Abstract

We propose a simple and robust phase demodulation algorithm for two-shot fringe patterns with random phase shifts. Based on a smoothness assumption, the phase to be recovered is decomposed into a linear combination of finite terms of orthogonal polynomials, and the expansion coefficients and the phase shift are exhaustively searched through global optimization. The technique is insensitive to noise or defects, and is capable of retrieving phase from low fringe-number (less than one) or low-frequency interferograms. It can also cope with interferograms with very small phase shifts. The retrieved phase is continuous and no further phase unwrapping process is required. The method is expected to be promising to process interferograms with regular fringes, which are common in optical shop testing. Computer simulation and experimental results are presented to demonstrate the performance of the algorithm.

© 2016 Optical Society of America

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References

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    [Crossref]
  54. C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
    [Crossref]

2015 (5)

2014 (8)

C.-S. Guo, B. Sha, Y.-Y. Xie, and X.-J. Zhang, “Zero difference algorithm for phase shift extraction in blind phase-shifting holography,” Opt. Lett. 39(4), 813–816 (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]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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

K. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).

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, 205–211 (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, 230–240 (2014).
[Crossref]

2013 (2)

2012 (6)

2011 (7)

2010 (3)

2009 (1)

2008 (1)

2007 (2)

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

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

2005 (1)

2004 (2)

2003 (1)

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (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)

1997 (2)

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
[Crossref] [PubMed]

R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces,” J. Glob. Optim. 11(4), 341–359 (1997).
[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 263, 263–273 (1992).
[Crossref]

1991 (1)

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

1990 (1)

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

1982 (1)

1976 (1)

Álvarez-Herrero, A.

Antonio Quiroga, J.

J. Antonio Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Belenguer, T.

Bernini, M. B.

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]

Carazo, J. M.

Carpio-Valadez, M.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Chai, L.

Chakraborty, U. K.

S. Das, A. Konar, and U. K. Chakraborty, “Two improved differential evolution schemes for faster global search,” in Proceedings of the 7th annual conference on Genetic and evolutionary computation, (ACM, 2005), pp. 991–998.
[Crossref]

Chen, W.

Chen, X.

Cuevas, F. J.

Das, S.

S. Das, A. Konar, and U. K. Chakraborty, “Two improved differential evolution schemes for faster global search,” in Proceedings of the 7th annual conference on Genetic and evolutionary computation, (ACM, 2005), pp. 991–998.
[Crossref]

Deng, J.

Du, Y.

Estrada, J. C.

Federico, A.

Feng, G.

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]

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]

Gramaglia, M.

Guerrero, J. A.

Guo, C.-S.

Han, B.

Hao, J.

Ina, H.

Jueptner, W. P. O.

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

Kaufmann, G. H.

Kemao, Q.

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

Kobayashi, S.

Konar, A.

S. Das, A. Konar, and U. K. Chakraborty, “Two improved differential evolution schemes for faster global search,” in Proceedings of the 7th annual conference on Genetic and evolutionary computation, (ACM, 2005), pp. 991–998.
[Crossref]

Kreis, T. M.

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

Kujawinska, 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]

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

Kulkarni, R.

Larkin, K. G.

Li, C.

Li, H.

Li, Y.

Ling, T.

Liu, D.

Liu, F.

Lu, X.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

Luo, C.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

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]

Luo, Y.

Lv, X.

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, 205–211 (2014).
[Crossref]

Ma, S.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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]

Mahajan, V. N.

Malacara-Hernandez, D.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Marroquin, J. L.

Noll, R. J.

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, 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, 205–211 (2014).
[Crossref]

Patorski, K.

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[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, 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. Patorski, M. Trusiak, and K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).

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

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. 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).
[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]

K. Patorski and K. Pokorski, “Examination of singular scalar light fields using wavelet processing of fork fringes,” Appl. Opt. 50(5), 773–781 (2011).
[Crossref] [PubMed]

Peng, H.

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.

Price, K.

