Abstract

We propose an application for a bidimensional empirical mode decomposition and a Hilbert transform algorithm (BEMD-HT) in processing amplitude modulated fringe patterns. In numerical studies we investigate the influence of parameters of the algorithm and a fringe pattern under study on the demodulation results to optimize the procedure. A spiral phase method and the angle-oriented partial Hilbert transform are introduced to the BEMD-HT and tested. A postprocessing filtration method for BEMD-HT is proposed. Results of processing experimental data, such as vibration mode patterns obtained by time-average interferometry, correspond richly with numerical findings. They compare very well with the results of our previous investigations using the temporal phase-shifting (TPS) method and the continuous wavelet transform (CWT). Not needing to perform phase-shifting represents significant simplification of the experimental procedure in comparison with the TPS method.

© 2011 Optical Society of America

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2011

2010

K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651(2010).
[CrossRef] [PubMed]

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2010).
[CrossRef]

2009

2008

2007

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46, 4613–4624(2007).
[CrossRef] [PubMed]

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).
[CrossRef]

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Commun. 45, 723–729 (2007).

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

2006

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45, 085602 (2006).
[CrossRef]

2005

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44, 065601 (2005).
[CrossRef]

M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).
[CrossRef]

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

2003

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE 5145, 1–16(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]

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

2001

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1871–18810 (2001).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[CrossRef]

T. Bülow and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[CrossRef]

1998

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

1996

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

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

1994

1992

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal,” Proc. IEEE 80, 520–568(1992).
[CrossRef]

1987

1976

1973

B. Chatelain, “Holographic photo-elasticity: independent observation of the isochromatic and isopachic fringes for a single model subjected to only one process,” Opt. Laser Technol. 5, 201–204 (1973).
[CrossRef]

1971

R. J. Sanford and A. J. Durelli, “Interpretation of fringes in stress-holo-interferometry,” Exp. Mech. 11, 161–166 (1971).
[CrossRef]

J. D. Hovanesian and Y. Hung, “Moiré contour-sum, contour difference and vibration analysis of arbitrary objects,” Appl. Opt. 10, 2734–2738 (1971).
[CrossRef] [PubMed]

1964

M. Nishida and H. Saito, “A new interferometric method of two-dimensional stress analysis,” Exp. Mech. 4, 366–376(1964).
[CrossRef]

1932

Barber, C.

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

Bernini, M.

Bernini, M. B.

Boashash, B.

B. Boashash, “Estimating and interpreting the instantaneous frequency of a signal,” Proc. IEEE 80, 520–568(1992).
[CrossRef]

Bone, D. J.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1871–18810 (2001).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[CrossRef]

Bosseboeuf, A.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE 5145, 1–16(2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[CrossRef]

Bouaoune, Y.

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

Brunning, J. H.

J. E. Greivenkamp and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara ed. (Wiley, 1992), pp. 501–598.

Bryngdahl, O.

Bülow, T.

T. Bülow and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

T. Bülow, D. Pallek, and G. Sommer, “Riesz transforms for the isotropic estimation of the local phase of moiré interferograms,” presented at the 22nd DAGM Symposium Mustererkennung, (Heidelberg, 2000).

Bunel, .Ph.

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

Chatelain, B.

B. Chatelain, “Holographic photo-elasticity: independent observation of the isochromatic and isopachic fringes for a single model subjected to only one process,” Opt. Laser Technol. 5, 201–204 (1973).
[CrossRef]

Czarnek, R.

Damerval, C.

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

Danaie, K.

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[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 Vision Comput. 21, 1019–1026 (2003).
[CrossRef]

Dobkin, D.

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

Durelli, A. J.

R. J. Sanford and A. J. Durelli, “Interpretation of fringes in stress-holo-interferometry,” Exp. Mech. 11, 161–166 (1971).
[CrossRef]

Equis, S.

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2010).
[CrossRef]

S. Equis and P. Jacquot, “The empirical mode decomposition: a must have tool in speckle interferometry?” Opt. Express 17, 611–623 (2009).
[CrossRef] [PubMed]

Federico, A.

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

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48, 6862–6869 (2009).
[CrossRef] [PubMed]

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] [PubMed]

A. Federico and G. H. Kaufmann, “Phase recovery in temporal speckle pattern interferometry using the generalized S-transform,” Opt. Lett. 33, 866–868 (2008).
[CrossRef] [PubMed]

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Hilbert transform analysis of a time series of speckle interferograms with a temporal carrier,” Appl. Opt. 47, 1310–1316(2008).
[CrossRef]

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Commun. 45, 723–729 (2007).

