Abstract

Single-shot crossed-type fringe pattern processing and analysis method called Hilbert-Huang grating interferometry (HHGI) is proposed. It consist of three main procedures: (1) crossed pattern is resolved into two fringe families using novel orthogonal empirical mode decomposition approach, (2) separated fringe sets are filtered using modified automatic selective reconstruction aided by enhanced fast empirical mode decomposition and mutual information detrending, and (3) Hilbert spiral transform is employed for fringe phase demodulation. Numerical and experimental studies corroborate the validity, versatility and robustness of the proposed HHGI technique. It can be successfully applied to multiplicative and additive type crossed patterns with sinusoidal and binary orthogonal component structures. Efficient adaptive filtering enables successful fast processing and analysis of complex and defected patterns.

© 2013 Optical Society of America

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2014 (1)

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]

2013 (5)

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt.52(26), 6537–6542 (2013).
[CrossRef] [PubMed]

Y. Zhou, S.-T. Zhou, Z.-Y. Zhong, and H.-G. Li, “A de-illumination scheme for face recognition based on fast decomposition and detail feature fusion,” Opt. Express21(9), 11294–11308 (2013).
[CrossRef] [PubMed]

Z. Yang, B. W.-K. Ling, and C. Bingham, “Trend extraction based on separations of consecutive empirical mode decomposition components in Hilbert marginal spectrum,” Measurement46(8), 2481–2491 (2013).
[CrossRef]

S. Osman and W. Wang, “An enhanced Hilbert-Huang transform technique for bearing condition monitoring,” Meas. Sci. Technol.24(8), 085004 (2013).
[CrossRef]

2012 (6)

2011 (7)

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Phase measurement in temporal speckle pattern interferometry signals presenting low-modulated regions by means of the bidimensional empirical mode decomposition,” Appl. Opt.50(5), 641–647 (2011).
[CrossRef] [PubMed]

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(28), 5513–5523 (2011).
[CrossRef] [PubMed]

Y. Zhou and H. Li, “Adaptive noise reduction method for DSPI fringes based on bi-dimensional ensemble empirical mode decomposition,” Opt. Express19(19), 18207–18215 (2011).
[CrossRef] [PubMed]

H. Itoh, K. Nagai, G. Sato, K. Yamaguchi, T. Nakamura, T. Kondoh, C. Ouchi, T. Teshima, Y. Setomoto, and T. Den, “Two-dimensional grating-based X-ray phase-contrast imaging using Fourier transform phase retrieval,” Opt. Express19(4), 3339–3346 (2011).
[CrossRef] [PubMed]

K. S. Morgan, D. M. Paganin, and K. K. Siu, “Quantitative single-exposure x-ray phase contrast imaging using a single attenuation grid,” Opt. Express19(20), 19781–19789 (2011).
[CrossRef] [PubMed]

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
[CrossRef]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express19(27), 26065–26078 (2011).
[CrossRef] [PubMed]

2010 (3)

H. H. Wen, E. E. Bennett, R. Kopace, A. F. Stein, and V. Pai, “Single-shot x-ray differential phase-contrast and diffraction imaging using two-dimensional transmission gratings,” Opt. Lett.35(12), 1932–1934 (2010).
[CrossRef] [PubMed]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
[CrossRef] [PubMed]

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
[CrossRef]

2009 (4)

2008 (3)

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(21), 728356 (2008).
[CrossRef]

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[CrossRef]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Noise reduction in digital speckle pattern interferometry using bidimensional empirical mode decomposition,” Appl. Opt.47(14), 2592–2598 (2008).
[CrossRef] [PubMed]

2007 (1)

Z. Wu, N. E. Huang, S. R. Long, and C. K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proc. Natl. Acad. Sci. U.S.A.104(38), 14889–14894 (2007).
[CrossRef] [PubMed]

2005 (1)

C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

2003 (1)

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

2002 (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

2001 (2)

2000 (1)

1999 (3)

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng.38(6), 974–982 (1999).
[CrossRef]

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE3744, 231–240 (1999).
[CrossRef]

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit.32(1), 71–86 (1999).
[CrossRef]

