Abstract

We propose a 2D generalization to the midpoint-based empirical mode decomposition algorithm (MBEMD). Unlike with the regular bidimensional empirical mode decomposition algorithm (BEMD), we do not interpolate the upper and lower envelopes, but rather directly find the mean envelope, utilizing well-defined points between two extrema of different kinds (midpoints). This approach has several advantages, such as improved spectral selectivity and better time performance over the regular BEMD process. The MBEMD algorithm is then applied to the task of the interferometric fringe pattern analysis, to identify its distinct components. This allows separating the oscillatory pattern component, which is of interest, from the background, noise, and possibly other spurious interferometric patterns. Such an enhancement is meant to aid further phase demodulation and reduce its errors. Flexibility of the adaptive method allows for processing correlation fringe patterns met in the digital speckle pattern interferometry as well as the regular interferometric fringe patterns without any special tuning of the algorithm.

© 2014 Optical Society of America

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    [CrossRef]
  5. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  6. J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
    [CrossRef]
  7. L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distribution by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [CrossRef]
  8. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20, 23463–23479 (2012).
    [CrossRef]
  14. M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
    [CrossRef]
  15. M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
    [CrossRef]
  16. Q. He, R. X. Gao, and P. Freedson, “Midpoint-based empirical decomposition for nonlinear trend estimation,” Proceedings of IEEE Engineering in Medicine and Biology Society (IEEE, 2009), pp. 2228–2231.
  17. Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
    [CrossRef]
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    [CrossRef]
  20. S. M. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2008), pp. 1313–1316.
  21. M. Wielgus, A. Antoniewicz, M. Bartyś, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion,” in Proceedings of 15th International Conference on Information Fusion FUSION 2012 (2012), pp. 649–654.
  22. K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19, 26065–26078 (2011).
    [CrossRef]
  23. P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
    [CrossRef]
  24. Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. A 460, 1597–1611 (2004).
    [CrossRef]
  25. M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt. 47, 2592–2598 (2008).
    [CrossRef]
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  27. L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
    [CrossRef]
  28. K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng. 36, 2010–2015 (1997).
    [CrossRef]

2014

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[CrossRef]

2012

2011

Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
[CrossRef]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19, 26065–26078 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Wielgus, “Information retrieval from amplitude modulated fringe patterns using single frame processing methods,” Proc. SPIE 8338, 833802 (2011).
[CrossRef]

2008

2007

2006

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

2005

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005).
[CrossRef]

2004

P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
[CrossRef]

Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. A 460, 1597–1611 (2004).
[CrossRef]

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

2003

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

A. Federico and G. H. Kaufmann, “Phase retrieval in digital speckle pattern interferometry by use of a smoothed space-frequency distribution,” Appl. Opt. 42, 7066–7071 (2003).
[CrossRef]

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

1999

1998

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

1997

K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng. 36, 2010–2015 (1997).
[CrossRef]

Adhami, R. R.

S. M. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2008), pp. 1313–1316.

Antoniewicz, A.

M. Wielgus, A. Antoniewicz, M. Bartyś, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion,” in Proceedings of 15th International Conference on Information Fusion FUSION 2012 (2012), pp. 649–654.

Barnes, T. H.

Bartys, M.

M. Wielgus, A. Antoniewicz, M. Bartyś, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion,” in Proceedings of 15th International Conference on Information Fusion FUSION 2012 (2012), pp. 649–654.

Bernini, M. B.

Bhuiyan, S. M.

S. M. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2008), pp. 1313–1316.

Bouaoune, Y.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Bunel, Ph.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Burton, D. R.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Coskun, E.

Damerval, C.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005).
[CrossRef]

Dean, T.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

Delechelle, E.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Federico, A.

Flandrin, P.

P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
[CrossRef]

Freedson, P.

Q. He, R. X. Gao, and P. Freedson, “Midpoint-based empirical decomposition for nonlinear trend estimation,” Proceedings of IEEE Engineering in Medicine and Biology Society (IEEE, 2009), pp. 2228–2231.

Gao, R. X.