Felsberg, M.

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

Freedson, P.

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

Fu, S.

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).
[CrossRef]

Galizzi, G. E.

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Commun. 45, 723–729 (2007).

Gao, R.

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

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]

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara ed. (Wiley, 1992), pp. 501–598.

Guo, Y.

He, Q.

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

Hovanesian, J. D.

Huang, N.

Z. Wu and N. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Tech. Rep. No. 193, Centre for Ocean-Land-Atmosphere Studies(2005).

Huang, N. E.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Huhdanpaa, H.

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

Hung, Y.

Ishii, Y.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Fringe 2005, the 5th International Workshop on Automatic Processing of Fringe Patterns, W.Osten ed. (Springer, 2006), pp. 82–89.

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]

Jacquot, P.

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2010).
[CrossRef]

S. Equis and P. Jacquot, “The empirical mode decomposition: a must have tool in speckle interferometry?” Opt. Express 17, 611–623 (2009).
[CrossRef] [PubMed]

Jiang, T.

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.

Kaufmann, G. H.

Larkin, K. G.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1871–18810 (2001).
[CrossRef]

K. G. Larkin, “Efficient nonlinear algorithm for envelope detection in white light interferometry,” J. Opt. Soc. Am. A 13, 832–843 (1996).
[CrossRef]

Liu, H. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Liu, Z.

Z. Liu, H. Wang, and S. Peng, “Texture classification through empirical mode decomposition,” in Proceedings of the 17th International Conference on Pattern Recognition (ICPR 2004) (IEEE, 2004), pp. 803–806.

Long, S. R.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. 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).

Marengo-Rodriguez, F. A.

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Hilbert transform analysis of a time series of speckle interferograms with a temporal carrier,” Appl. Opt. 47, 1310–1316(2008).
[CrossRef]

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

Maskeliunas, R.

M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).
[CrossRef]

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 Vision Comput. 21, 1019–1026 (2003).
[CrossRef]

Nishida, M.

M. Nishida and H. Saito, “A new interferometric method of two-dimensional stress analysis,” Exp. Mech. 4, 366–376(1964).
[CrossRef]

Nunes, J. C.

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

Oldfield, M. A.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1871–18810 (2001).
[CrossRef]

Onodera, R.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Fringe 2005, the 5th International Workshop on Automatic Processing of Fringe Patterns, W.Osten ed. (Springer, 2006), pp. 82–89.

Osterberg, H.

Pallek, D.

T. Bülow, D. Pallek, and G. Sommer, “Riesz transforms for the isotropic estimation of the local phase of moiré interferograms,” presented at the 22nd DAGM Symposium Mustererkennung, (Heidelberg, 2000).

Patorski, K.

K. Patorski and K. Pokorski, “Examination of singular scalar fields using wavelet processing of fork fringes,” Appl. Opt. 50, 773–781 (2011).
[CrossRef] [PubMed]

K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651(2010).
[CrossRef] [PubMed]

A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE 7063, 70630P (2008).
[CrossRef]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46, 4613–4624(2007).
[CrossRef] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45, 085602 (2006).
[CrossRef]

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44, 065601 (2005).
[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, D. Post, R. Czarnek, and Y. Guo, “Real-time optical differentiation for moire interferometry,” Appl. Opt. 26, 1977–1982 (1987).
[CrossRef] [PubMed]

K. Patorski, Handbook of the Moiré Fringe Technique(Elsevier, 1993).

Peng, S.

Z. Liu, H. Wang, and S. Peng, “Texture classification through empirical mode decomposition,” in Proceedings of the 17th International Conference on Pattern Recognition (ICPR 2004) (IEEE, 2004), pp. 803–806.

Perrier, V.

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

Petitgrand, S.

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE 5145, 1–16(2003).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[CrossRef]

Pokorski, K.

Post, D.

Ragulskis, M.

M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).
[CrossRef]

Rosvold, G.

Saito, H.

M. Nishida and H. Saito, “A new interferometric method of two-dimensional stress analysis,” Exp. Mech. 4, 366–376(1964).
[CrossRef]

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]

Sanford, R. J.