1998 (2)

H. Canabal, J. A. Quiroga, and E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt.37(26), 6227–6233 (1998).
[CrossRef] [PubMed]

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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

1997 (1)

J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng.36(8), 2240–2248 (1997).
[CrossRef]

1996 (1)

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

1995 (1)

M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng.34(8), 2459–2466 (1995).
[CrossRef]

1986 (4)

1983 (1)

1980 (1)

1972 (2)

A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun.4(5), 326–328 (1972).
[CrossRef]

D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt.11(11), 2613–2624 (1972).
[CrossRef] [PubMed]

1971 (3)

1964 (1)

Adhami, R. R.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
[CrossRef]

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(21), 728356 (2008).
[CrossRef]

Antoniewicz, A.

M. Wielgus, M. Bartys, A. Antoniewicz, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion, ” in Proc. 15th Int. Conf. Inform. Fusion (FUSION,2012), 649–654.

Attoh-Okine, N. O.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
[CrossRef]

Ayenu-Prah, A. Y.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
[CrossRef]

Barber, C. B.

C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

Barner, K. E.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
[CrossRef]

Bartys, M.

M. Wielgus, M. Bartys, A. Antoniewicz, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion, ” in Proc. 15th Int. Conf. Inform. Fusion (FUSION,2012), 649–654.

Bar-Ziv, E.

Bennett, E. E.

Bernabeu, E.

J. A. Quiroga, D. Crespo, and E. Bernabeu, “Fourier transform method for automatic processing of moiré deflectograms,” Opt. Eng.38(6), 974–982 (1999).
[CrossRef]

H. Canabal and E. Bernabeu, “Phase extraction methods for analysis of crossed fringe patterns,” Proc. SPIE3744, 231–240 (1999).
[CrossRef]

H. Canabal, J. A. Quiroga, and E. Bernabeu, “Improved phase-shifting method for automatic processing of moiré deflectograms,” Appl. Opt.37(26), 6227–6233 (1998).
[CrossRef] [PubMed]

Bernini, M. B.

Berujon, S.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
[CrossRef] [PubMed]

Bhuiyan, S. M. A.

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
[CrossRef]

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Bouaoune, Y.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
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Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
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Bunel, P.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
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J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng.36(8), 2240–2248 (1997).
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C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
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H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
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S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
[CrossRef] [PubMed]

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
[CrossRef]

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J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
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Denecke, M.

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
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S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
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I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
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Hawkes, D. J.

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit.32(1), 71–86 (1999).
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Hill, D. L. G.

C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit.32(1), 71–86 (1999).
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Z. Wu, N. E. Huang, S. R. Long, and C. K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proc. Natl. Acad. Sci. U.S.A.104(38), 14889–14894 (2007).
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C. B. Barber, D. P. Dobkin, and H. Huhdanpaa, “The quickhull algorithm for convex hulls,” ACM Trans. Math. Softw.22(4), 469–483 (1996).
[CrossRef]

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Jia, S.

Jiang, T.

Kafri, O.

Kaufmann, G. H.

Keren, E.

Khan, J. F.

S. M. A. Bhuiyan, R. R. Adhami, and J. F. Khan, “Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation,” EURASIP J. Adv. Signal Process.2008(21), 728356 (2008).
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Kopace, R.

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Li, H.

Li, H.-G.

Li, Z.-H.

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Z. Yang, B. W.-K. Ling, and C. Bingham, “Trend extraction based on separations of consecutive empirical mode decomposition components in Hilbert marginal spectrum,” Measurement46(8), 2481–2491 (2013).
[CrossRef]

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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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun.4(5), 326–328 (1972).
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A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun.2(9), 413–415 (1971).
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Z. Wu, N. E. Huang, S. R. Long, and C. K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proc. Natl. Acad. Sci. U.S.A.104(38), 14889–14894 (2007).
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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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
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Nagai, K.

Nakamura, T.

Niang, O.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
[CrossRef]

Nunes, J. C.

