Abstract

Fringe analysis in two-shot phase-shifting interferometry is important but meets challenges due to a limited number of images, corrupting noise, and background modulation. Here we propose an effective algorithm for phase retrieval from two interferograms with unknown phase shifts. The algorithm first evaluates the phase shift in a local mask through phase fitting and global optimization and then computes a full-field phase map using an arctangent function. Since the phase shift evaluation is performed within a local mask, the algorithm is fast compared with conventional optimization-based algorithms and typically needs tens of seconds to complete the processing. Computer simulation and experimental results show that the proposed algorithm has excellent performance compared with state-of-the-art algorithms. A complete software package of the algorithm in MATLAB is available at http://two-shot.sourceforge.io/.

© 2017 Optical Society of America

Full Article  |  PDF Article
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2017 (3)

2016 (8)

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers Eng. 85, 29–37 (2016).
[Crossref]

O. Dalmau, M. Rivera, and A. Gonzalez, “Phase shift estimation in interferograms with unknown phase step,” Opt. Commun. 372, 37–43 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express 24(4), 3202–3215 (2016).
[Crossref] [PubMed]

C. Tian, X. Chen, and S. Liu, “Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes,” Opt. Express 24(4), 3572–3583 (2016).
[Crossref] [PubMed]

M. Trusiak, Ł. Służewski, and K. Patorski, “Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis,” Opt. Express 24(4), 4221–4238 (2016).
[Crossref] [PubMed]

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

Y. Lu, R. Li, and R. Lu, “Gram-Schmidt orthonormalization for retrieval of amplitude images under sinusoidal patterns of illumination,” Appl. Opt. 55(25), 6866–6873 (2016).
[Crossref] [PubMed]

2015 (5)

2014 (3)

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (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]

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]

2013 (2)

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[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).
[Crossref] [PubMed]

2012 (3)

2011 (5)

2010 (5)

2009 (1)

2007 (1)

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

2006 (1)

2005 (1)

2004 (1)

2003 (1)

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

2001 (1)

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

1997 (1)

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]

1992 (1)

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

1988 (1)

K. Creath, “V phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

1976 (1)

Andreiko, I.

L. Muravsky, O. Ostash, T. Voronyak, and I. Andreiko, “Two-frame phase-shifting interferometry for retrieval of smooth surface and its displacements,” Opt. Lasers Eng. 49(3), 305–312 (2011).
[Crossref]

Belenguer, T.

Cai, L. Z.

Cai, L.-Z.

X.-F. Xu, L.-Z. Cai, Y.-R. Wang, and D.-L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

Carazo, J. M.

Chai, L.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Chen, X.

Creath, K.

K. Creath, “V phase-measurement interferometry techniques,” Prog. Opt. 26, 349–393 (1988).
[Crossref]

Dalmau, O.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers Eng. 85, 29–37 (2016).
[Crossref]

O. Dalmau, M. Rivera, and A. Gonzalez, “Phase shift estimation in interferograms with unknown phase step,” Opt. Commun. 372, 37–43 (2016).
[Crossref]

Deng, J.

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]

Dong, G. Y.

Du, H.

Estrada, J. C.

Fei, L.

Feng, L.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Gao, P.

Gao, W.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

García-Botella, Á.

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

Geist, E.

Gómez-Pedrero, J. A.

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

Gonzalez, A.

O. Dalmau, M. Rivera, and A. Gonzalez, “Phase shift estimation in interferograms with unknown phase step,” Opt. Commun. 372, 37–43 (2016).
[Crossref]

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers Eng. 85, 29–37 (2016).
[Crossref]

Guerrero, J. A.

Han, B.

Harder, I.

Hernandez-Lopez, F.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers Eng. 85, 29–37 (2016).
[Crossref]

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Jin, W.

Jueptner, W. P.

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

Kemao, Q.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[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]

Kreis, T. M.

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

Kulkarni, R.

R. Kulkarni and P. Rastogi, “Direct unwrapped phase estimation in phase shifting interferometry using Levenberg–Marquardt algorithm,” J. Opt. 19(1), 015608 (2017).
[Crossref]

Li, D.-L.

X.-F. Xu, L.-Z. Cai, Y.-R. Wang, and D.-L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

Li, J.

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[Crossref]

Li, R.

Li, Y.

Lindlein, N.

Liu, D.

Liu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[Crossref] [PubMed]

Liu, S.

Lu, R.

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]

W. Niu, L. Zhong, P. Sun, W. Zhang, and X. Lu, “Two-step phase retrieval algorithm based on the quotient of inner products of phase-shifting interferograms,” J. Opt. 17(8), 085703 (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]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (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]

Lu, Y.