R. J. Sanford and A. J. Durelli, “Interpretation of fringes in stress-holo-interferometry,” Exp. Mech. 11, 161–166 (1971).
[CrossRef]

Schwider, J.

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

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. 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. A 454, 903–995 (1998).
[CrossRef]

Shih, W. H.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Sienicki, Z.

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44, 065601 (2005).
[CrossRef]

Sommer, G.

T. Bülow and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

T. Bülow, D. Pallek, and G. Sommer, “Riesz transforms for the isotropic estimation of the local phase of moiré interferograms,” presented at the 22nd DAGM Symposium Mustererkennung, (Heidelberg, 2000).

Styk, A.

A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE 7063, 70630P (2008).
[CrossRef]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46, 4613–4624(2007).
[CrossRef] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45, 085602 (2006).
[CrossRef]

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44, 065601 (2005).
[CrossRef]

Tung, C. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Turla, V.

M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).
[CrossRef]

Venouziou, M.

M. Venouziou and H. Zhang, “Characterizing the Hilbert transform by the Bedrosian theorem,” J. Math. Anal. Appl. 338, 1477–1481 (2008).
[CrossRef]

Wang, H.

Z. Liu, H. Wang, and S. Peng, “Texture classification through empirical mode decomposition,” in Proceedings of the 17th International Conference on Pattern Recognition (ICPR 2004) (IEEE, 2004), pp. 803–806.

Wu, M. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Wu, Z.

Z. Wu and N. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Tech. Rep. No. 193, Centre for Ocean-Land-Atmosphere Studies(2005).

Yahiaoui, R.

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[CrossRef]

Yamamoto, Y.

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Fringe 2005, the 5th International Workshop on Automatic Processing of Fringe Patterns, W.Osten ed. (Springer, 2006), pp. 82–89.

Yang, X.

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).
[CrossRef]

Yen, N. C.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Yu, Q.

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).
[CrossRef]

Zhang, H.

M. Venouziou and H. Zhang, “Characterizing the Hilbert transform by the Bedrosian theorem,” J. Math. Anal. Appl. 338, 1477–1481 (2008).
[CrossRef]

Zhao, H.

Zheng, Q.

N. E. Huang, Z. Sheng, S. R. Long, M. C. Wu, W. 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. A 454, 903–995 (1998).
[CrossRef]

Zhou, X.

ACM Trans. Math. Softw.

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Appl. Opt.

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] [PubMed]

M. B. Bernini, A. Federico, and G. H. Kaufmann, “Normalization of fringe patterns using the bidimensional empirical mode decomposition and the Hilbert transform,” Appl. Opt. 48, 6862–6869 (2009).
[CrossRef] [PubMed]

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

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[CrossRef] [PubMed]

A. Styk and K. Patorski, “Analysis of systematic errors in spatial carrier phase shifting applied to interferogram intensity modulation determination,” Appl. Opt. 46, 4613–4624(2007).
[CrossRef] [PubMed]

K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49, 3640–3651(2010).
[CrossRef] [PubMed]

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Hilbert transform analysis of a time series of speckle interferograms with a temporal carrier,” Appl. Opt. 47, 1310–1316(2008).
[CrossRef]

G. Rosvold, “Video-based vibration analysis using projected fringes,” Appl. Opt. 33, 775–786 (1994).
[CrossRef] [PubMed]

K. Patorski and K. Pokorski, “Examination of singular scalar fields using wavelet processing of fork fringes,” Appl. Opt. 50, 773–781 (2011).
[CrossRef] [PubMed]

K. Patorski, D. Post, R. Czarnek, and Y. Guo, “Real-time optical differentiation for moire interferometry,” Appl. Opt. 26, 1977–1982 (1987).
[CrossRef] [PubMed]

Exp. Mech.

M. Nishida and H. Saito, “A new interferometric method of two-dimensional stress analysis,” Exp. Mech. 4, 366–376(1964).
[CrossRef]

R. J. Sanford and A. J. Durelli, “Interpretation of fringes in stress-holo-interferometry,” Exp. Mech. 11, 161–166 (1971).
[CrossRef]

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]

IEEE Trans. Signal Process.

T. Bülow and G. Sommer, “Hypercomplex signals—a novel extension of the analytic signal to the multidimensional case,” IEEE Trans. Signal Process. 49, 2844–2852 (2001).
[CrossRef]

M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Trans. Signal Process. 49, 3136–3144 (2001).
[CrossRef]

Image Vision Comput.