R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces,” J. Glob. Optim. 11(4), 341–359 (1997).
[Crossref]

Qin, J.

Quiroga, J. A.

Rastogi, P.

Rivera, M.

Sanchez-Mondragon, J. J.

D. Malacara-Hernandez, M. Carpio-Valadez, and J. J. Sanchez-Mondragon, “Wavefront fitting with discrete orthogonal polynomials in a unit radius circle,” Opt. Eng. 29(6), 672–675 (1990).
[Crossref]

Servin, M.

Servín, M.

Sha, B.

Silva-Moreno, A. 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]

Sorzano, C. O. S.

Storn, R.

R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for global Optimization over Continuous Spaces,” J. Glob. Optim. 11(4), 341–359 (1997).
[Crossref]

Sun, P.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Takeda, M.

Tian, C.

Tian, J.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

Tkaczyk, T.

Trusiak, M.

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[Crossref] [PubMed]

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. 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 K. Pokorski, “Single-shot two-channel Talbot interferometry using checker grating and Hilbert-Huang fringe pattern processing,” Proc. SPIE 9132, 91320Z (2014).

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

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. 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).
[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]

Vargas, J.

Wang, H.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
[Crossref] [PubMed]

Wang, K.

Wang, Z.

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, 205–211 (2014).
[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]

Wei, T.

Weng, J.

Wielgus, M.

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

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

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]

Wojciak, J.

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

Wu, D.

Wu, F.

Wu, Y.

Xie, Y.-Y.

Xu, J.

Xu, Q.

Yang, Y.

Yeazell, J. A.

Zhang, D.

Zhang, F.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
[Crossref] [PubMed]

Zhang, S.

Zhang, W.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Zhang, X.-J.

Zhong, J.

Zhong, L.

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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]

J. Deng, H. Wang, F. Zhang, D. Zhang, L. Zhong, and X. Lu, “Two-step phase demodulation algorithm based on the extreme value of interference,” Opt. Lett. 37(22), 4669–4671 (2012).
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Zhou, S.

Zhuo, Y.

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Appl. Phys. B (1)

C. Luo, L. Zhong, P. Sun, H. Wang, J. Tian, and X. Lu, “Two-step demodulation algorithm based on the orthogonality of diamond diagonal vectors,” Appl. Phys. B 119(2), 387–391 (2015).
[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).
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J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
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Opt. Express (12)

J. Xu, Q. Xu, L. Chai, Y. Li, and H. Wang, “Direct phase extraction from interferograms with random phase shifts,” Opt. Express 18(20), 20620–20627 (2010).
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J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express 19(2), 584–596 (2011).
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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).
[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]

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).
[Crossref]

M. Trusiak, K. Patorski, and M. Wielgus, “Adaptive enhancement of optical fringe patterns by selective reconstruction using FABEMD algorithm and Hilbert spiral transform,” Opt. Express 20(21), 23463–23479 (2012).
[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]

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[Crossref] [PubMed]

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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, 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).
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Opt. Lett. (10)

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
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Figures (8)