Q. He, R. X. Gao, and P. Freedson, “Midpoint-based empirical decomposition for nonlinear trend estimation,” Proceedings of IEEE Engineering in Medicine and Biology Society (IEEE, 2009), pp. 2228–2231.

Gdeisat, M. A.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Goktas, H.

Goncalves, P.

P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts, 2007).

Gorecki, C.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

He, Q.

Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
[CrossRef]

Q. He, R. X. Gao, and P. Freedson, “Midpoint-based empirical decomposition for nonlinear trend estimation,” Proceedings of IEEE Engineering in Medicine and Biology Society (IEEE, 2009), pp. 2228–2231.

Huang, N. E.

Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. A 460, 1597–1611 (2004).
[CrossRef]

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Jacobelli, A.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

Jozwik, M.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

Kacperski, J.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

Kaufmann, G. H.

Kemao, Q.

Khan, J. F.

S. M. Bhuiyan, R. R. Adhami, and J. F. Khan, “A novel approach of fast and adaptive bidimensional empirical mode decomposition,” Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2008), pp. 1313–1316.

Kocahan, O.

Kong, F.

Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
[CrossRef]

Lalor, M. J.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Liu, H. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Liu, Y.

Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
[CrossRef]

Long, S. R.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Meignen, S.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005).
[CrossRef]

Niang, O.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Nunes, J. C.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Olszak, A.

K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng. 36, 2010–2015 (1997).
[CrossRef]

Ozder, S.

Patorski, K.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[CrossRef]

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20, 23463–23479 (2012).
[CrossRef]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Wielgus, “Information retrieval from amplitude modulated fringe patterns using single frame processing methods,” Proc. SPIE 8338, 833802 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19, 26065–26078 (2011).
[CrossRef]

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng. 36, 2010–2015 (1997).
[CrossRef]

Perrier, V.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005).
[CrossRef]

Podoleanu, A. Gh.

Pokorski, K.

K. Patorski, K. Pokorski, and M. Wielgus, “Information retrieval from amplitude modulated fringe patterns using single frame processing methods,” Proc. SPIE 8338, 833802 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19, 26065–26078 (2011).
[CrossRef]

Putz, B.

M. Wielgus, A. Antoniewicz, M. Bartyś, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion,” in Proceedings of 15th International Conference on Information Fusion FUSION 2012 (2012), pp. 649–654.

Reid, G.

D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics, 1993).

Rilling, G.

P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
[CrossRef]

Robinson, D. W.

D. W. Robinson and G. Reid, Interferogram Analysis: Digital Fringe Pattern Measurement (Institute of Physics, 1993).

Salbut, L.

L. Salbut, K. Patorski, M. Jozwik, J. Kacperski, C. Gorecki, A. Jacobelli, and T. Dean, “Active micro-elements testing by interferometry using time-average and quasi-stroboscopic techniques,” Proc. SPIE 5145, 23–32 (2003).
[CrossRef]

Schwider, J.

J. Schwider, “Advanced evaluation techniques in interferometry,” in Progress in Optics, E. Wolf, ed. (Elsevier, 1990), Vol. 28, pp. 271–359.

Servin, M.

D. Malacara, M. Servin, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

Sheng, Z.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Shih, W. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Tan, S. M.

Trusiak, M.

Tung, C. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Watkins, L. R.

Weng, J.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Wielgus, M.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[CrossRef]

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20, 23463–23479 (2012).
[CrossRef]

M. Wielgus and K. Patorski, “Evaluation of amplitude encoded fringe patterns using the bidimensional empirical mode decomposition and the 2D Hilbert transform generalizations,” Appl. Opt. 50, 5513–5523 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Wielgus, “Information retrieval from amplitude modulated fringe patterns using single frame processing methods,” Proc. SPIE 8338, 833802 (2011).
[CrossRef]

M. Wielgus, A. Antoniewicz, M. Bartyś, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion,” in Proceedings of 15th International Conference on Information Fusion FUSION 2012 (2012), pp. 649–654.

Wu, M. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Wu, Z.