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

J. Math. Anal. Appl.

M. Venouziou and H. Zhang, “Characterizing the Hilbert transform by the Bedrosian theorem,” J. Math. Anal. Appl. 338, 1477–1481 (2008).
[CrossRef]

J. Opt. Soc. Am.

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. II. Stationary phase analysis of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1871–18810 (2001).
[CrossRef]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform,” J. Opt. Soc. Am. 18, 1862–1870 (2001).
[CrossRef]

O. Bryngdahl, “Characteristics of superposed patterns in optics,” J. Opt. Soc. Am. 66, 87–94 (1976).
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H. Osterberg, “An interferometer method of studying the vibrations of an oscillating quartz plate,” J. Opt. Soc. Am. 22, 19–35 (1932).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

M. B. Bernini, G. E. Galizzi, A. Federico, and G. H. Kaufmann, “Evaluation of the 1D empirical mode decomposition method to smooth digital speckle pattern interferometry fringes,” Opt. Commun. 45, 723–729 (2007).

F. A. Marengo-Rodriguez, A. Federico, and G. H. Kaufmann, “Phase measurement improvement in temporal speckle pattern interferometry using empirical mode decomposition,” Opt. Commun. 275, 38–41 (2007).
[CrossRef]

X. Yang, Q. Yu, and S. Fu, “A combined method for obtaining fringe orientations of ESPI,” Opt. Commun. 273, 60–66(2007).
[CrossRef]

Opt. Eng.

K. Patorski, Z. Sienicki, and A. Styk, “Phase-shifting method contrast calculations in time-averaged interferometry: error analysis,” Opt. Eng. 44, 065601 (2005).
[CrossRef]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45, 085602 (2006).
[CrossRef]

Opt. Express

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Opt. Lasers Eng.

M. Ragulskis, R. Maskeliunas, and V. Turla, “Investigation of dynamic displacements of lithographic press rubber roller by time average geometric moiré,” Opt. Lasers Eng. 43, 951–962(2005).
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[CrossRef]

Proc. SPIE

A. Bosseboeuf and S. Petitgrand, “Application of microscopic interferometry in the MEMS field,” Proc. SPIE 5145, 1–16(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]

A. Styk and K. Patorski, “Fizeau interferometer for quasi parallel optical plate testing,” Proc. SPIE 7063, 70630P (2008).
[CrossRef]

S. Petitgrand, R. Yahiaoui, A. Bosseboeuf, and K. Danaie, “Quantitative time-averaged microscopic interferometry for micromechanical device vibration mode characterization,” Proc. SPIE 4400, 51–60 (2001).
[CrossRef]

Strain

S. Equis and P. Jacquot, “Phase extraction in dynamic speckle interferometry with empirical mode decomposition and Hilbert transform,” Strain 46, 550–558 (2010).
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Other

K. Patorski, Handbook of the Moiré Fringe Technique(Elsevier, 1993).

Z. Wu and N. Huang, “Ensemble empirical mode decomposition: a noise assisted data analysis method,” Tech. Rep. No. 193, Centre for Ocean-Land-Atmosphere Studies(2005).

Z. Liu, H. Wang, and S. Peng, “Texture classification through empirical mode decomposition,” in Proceedings of the 17th International Conference on Pattern Recognition (ICPR 2004) (IEEE, 2004), pp. 803–806.

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

J. E. Greivenkamp and J. H. Brunning, “Phase shifting interferometry,” in Optical Shop Testing, D.Malacara ed. (Wiley, 1992), pp. 501–598.

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

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

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

R. Onodera, Y. Yamamoto, and Y. Ishii, “Signal processing of interferogram using a two-dimensional discrete Hilbert transform,” in Fringe 2005, the 5th International Workshop on Automatic Processing of Fringe Patterns, W.Osten ed. (Springer, 2006), pp. 82–89.

T. Bülow, D. Pallek, and G. Sommer, “Riesz transforms for the isotropic estimation of the local phase of moiré interferograms,” presented at the 22nd DAGM Symposium Mustererkennung, (Heidelberg, 2000).

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Figures (14)

Fig. 1
Fig. 1

Top: (a) synthetic interferogram; (b), (c), (d) first three IMFs. Bottom: (e) modulation function modulus, (f) results of the detrending-demodulation procedure using PHT, (g) OPHT, and (h) HS algorithms.