J. C. Nunes, Y. Bouaoune, E. Delechelle, O. Niang, and P. Bunel, “Image analysis by bidimensional empirical mode decomposition,” Image Vis. Comput.21(12), 1019–1026 (2003).
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H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
[CrossRef] [PubMed]

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]

K. Pokorski and K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt.51(35), 8433–8439 (2012).
[CrossRef] [PubMed]

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express20(21), 23463–23479 (2012).
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Z. Wu, N. E. Huang, S. R. Long, and C. K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proc. Natl. Acad. Sci. U.S.A.104(38), 14889–14894 (2007).
[CrossRef] [PubMed]

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C. Damerval, S. Meignen, and V. Perrier, “A fast algorithm for bidimensional EMD,” IEEE Signal Process. Lett.12(10), 701–704 (2005).
[CrossRef]

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M. Pirga and M. Kujawinska, “Two directional spatial-carrier phase-shifting method for analysis of crossed and closed fringe patterns,” Opt. Eng.34(8), 2459–2466 (1995).
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Pokorski, K.

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M. Wielgus, M. Bartys, A. Antoniewicz, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion, ” in Proc. 15th Int. Conf. Inform. Fusion (FUSION,2012), 649–654.

Quiroga, J. A.

Rilling, G.

G. Rilling and P. Flandrin, “One or two frequencies? The empirical mode decomposition answers,” IEEE Trans. Signal Process.56(1), 85–95 (2008).
[CrossRef]

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H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
[CrossRef] [PubMed]

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
[CrossRef] [PubMed]

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
[CrossRef]

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Sawhney, K.

H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
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J. Schmit, K. Patorski, and K. Creath, “Simultaneous registration of in- and out-of-plane displacements in modified grating interferometry,” Opt. Eng.36(8), 2240–2248 (1997).
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Setomoto, Y.

Sgulim, S.

Shen, Z.

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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
[CrossRef]

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A. W. Lohmann and D. E. Silva, “A Talbot interferometer with circular gratings,” Opt. Commun.4(5), 326–328 (1972).
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D. E. Silva, “Talbot interferometer for radial and lateral derivatives,” Appl. Opt.11(11), 2613–2624 (1972).
[CrossRef] [PubMed]

A. W. Lohmann and D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun.2(9), 413–415 (1971).
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Song, Y.

Stein, A. F.

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C. Studholme, D. L. G. Hill, and D. J. Hawkes, “An overlap invariant entropy measure of 3D medical image alignment,” Pattern Recognit.32(1), 71–86 (1999).
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Suzuki, T.

Teshima, T.

Trusiak, M.

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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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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Walker, C. T.

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
[CrossRef]

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H. Wang, S. Berujon, I. Pape, S. Rutishauser, C. David, and K. Sawhney, “At-wavelength metrology using the moiré fringe analysis method based on a two-dimensional grating interferometer,” Nucl. Instrum. Methods Phys. Res. A710, 78–81 (2013).
[CrossRef]

S. Berujon, H. Wang, I. Pape, K. Sawhney, S. Rutishauser, and C. David, “X-ray submicrometer phase contrast imaging with a Fresnel zone plate and a two dimensional grating interferometer,” Opt. Lett.37(10), 1622–1624 (2012).
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Wang, W.

S. Osman and W. Wang, “An enhanced Hilbert-Huang transform technique for bearing condition monitoring,” Meas. Sci. Technol.24(8), 085004 (2013).
[CrossRef]

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Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett.9(3), 81–84 (2002).
[CrossRef]

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Weitkamp, T.

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
[CrossRef]

I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
[CrossRef] [PubMed]

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
[CrossRef]

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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. Express20(21), 23463–23479 (2012).
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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(28), 5513–5523 (2011).
[CrossRef] [PubMed]

M. Wielgus, M. Bartys, A. Antoniewicz, and B. Putz, “Fast and adaptive bidimensional empirical mode decomposition for the real-time video fusion, ” in Proc. 15th Int. Conf. Inform. Fusion (FUSION,2012), 649–654.

Wu, M. C.

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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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Wu, Z.

