Abstract

Gram-Schmidt orthonormalization is a very fast and efficient method for the fringe pattern phase demodulation. It requires only two arbitrarily phase-shifted frames. Images are treated as vectors and upon orthogonal projection of one fringe vector onto another the quadrature fringe pattern pair is obtained. Orthonormalization process is very susceptible, however, to noise, uneven background and amplitude modulation fluctuations. The Hilbert-Huang transform based preprocessing is proposed to enhance fringe pattern phase demodulation by filtering out the spurious noise and background illumination and performing fringe normalization. The Gram-Schmidt orthonormalization process error analysis is provided and its filtering-expanded capabilities are corroborated analyzing DSPI fringes and performing amplitude demodulation of Bessel fringes. Synthetic and experimental fringe pattern analyses presented to validate the proposed technique show that it compares favorably with other pre-filtering schemes, i.e., Gaussian filtering and continuous wavelet transform.

© 2015 Optical Society of America

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References

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2015 (2)

Q. Kemao, “Applications of windowed Fourier fringe analysis in optical measurement: a review,” Opt. Lasers Eng. 66(3), 67–73 (2015).
[Crossref]

M. Zhao and Q. Kemao, “Multicore implementation of the windowed Fourier transform algorithms for fringe pattern analysis,” Appl. Opt. 54(3), 587–594 (2015).
[Crossref]

2014 (9)

M. Wielgus and K. Patorski, “Denoising and extracting background from fringe patterns using midpoint-based bidimensional empirical mode decomposition,” Appl. Opt. 53(10), B215–B222 (2014).
[Crossref] [PubMed]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and T. Tkaczyk, “Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing,” Opt. Express 22(8), 9517–9527 (2014).
[Crossref] [PubMed]

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
[Crossref]

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(1), 230–240 (2014).
[Crossref]

K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

2013 (5)

2012 (6)

2011 (4)

2010 (2)

K. Pokorski and K. Patorski, “Visualization of additive-type moiré and time-average fringe patterns using the continuous wavelet transform,” Appl. Opt. 49(19), 3640–3651 (2010).
[Crossref] [PubMed]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

2008 (2)

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. 164, 725356 (2008).

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47(29), 5408–5419 (2008).
[Crossref] [PubMed]

2007 (2)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

2006 (1)

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[Crossref]

2005 (1)

2004 (1)

2003 (1)

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]

2002 (2)

2001 (5)

1999 (1)

1998 (1)

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. A 454(1971), 903–995 (1998).
[Crossref]

1997 (1)

1996 (1)

K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24(5–6), 365–379 (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]

1992 (1)

T. M. Kreis and W. P. O. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

1991 (1)

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[Crossref]

1986 (1)

1982 (1)

Adhami, R. R.

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. 164, 725356 (2008).

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

Bachor, H.-A.

Barnes, T. H.

Belenguer, T.

Bhuiyan, S. M. A.

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. 164, 725356 (2008).

Bone, D. J.

Bovik, A. C.

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

Burton, D. R.

Carazo, J. M.

Chen, Z.

Creath, K.

K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24(5–6), 365–379 (1996).
[Crossref]

Cuevas, F. J.

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]

Debnath, S. K.

Du, Y.

Dudek, M.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

Estrada, J. C.

Feng, G.

Gao, W.

Garcia-Botella, A.

J. A. Quiroga, J. A. Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1–3), 43–51 (2001).
[Crossref]

Gdeisat, M. A.

Gomez-Pedrero, J. A.

J. A. Quiroga, J. A. Gomez-Pedrero, and A. Garcia-Botella, “Algorithm for fringe pattern normalization,” Opt. Commun. 197(1–3), 43–51 (2001).
[Crossref]

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]

Guerrero, J. A.

Heppner, J.

Herráez, M. A.

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

Huang, N. E.

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. A 454(1971), 903–995 (1998).
[Crossref]

Ina, H.

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]

Jing, Z.

Z. Zhang, Z. Jing, Z. Wang, and D. Kuang, “Comparison of Fourier transform, windowed Fourier transform and wavelet transform methods for phase calculation at discontinuities in fringe projection profilometry,” Opt. Lasers Eng. 50(8), 1152–1160 (2012).
[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]

Jueptner, W. P. O.

T. M. Kreis and W. P. O. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[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]

Kai, L.

Kemao, Q.