Fig. 1
Fig. 1 Definition of unit square and unit circle. Q(x, y) or Q(ρ, θ) is a point within the unit square and the unit circle.
Fig. 2
Fig. 2 Illustration of the crossover process. When randi(0, 1) ≤ CR or i = irand, copy the ith mutant element to the trial vector; Otherwise, copy the ith element in the target vector to the trial vector.
Fig. 3
Fig. 3 Simulated two-shot interferograms with a phase shift 0.01π. (a) and (b) two-shot interferograms with noise and defects; (c) and (d) 2D (wrapped) and 3D (unwrapped) true phase. Colorbar unit: rad.
Fig. 4
Fig. 4 Extracted 8th order coefficients. (a) Legendre coefficients estimated by the proposed two-shot method, (b) Zernike coefficients estimated by the single-shot method in [17].
Fig. 5
Fig. 5 Demodulation results of the two-shot interferograms in Fig. 3 using the proposed two-shot method, the single-shot method in [17], the Kreis method, the OF method and the GS method. First row: demodulated phase map, second row: corresponding error map.
Fig. 6
Fig. 6 Phase extraction of two-shot interferograms with open fringes. (a) and (b) two experimentally acquired fringe patterns; (c) and (d) estimated 3rd order Zernike coefficients and reconstructed phase map (rewrapped for comparison).
Fig. 7
Fig. 7 Phase extraction of two-shot interferograms with closed fringes. (a) and (b) two experimental fringe patterns; (c) retrieved phase by the AIA method using 11 frame images serving as the reference; (d) – (f) estimated 7th order Zernike coefficients, reconstructed phase map and residual error map by the proposed method, respectively; (g) and (h) reconstructed phase map and residual error map by the Kreis method. Colorbar unit: rad.
Fig. 8
Fig. 8 Phase retrieval of low fringe-number interferograms. (a) and (b) two experimental fringe patterns with less-than-one fringe; (c) retrieved phase by the AIA method using 11 frame images serving as the reference; (d) - (f) estimated 5th order Zernike coefficients, reconstructed phase map and residual error map by the proposed method, respectively; (g) and (h) reconstructed phase map and residual error map by the GS method. Colorbar unit: rad.

Tables (4)

Tables Icon

Table 1 2D Legendre polynomials up to 4th order

Tables Icon

Table 2 Standard non-orthonormal Zernike polynomials up to 4th order

Tables Icon

Table 3 Summary of Q indices and RMS values of Figs. 5, 7 and 8

Tables Icon

Table 4 Statistical results of all four experiments by the proposed methoda

Equations (21)

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

I 1 (x,y)= a 1 (x,y)+ b 1 (x,y)cos[ϕ(x,y)]+ n 1 (x,y),
I 2 (x,y)= a 2 (x,y)+ b 2 (x,y)cos[ϕ(x,y)+δ]+ n 2 (x,y),
I 1 n (x,y)=cos[ϕ(x,y)],
I 2 n (x,y)=cos[ϕ(x,y)+δ],
ϕ(x,y) ϕ f (x,y)= k c k ψ k (x,y) ,
f(X)= (x,y)mask ( { I 1 n (x,y)cos[ ϕ f (x,y)] } 2 + { I 2 n (x,y)cos[ ϕ f (x,y)+δ] } 2 ) ,
X= argmin X f(X).
P m (x)= 2m1 m x P m1 (x) m1 m P m2 (x)
1 1 P m (x) P m (x) dx= 2 2m+1 δ m m
L j (x,y)= P l (x) P m (y)
1 1 1 1 L j (x,y) L j (x,y)dxdy= 2 2l+1 2 2m+1 δ l l δ m m
Z j ={ R n m (ρ),m=0, R n m (ρ)cosmθ,m0andevenj, R n m (ρ)sinmθ,m0andoddj,
R n m (ρ)= s=0 (nm)/2 (1) s (ns)! s![(n+m)/2s]![(nm)/2s]! ρ n2s ,
0 1 0 2π Z j (ρ,θ) Z j (ρ,θ)ρdρdθ =π δ j j .
X j G = [ c 1,j G , c 2,j G ,, c k,j G , δ j G ] T = [ x 1,j G , x 2,j G ,, x D1,j G , x D,j G ] T ,
x i,j 0 = x imin +rand(0,1)×( x imax x imin ),
V j G = X r 1 G +F( X r 2 G X r 3 G ),
u i,j G ={ v i,j G ,if rand i (0,1)CRori= i rand , x i,j G ,otherwise,
X j G+1 ={ U j G ,iff( U j G )f( X j G ) X j G ,otherwise .
ϕ(x,y)=π×0.5{ ( x 2 + y 2 )10[ cos(2πx)+cos(2πy) ] },
Q= 4 σ AB μ A μ B ( σ A 2 + σ B 2 )( μ A 2 + μ B 2 ) ,

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