Z. Wu, N. E. Huang, S. R. Long, and C. K. Peng, “On the trend, detrending, and variability of nonlinear and nonstationary time series,” Proc. Natl. Acad. Sci. U.S.A.104(38), 14889–14894 (2007).
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Yamaguchi, K.

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Yang, Z.

Z. Yang, B. W.-K. Ling, and C. Bingham, “Trend extraction based on separations of consecutive empirical mode decomposition components in Hilbert marginal spectrum,” Measurement46(8), 2481–2491 (2013).
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Zanette, I.

S. Rutishauser, I. Zanette, T. Weitkamp, T. Donath, and C. David, “At-wavelength characterization of refractive x-ray lenses using a two-dimensional grating interferometer,” Appl. Phys. Lett.99(22), 221104 (2011).
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I. Zanette, T. Weitkamp, T. Donath, S. Rutishauser, and C. David, “Two-dimensional x-ray grating interferometer,” Phys. Rev. Lett.105(24), 248102 (2010).
[CrossRef] [PubMed]

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
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Zhao, H.

Zheng, Q.

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 non-linear and non-stationary time series analysis,” Proc. R. Soc. Lond. A454(1971), 903–995 (1998).
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Zhou, S.-T.

Zhou, X.

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Adv. Adapt. Data Anal. (1)

S. M. A. Bhuiyan, N. O. Attoh-Okine, K. E. Barner, A. Y. Ayenu-Prah, and R. R. Adhami, “Bidimensional empirical mode decomposition using various interpolation techniques,” Adv. Adapt. Data Anal.1(2), 309–338 (2009).
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AIP Conf. Proc. (1)

I. Zanette, C. David, S. Rutishauser, T. Weitkamp, M. Denecke, and C. T. Walker, “2D grating simulation for X-ray phase-contrast and dark-field imaging with a Talbot interferometer,” AIP Conf. Proc.1221, 73–79 (2010).
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Figures (17)

Fig. 1
Fig. 1

Simulated vertical (a) and horizontal (b) fringe patterns and their additive superimposition (c).

Fig. 2
Fig. 2

Left - plot representing the relationship between coefficient a and Q index values for extracted patterns (blue line - horizontal, red line - vertical term); right - plot representing the relationship between normalized carrier frequency of both fringe families and Q index for extracted patterns (blue line - horizontal, red line - vertical term).

Fig. 3
Fig. 3

On the left - plot representing relations between coefficient a and Q index values for extracted patterns in the case of a noisy interferogram; on the right - plot representing relations between added white Gaussian noise variance and obtained Q index values for extracted patterns, see text for explanations.

Fig. 4
Fig. 4

Proposed algorithm diagram. Additive superimposition of noisy fringe sets is considered. The OEMD method is used for resolving two orthogonal fringe families, the ASR-EFEMD technique is applied for extracted single fringe pattern filtering and normalization, and the Hilbert spiral transform is employed for phase demodulation.

Fig. 5
Fig. 5

Phase error maps obtained in the case of vertical (a-b) and horizontal (c-d) fringe set extraction from noisy crossed pattern, Fig. 4(a), using regular OEMD method (a,c) and additional directional averaging (b,d). Additional directional filtering decreases the error level in the final phase distribution.

Fig. 6
Fig. 6

(a) Simulated simple fringe pattern with uneven background illumination, (b)-(j) first eight BIMFs and the residue obtained using the EFEMD method, (k) background intensity term calculated using the MID approach, (l) fringe pattern normalized using the ASR-EFEMD method aided by MID.

Fig. 7
Fig. 7

Two plots illustrating the functions (a) f(x) = MI(BIMFx, BIMFx + 1) and (b) g(x) = f(x + 1)/f(x) calculated using MID for BIMF set depicted in Fig. 6. We established the flag number equal to 4.

Fig. 8
Fig. 8

(a) Simulated multiplicative superimposition type crossed pattern, (b) vertical fringes extracted using OEMD with additional directional filtering, note strong uneven background term, (c) pattern (b) after normalization, (d) background intensity term extracted from (b) using MID, (e) vertical fringes enhanced and normalized with ASR-EFEMD aided by MID (with uneven background term removed), (f) horizontal fringe set extracted from (a).