M. Zhao and Q. Kemao, “Multicore implementation of the windowed Fourier transform algorithms for fringe pattern analysis,” Appl. Opt. 54(3), 587–594 (2015).
[Crossref]

Q. Kemao, “Applications of windowed Fourier fringe analysis in optical measurement: a review,” Opt. Lasers Eng. 66(3), 67–73 (2015).
[Crossref]

L. Kai and Q. Kemao, “Improved generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 21(20), 24385–24397 (2013).
[Crossref] [PubMed]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

Q. Kemao, H. Wang, and W. Gao, “Windowed Fourier transform for fringe pattern analysis: theoretical analyses,” Appl. Opt. 47(29), 5408–5419 (2008).
[Crossref] [PubMed]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
[Crossref]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[Crossref] [PubMed]

Kemper, B.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

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. 164, 725356 (2008).

Kobayashi, S.

Kreis, T. M.

T. M. Kreis and W. P. O. Jueptner, “Fourier transform evaluation of interference patterns: demodulation and sign ambiguity,” Proc. SPIE 1553, 263–273 (1992).
[Crossref]

Kuang, D.

Z. Zhang, Z. Jing, Z. Wang, and D. Kuang, “Comparison of Fourier transform, windowed Fourier transform and wavelet transform methods for phase calculation at discontinuities in fringe projection profilometry,” Opt. Lasers Eng. 50(8), 1152–1160 (2012).
[Crossref]

Kujawinska, M.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

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]

M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
[Crossref]

Kus, A.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

Lalor, M. J.

Larkin, K. G.

Li, B.

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
[Crossref]

Li, H.

Liu, H. H.

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. A 454(1971), 903–995 (1998).
[Crossref]

Long, S. R.

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. A 454(1971), 903–995 (1998).
[Crossref]

Lu, X.

Luo, C.

Ma, J.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Ma, S.

Marroquin, J. L.

Massig, J. H.

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

Oldfield, M. A.

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[Crossref]

Pan, T.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Park, Y.-K.

Patorski, K.

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

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(1), 230–240 (2014).
[Crossref]

K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Wielgus and K. Patorski, “Denoising and extracting background from fringe patterns using midpoint-based bidimensional empirical mode decomposition,” Appl. Opt. 53(10), B215–B222 (2014).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and T. Tkaczyk, “Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing,” Opt. Express 22(8), 9517–9527 (2014).
[Crossref] [PubMed]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

K. Pokorski and K. Patorski, “Processing and phase analysis of fringe patterns with contrast reversals,” Opt. Express 21(19), 22596–22609 (2013).
[PubMed]

K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21(14), 16863–16881 (2013).
[PubMed]

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. Express 20(21), 23463–23479 (2012).
[Crossref] [PubMed]

M. Trusiak and K. Patorski, “Space domain interpretation of incoherent moiré superimpositions using FABEMD,” Proc. SPIE 8697, 869704 (2012).
[Crossref]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19(27), 26065–26078 (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(19), 3640–3651 (2010).
[Crossref] [PubMed]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[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]

Pirga, M.

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]

Pokorski, K.

Quiroga, J. A.

Rivera, M.

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]

Sandeman, R. J.

Schmit, J.

K. Creath and J. Schmit, “N-point spatial phase-measurement techniques for non-destructive testing,” Opt. Lasers Eng. 24(5–6), 365–379 (1996).
[Crossref]

Servin, M.

Servín, M.

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. A 454(1971), 903–995 (1998).
[Crossref]

Shih, H. H.

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. A 454(1971), 903–995 (1998).
[Crossref]

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N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

Sorzano, C. O. S.

Styk, A.

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[Crossref]

Sun, C.

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
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Takeda, M.

Tan, S. M.

Tang, C.

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
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X. Zhu, Z. Chen, and C. Tang, “Variational image decomposition for automatic background and noise removal of fringe patterns,” Opt. Lett. 38(3), 275–277 (2013).
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Tkaczyk, T.

Trusiak, M.

K. Patorski, M. Trusiak, and T. Tkaczyk, “Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing,” Opt. Express 22(8), 9517–9527 (2014).
[Crossref] [PubMed]

K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

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(1), 230–240 (2014).
[Crossref]

K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21(14), 16863–16881 (2013).
[PubMed]

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[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. Express 20(21), 23463–23479 (2012).
[Crossref] [PubMed]

M. Trusiak and K. Patorski, “Space domain interpretation of incoherent moiré superimpositions using FABEMD,” Proc. SPIE 8697, 869704 (2012).
[Crossref]

K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19(27), 26065–26078 (2011).
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Tung, C. 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. A 454(1971), 903–995 (1998).
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Vollmer, A.