Z. Wu and N. E. Huang, “A study of the characteristics of white noise using the empirical mode decomposition method,” Proc. R. Soc. A 460, 1597–1611 (2004).
[CrossRef]

Yang, T.

Yang, Z.

Yen, N. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Zeng, Q.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. H. Shih, Q. Zeng, N. C. Yen, C. C. Tung, and H. H. Liu, “The empirical mode decomposition and the Hilbert spectrum for non-linear and nonstationary time series analysis,” Proc. R. Soc. A 454, 903–995 (1998).
[CrossRef]

Zhao, H.

Zhong, J.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

Zhou, X.

Appl. Opt.

IEEE Signal Process. Lett.

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett. 12, 701–704 (2005).
[CrossRef]

P. Flandrin, G. Rilling, and P. Goncalves, “Empirical mode decomposition as a filterbank,” IEEE Signal Process. Lett. 11, 112–114 (2004).
[CrossRef]

Image Vis. Comput.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and Ph. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput. 21, 1019–1026 (2003).

Meas. Sci. Technol.

Q. He, Y. Liu, and F. Kong, “Machine fault signature analysis by midpoint based empirical mode decomposition,” Meas. Sci. Technol. 22, 015702 (2011).
[CrossRef]

Opt. Commun.

M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Eliminating the zero spectrum in Fourier transform profilometry using two-dimensional continuous wavelet transform,” Opt. Commun. 266, 482–489 (2006).
[CrossRef]

Opt. Eng.

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895–899 (2004).
[CrossRef]

K. Patorski and A. Olszak, “Digital in-plane electronic speckle pattern shearing interferometry,” Opt. Eng. 36, 2010–2015 (1997).
[CrossRef]

Opt. Express

Opt. Lasers Eng.

M. Trusiak, M. Wielgus, and K. Patorski, “Advanced processing of optical fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Opt. Lasers Eng. 52, 230–240 (2014).
[CrossRef]

Opt. Lett.

Proc. R. Soc. A

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[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of the midpoint extraction method.

Fig. 2.
Fig. 2.

Modulus of the Fourier spectrum of the second IMF of the white noise for (a) BEMD and (b) MBEMD.

Fig. 3.
Fig. 3.

IMF spectral power radial density comparison for the BEMD and MBEMD algorithms.

Fig. 4.
Fig. 4.

Fringe pattern with (a) the nonlinear background, (b) nonlinear background, (c) BEMD estimation, and (d) MBEMD estimation.

Fig. 5.
Fig. 5.

(a) Fringe pattern with noise, (b) noiseless fringe pattern, (c) BEMD estimation, and (d) MBEMD estimation.

Fig. 6.
Fig. 6.

(a) ESPI image, (b) noiseless corresponding fringe pattern, (c) BEMD estimation, and (d) MBEMD estimation.

Fig. 7.
Fig. 7.

(a) Synthetic pattern spoiled with noise and background, (b) ideal fringe pattern, (c) background component, (d)–(g) BEMD decomposition, (h)–(m) MBEMD decomposition, (n) fringe pattern restored from the BEMD decomposition, and (o) fringe pattern restored from the MBEMD decomposition.

Fig. 8.
Fig. 8.

(a) Experimental interferogram of the vibrating micromembrane, (b) MBEMD background estimation, (c) MBEMD fringe component estimation, and (d) fringe pattern modulation calculated with the Hilbert transform using MBEMD fringe component estimation.

Fig. 9.
Fig. 9.

(a) Experimental DSPI pattern and (b) the pattern denoised with the MBEMD procedure.

Equations (10)

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x|Tij(x)Tij+1(x)|2x|Tij(x)|2<C,
s(x)=rN(x)+k=1NIMFk(x).
ve+f=2,
2e=3(f1)+δ,
e=3v3δ,
e2[v2,32v3].
m^=32v=32|AB|.
Gk(ω)=ω0ω02π|IMF^k(ω,φ)|2ωdφdω=ω02π|IMF^k(ω,φ)|2dφ.
I(x)=b(x)+A(x)cos[ϕ(x)]+n(x).
I(x)|sin[ψ(x)+Δϕ(x)2]||sin[Δϕ(x)2]|,

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