Fig. 2
Fig. 2

Detrending efficiency comparison of four EMD variants.

Fig. 3
Fig. 3

BEMD detrending NRS error comparison for different SD values, EXT = 8 .

Fig. 4
Fig. 4

BEMD detrending NRS error comparison for different SD values, EXT = 6 .

Fig. 5
Fig. 5

BEMD denoising NRS error as a function of the number n of subtracted IMFs.

Fig. 6
Fig. 6

Demodulation accuracy comparison for different carrier frequencies ω and different demodulation algorithms.

Fig. 7
Fig. 7

Demodulation accuracy comparison for different modulation frequencies and different demodulation algorithms.

Fig. 8
Fig. 8

Time-average interferograms of a (a) nonvibrating and (b) vibrating circular silicon micromembrane (resonance frequency 833 kHz ). The first two IMFs shown in (c) and (d) were extracted from (b).

Fig. 9
Fig. 9

Demodulation result for resonance frequency 724 kHz obtained with (a) HS and (b) PHT algorithms. Relative difference between (a) and (b) in the marked region is displayed in (c).

Fig. 10
Fig. 10

Top: demodulation results for resonance frequency 833 kHz obtained with (a) one IMF and (b) 15 IMFs. Bottom: result of postprocessing filtration by subtracting (c) two IMFs and (d) five IMFs from (b).

Fig. 11
Fig. 11

(a) and (b) Exemplary additive moiré patterns and (c) and (d) the corresponding BEMD-HT demodulation results.

Fig. 12
Fig. 12

(a) Demodulation results for resonance frequency 256 kHz , obtained using the TPS, (b) CWT [22], and (c) BEMD-HT methods.

Fig. 13
Fig. 13

Demodulation results for resonance frequency 172 kHz obtained using the (a) TPS, (b) CWT [22], and (c) BEMD-HT.

Fig. 14
Fig. 14

Comparison of TPS and BEMD-HT results for a single row from experimental data.

Tables (1)

Tables Icon

Table 1 Demodulation NRS Error Values

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

s ( x ) = r N ( x ) + k = 1 N IMF k ( x ) .
SD = x Ω | h i , j 1 ( x ) h i , j ( x ) | 2 h i , j 1 2 ( x ) < C ,
s H x ( x , y ) = 1 π PV R s ( u , y ) x u d u .
F { s H x ( x , y ) } = i sign ( ζ 1 ) F { s ( x , y ) } ,
s A x ( x , y ) = s ( x , y ) + i s H x ( x , y ) .
| A ( x , y ) | = | s A x ( x , y ) | + | s A y ( x , y ) | 2 .
| A ( x , y ) | = | s A x ( x , y ) | cos 2 β + | s A y ( x , y ) | sin 2 β .
F { s H ( x , y ) } = i sign ( ζ 1 cos β + ζ 2 sin β ) F { s ( x , y ) } .
s M ( x , y ) = s ( x , y ) + i R 1 { s ( x , y ) } + j R 2 { s ( x , y ) } ,
F { R σ { s ( x , y ) } } = i ζ σ ζ 1 2 + ζ 2 2 F { s ( x , y ) } ,
| A ( x , y ) | = s 2 ( x , y ) + R 1 2 { s ( x , y ) } + R 2 2 { s ( x , y ) } .
P ( ζ 1 , ζ 2 ) = ζ 1 + i ζ 2 ζ 1 2 + ζ 2 2 ,
s H ( x , y ) = i exp ( i β ) F 1 { P ( ζ 1 , ζ 2 ) F { s ( x , y ) } } .
| A ( x , y ) | = s 2 ( x , y ) + | F 1 { P ( ζ 1 , ζ 2 ) F { s ( x , y ) } } | 2 .
NRS = ( x , y ) Ω [ v ( x , y ) v ^ ( x , y ) ] 2 ( x , y ) Ω v 2 ( x , y ) ,
I v ( r ) = K ( r ) [ 1 + V st ( r ) J 0 ( 4 π λ a 0 ( r ) ) cos φ v ( r ) ] ,
I ( r ) = B ( r ) + C ( r ) M ( r ) + GN .
C ( r ) = cos ( 30 r ) ,
M ( r ) = J 0 ( 9 x 2 + y 2 ) ,

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