Fig. 9
Fig. 9

(a) Multiplicative type crossed pattern spoiled with white Gaussian noise of variance 0.02, (b) vertical and (c) horizontal fringe families extracted using OEMD with additional directional filtering (a = 7).

Fig. 10
Fig. 10

(a) Simulated additive type binary crossed pattern, (b) vertical binary fringes extracted using the OEMD method, (c) cosinusoidal vertical fringes obtained filtering out high frequency BIMFs, (d) horizontal binary fringes extracted using OEMD, (e) cosinusoidal horizontal fringes obtained filtering out high frequency BIMFs, (f) unwrapped phase map of (e) - the final outcome of the HHGI algorithm.

Fig. 11
Fig. 11

(a) Simulated multiplicative type binary crossed pattern, (b) vertical binary fringes extracted using the OEMD method, (c) cosinusoidal vertical fringes obtained filtering out high frequency BIMFs and the background term with MID, (d) horizontal binary fringes extracted using OEMD, (e) cosinusoidal horizontal fringes obtained filtering out high frequency BIMFs, (f) unwrapped phase map of (e) - the final outcome of the HHGI algorithm.

Fig. 12
Fig. 12

(a) Experimental crossed interferogram obtained in the grating (moiré) interferometry setup (310x410 pixels image corresponding to the 4,5x6 mm2 area of the sample under test; sample with transferred reflective diffraction grating 1200 lines/mm; carrier fringes introduced by tilting illumination beams [33]), (b) BIMF containing vertical fringes extracted from (a) using the OEMD algorithm (a = 10), (c) vertical fringes enhanced and normalized using modified ASR-EFEMD, (d) wrapped phase fringes obtained using Hilbert transform, (e-g) analogous processing results for the horizontal fringe set.

Fig. 13
Fig. 13

Unwrapped phase maps corresponding to the u(x,y) displacement field obtained using the proposed HHGI approach (a) and the 5-frame temporal phase shift method (b); plot illustrating cross-sections along 50th row of the u(x,y) displacement field obtained using HHGI (red line) and TPS (blue line) methods (c). Unwrapped phase maps corresponding to the v(x,y) displacement field obtained using HHGI scheme (d) and TPS (e); (f) plot illustrating the cross-section along 200th row of the v(x,y) displacement field obtained using HHGI (red line) and TPS (blue line).

Fig. 14
Fig. 14

Continuous Wavelet Transform processing results: extracted horizontal fringe set (a) and its unwrapped phase distribution with carrier term removed (b); extracted vertical fringe set (c) and its unwrapped phase distribution with carrier term removed (d).

Fig. 15
Fig. 15

(a) Recorded self-image (750x750 pixels) of crossed binary grating (two 8 lines/mm binary gratings with orthogonal lines superimposed multiplicatively); extracted (b) vertical and (c) horizontal fringe families using OEMD - observe contrast variations corresponding to the uneven background illumination distribution of (a); (d)-(e) two fringe families normalized (note that the proposed method is robust to strong background illumination variations and applicable to binary structures); (f-g) two sinusoidal detecting gratings (reference patterns) with fringe spacing designed to match the spatial frequency of information carrying orthogonal terms (d-e). Numerically simulated reference gratings will be used for generating digital moiré fringes.

Fig. 16
Fig. 16

Top: two moiré pattern pairs with pi/2 phase shift generated by multiplicative superimposition of the extracted term and simulated detecting (reference) gratings, Figs. 15(d)15(g); moiré phase shifting was obtained by numerical error-free phase shifting of the reference grating. Bottom: moiré fringes filtered using the EFEMD decomposition and Hilbert transform normalization.

Fig. 17
Fig. 17

Wrapped moiré phase maps obtained using well-established analytic signal tool: (a) calculated from Figs. 16(e) and 16(f) and (b) calculated from Figs. 16(g) and 16(h); unwrapped phase maps obtained from (c) vertical and (d) horizontal moiré fringes - note characteristic distributions of the orthogonal spatial derivatives of the spherical wave front under test.

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