A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
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Wang, L.

X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
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J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
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Z. Zhang, Z. Jing, Z. Wang, and D. Kuang, “Comparison of Fourier transform, windowed Fourier transform and wavelet transform methods for phase calculation at discontinuities in fringe projection profilometry,” Opt. Lasers Eng. 50(8), 1152–1160 (2012).
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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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Watkins, L. R.

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(1), 230–240 (2014).
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M. Trusiak, K. Patorski, and M. Wielgus, “Hilbert-Huang processing and analysis of complex fringe patterns,” Proc. SPIE 9203, 92030K (2014).
[Crossref]

K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

M. Wielgus and K. Patorski, “Denoising and extracting background from fringe patterns using midpoint-based bidimensional empirical mode decomposition,” Appl. Opt. 53(10), B215–B222 (2014).
[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. Express 20(21), 23463–23479 (2012).
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M. Kujawinska and J. Wojciak, “Spatial-carrier phase-shifting technique of fringe pattern analysis,” Proc. SPIE 1508, 61–67 (1991).
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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. A 454(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. A 454(1971), 903–995 (1998).
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K. Pokorski and K. Patorski, “Separation of complex fringe patterns using two-dimensional continuous wavelet transform,” Appl. Opt. 51(35), 8433–8439 (2012).
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Fringe (1)

K. Patorski, M. Trusiak, and M. Wielgus, “Fast adaptive processing of low quality fringe patterns by automated selective reconstruction and enhanced fast empirical mode decomposition,” Fringe 2013, 185–190 (2014).

IEEE Signal Process. Lett. (1)

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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A. Kuś, M. Dudek, B. Kemper, M. Kujawińska, and A. Vollmer, “Tomographic phase microscopy of living three-dimensional cell cultures,” J. Biomed. Opt. 19(4), 046009 (2014).
[Crossref] [PubMed]

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Opt. Eng. (2)

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]

K. Patorski and A. Styk, “Interferogram intensity modulation calculations using temporal phase shifting: error analysis,” Opt. Eng. 45(8), 085602 (2006).
[Crossref]

Opt. Express (10)

K. Patorski, M. Trusiak, and T. Tkaczyk, “Optically-sectioned two-shot structured illumination microscopy with Hilbert-Huang processing,” Opt. Express 22(8), 9517–9527 (2014).
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Y. Du, G. Feng, H. Li, J. Vargas, and S. Zhou, “Spatial carrier phase-shifting algorithm based on principal component analysis method,” Opt. Express 20(15), 16471–16479 (2012).

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(21), 23463–23479 (2012).
[Crossref] [PubMed]

K. Patorski and M. Trusiak, “Highly contrasted Bessel fringe minima visualization for time-averaged vibration profilometry using Hilbert transform two-frame processing,” Opt. Express 21(14), 16863–16881 (2013).
[PubMed]

K. Pokorski and K. Patorski, “Processing and phase analysis of fringe patterns with contrast reversals,” Opt. Express 21(19), 22596–22609 (2013).
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L. Kai and Q. Kemao, “Improved generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 21(20), 24385–24397 (2013).
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M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
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H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
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K. Patorski, K. Pokorski, and M. Trusiak, “Fourier domain interpretation of real and pseudo-moiré phenomena,” Opt. Express 19(27), 26065–26078 (2011).
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J. Vargas, J. A. Quiroga, T. Belenguer, M. Servín, and J. C. Estrada, “Two-step self-tuning phase-shifting interferometry,” Opt. Express 19(2), 638–648 (2011).
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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(1), 230–240 (2014).
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L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
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Z. Zhang, Z. Jing, Z. Wang, and D. Kuang, “Comparison of Fourier transform, windowed Fourier transform and wavelet transform methods for phase calculation at discontinuities in fringe projection profilometry,” Opt. Lasers Eng. 50(8), 1152–1160 (2012).
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X. Zhu, C. Tang, B. Li, C. Sun, and L. Wang, “Phase retrieval for single frame projection fringe pattern with variational image decomposition,” Opt. Lasers Eng. 59(8), 25–33 (2014).
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Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: Principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
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M. Trusiak and K. Patorski, “Space domain interpretation of incoherent moiré superimpositions using FABEMD,” Proc. SPIE 8697, 869704 (2012).
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Figures (11)

