and K. L. Fan, A confidence limit for the empirical mode decomposition and Hilbert spectral analysis, Proc. Roy. Soc. Lond. Ser A 459 (2003), no. 2037, 2317–2345. 10.1098/rspa.2003.1123Search in Google Scholar

[6] N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung and H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. Roy. Soc. Lond. Ser A 454 (1998), no. 1971, 903–995. 10.1098/rspa.1998.0193Search in Google Scholar

[7] N. E. Huang and Z. Wu, A review on Hilbert–Huang transform: Method and its applications to geophysical studies, Rev. Geophys. 46 (2008), 10.1029/2007RG000228. 10.1029/2007RG000228Search in Google Scholar

[8] Y. Li, M. Xu, Y. Wei and W. Huang, An improvement EMD method based on the optimized rational hermite interpolation approach and its application to gear fault diagnosis, Measurement 63 (2015), 330–345. 10.1016/j.measurement.2014.12.021Search in Google Scholar

[9] J. C. Nunes, S. Guyot and E. Deléchelle, Texture analysis based on local analysis of the bidimensional empirical mode decomposition, Machine Vis. Appl. 16 (2005), no. 3, 177–188. 10.1007/s00138-004-0170-5Search in Google Scholar

[10] X. Qin, S. Liu, Z. Wu and H. Jun, Medical image enhancement method based on 2d empirical mode decomposition, 2nd International Conference on Bioinformatics and Biomedical Engineering, IEEE Press, Piscataway (2008), 2533–2536. 10.1109/ICBBE.2008.967Search in Google Scholar

[11] L. Schumaker, On shape preserving quadratic spline interpolation, SIAM J. Numer. Anal. 20 (1983), no. 4, 854–864. 10.1137/0720057Search in Google Scholar

[12] S. Sinclair and G. G. S. Pegram, Empirical mode decomposition in 2-d space and time: A tool for space-time rainfall analysis and nowcasting, Hydrol. Earth Syst. Sci. 9 (2005), no. 3, 127–137. 10.5194/hess-9-127-2005Search in Google Scholar

[13] M. D. van der Walt, Empirical mode decomposition with shape-preserving spline interpolation, Results Appl. Math. 5 (2020), Article ID 100086. 10.1016/j.rinam.2019.100086Search in Google Scholar

[14] Z. Wu, N. E. Huang and X. Chen, The multi-dimensional ensemble empirical mode decomposition method, Adv. Adaptive Data Anal. 1 (2009), no. 3, 339–372. 10.1142/S1793536909000187Search in Google Scholar

[15] Y. Xu, B. Liu, J. Liu and S. Riemenschneider, Two-dimensional empirical mode decomposition by finite elements, Proc. Roy. Soc. A 462 (2006), no. 2074, 3081–3096. 10.1098/rspa.2006.1700Search in Google Scholar

Received: 2022-09-12
Accepted: 2023-01-21
Published Online: 2023-02-23

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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