Fig. 1
Fig. 1 Simulated cosine term (a), cosine term phase shifted by 9π/14 (b), background (c), two modulation functions (d)-(e), ideal wrapped phase (f), synthetic fringes spoiled with background (g), two phase shifted synthetic fringe patterns spoiled with different modulation functions (h)-(i), synthetic fringe pattern spoiled with modulation, background and noise (j), two synthetic speckle pattern (cosines, Figs. 1(a) and 1(b), spoiled with speckle noise) (k)-(l).
Fig. 2
Fig. 2 Wrapped phase maps obtained using the GS algorithm applied to synthetic patterns spoiled with background (a), modulation (b), modulation and background (c), modulation, background and noise (d), and speckle noise (e); corresponding wrapped phases obtained using the GS1 algorithm (f)-(j).
Fig. 3
Fig. 3 Wrapped phase maps obtained using the GS2 algorithm applied to synthetic patterns spoiled with background (a), modulation (b), modulation and background (c), modulation, background and noise (d), and speckle noise (e); corresponding wrapped phases obtained using the CWT-GS algorithm (f)-(j).
Fig. 4
Fig. 4 (a)-(e) First four BIMFs and the residue obtained decomposing the synthetic pattern with background, Fig. 1(g), using the EFEMD algorithm; (f)-(o) first nine BIMFs and the residue obtained decomposing the synthetic speckle pattern, Fig. 1(k), using the accelerated FABEMD algorithm.
Fig. 5
Fig. 5 Wrapped phase maps obtained using the HHT-GS algorithm applied to synthetic patterns spoiled with background (a), modulation (b), modulation and background (c), modulation, background and noise (d), and speckle noise (e).
Fig. 6
Fig. 6 Two phase-shifted real u(x,y) fringe patterns recorded using the grating interferometry method (a)-(b), wrapped phases obtained using the GS algorithm (c), 4-frame temporal phase shifting method (d), Gaussian filtered TPS (e), GS1 (f), GS2 (g), CWT-GS (h) and HHT-GS (i) algorithms.
Fig. 7
Fig. 7 Two phase-shifted real v(x,y) fringe patterns recorded using the grating interferometry method (a)-(b), wrapped phases obtained using the GS algorithm (c), 4-frame temporal phase shifting method (d), Gaussian filtered TPS (e), GS1 (f), GS2 (g), CWT-GS (h) and HHT-GS (i) algorithms.
Fig. 8
Fig. 8 Two frames of real DSPI fringes (a-b), wrapped phase demodulated using the GS (c), GS1 (d), GS2 (e) CWT-GS (f), HHT-GS (g).
Fig. 9
Fig. 9 Simulated cosine carrier pattern (a), Bessel amplitude modulation function (b), synthetic fringe pattern with carrier term modulated by amplitude function (c), the ideal modulus of the modulation function (d), amplitude distribution demodulated using the GS (e), GS1 (f) and the HHT-GS algorithm (g) in case of non-zero DC term; amplitude distribution demodulated using the GS (h), GS1 (i) and the HHT-GS (j) algorithm in case of noise presence.
Fig. 10
Fig. 10 Two experimental TAI frames (a)-(b), amplitude demodulation using the two-frame reference algorithm [59] (c), 2-frame GS method (d), 2-frame GS1 (e), 2-frame HHT-GS (f) and modified 4-frame GS (g), GS1 (h) and HHT-GS (i) approach.
Fig. 11
Fig. 11 Amplitude demodulation error computed upon subtracting from the reference Bessel fringes obtained using the 2-frame technique proposed in [59], Fig. 10(c), the results of (a) 2-frame GS Fig. 10(d), (b) 2-frame HHT-GS, Fig. 10(d), (c) 4-frame GS, Fig. 10(g) and (d) 4-frame HHT-GS, Fig. 10(h).

Tables (9)

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Table 1 Wrapped phase demodulation quality evaluation (Q values)

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Table 2 Unwrapped phase demodulation quality evaluation (Q values)

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Table 3 Wrapped phase demodulation quality evaluation (RMS values in radians)

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Table 4 Unwrapped phase demodulation quality evaluation (RMS values in radians)

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Table 5 Computation time assessment (seconds)

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Table 6 Experimental phase demodulation quality evaluation (RMS values in radians) -u(x,y)

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Table 7 Experimental phase demodulation quality evaluation (RMS values in radians) - v(x,y)

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Table 8 Amplitude demodulation quality evaluation (Q values/RMS values in radians)

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Table 9 Experimental amplitude demodulation evaluation (Q values/RMS values in radians)

Equations (1)

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I(x,y)=a(x,y)+b(x,y)cos(φ(x,y))+n(x,y)

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