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]

Luo, Y.

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]

Mantel, K.

Marroquin, J. L.

Meng, X.

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[Crossref]

Meng, X. F.

Muravsky, L.

L. Muravsky, O. Ostash, T. Voronyak, and I. Andreiko, “Two-frame phase-shifting interferometry for retrieval of smooth surface and its displacements,” Opt. Lasers Eng. 49(3), 305–312 (2011).
[Crossref]

Niu, W.

W. Niu, L. Zhong, P. Sun, W. Zhang, and X. Lu, “Two-step phase retrieval algorithm based on the quotient of inner products of phase-shifting interferograms,” J. Opt. 17(8), 085703 (2015).
[Crossref]

Noll, R. J.

Ostash, O.

L. Muravsky, O. Ostash, T. Voronyak, and I. Andreiko, “Two-frame phase-shifting interferometry for retrieval of smooth surface and its displacements,” Opt. Lasers Eng. 49(3), 305–312 (2011).
[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, 205–211 (2014).
[Crossref]

Patorski, 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]

Quiroga, J. A.

Rastogi, P.

R. Kulkarni and P. Rastogi, “Direct unwrapped phase estimation in phase shifting interferometry using Levenberg–Marquardt algorithm,” J. Opt. 19(1), 015608 (2017).
[Crossref]

Rivera, M.

M. Rivera, O. Dalmau, A. Gonzalez, and F. Hernandez-Lopez, “Two-step fringe pattern analysis with a Gabor filter bank,” Opt. Lasers Eng. 85, 29–37 (2016).
[Crossref]

O. Dalmau, M. Rivera, and A. Gonzalez, “Phase shift estimation in interferograms with unknown phase step,” Opt. Commun. 372, 37–43 (2016).
[Crossref]

J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30(22), 3018–3020 (2005).
[Crossref] [PubMed]

Saide, D.

Servin, M.

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

Servín, M.

Shen, X. X.

Sluzewski, L.

Soon, S. H.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Sorzano, C. O.

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]

W. Niu, L. Zhong, P. Sun, W. Zhang, and X. Lu, “Two-step phase retrieval algorithm based on the quotient of inner products of phase-shifting interferograms,” J. Opt. 17(8), 085703 (2015).
[Crossref]

Sunderland, Z.

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]

Trusiak, M.

Vargas, J.

Voronyak, T.

L. Muravsky, O. Ostash, T. Voronyak, and I. Andreiko, “Two-frame phase-shifting interferometry for retrieval of smooth surface and its displacements,” Opt. Lasers Eng. 49(3), 305–312 (2011).
[Crossref]

Wan, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (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, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express 18(23), 24368–24378 (2010).
[Crossref] [PubMed]

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Lasers Eng. 48(6), 684–689 (2010).
[Crossref]

Wang, J.

H. Du, J. Yan, and J. Wang, “Random phase-shifting algorithm by constructing orthogonal phase-shifting fringe patterns,” Appl. Opt. 56(11), 3071–3076 (2017).
[Crossref] [PubMed]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Wang, Q.

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[Crossref]

Wang, Y.

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[Crossref]

Wang, Y. R.

Wang, Y.-R.

X.-F. Xu, L.-Z. Cai, Y.-R. Wang, and D.-L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

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]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Wei, T.

Wielgus, M.

Wu, F.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[Crossref] [PubMed]

Wu, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[Crossref] [PubMed]

Xu, J.

Xu, Q.

Xu, X. F.

Xu, X.-F.

X.-F. Xu, L.-Z. Cai, Y.-R. Wang, and D.-L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

Yan, J.

Yang, X.

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
[Crossref]

Yang, X. L.

Yang, Y.

Yao, B.

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.

W. Niu, L. Zhong, P. Sun, W. Zhang, and X. Lu, “Two-step phase retrieval algorithm based on the quotient of inner products of phase-shifting interferograms,” J. Opt. 17(8), 085703 (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]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (2014).
[Crossref] [PubMed]

Zhao, H.

Zhong, L.

W. Niu, L. Zhong, P. Sun, W. Zhang, and X. Lu, “Two-step phase retrieval algorithm based on the quotient of inner products of phase-shifting interferograms,” J. Opt. 17(8), 085703 (2015).
[Crossref]

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]

W. Zhang, X. Lu, L. Fei, H. Zhao, H. Wang, and L. Zhong, “Simultaneous phase-shifting dual-wavelength interferometry based on two-step demodulation algorithm,” Opt. Lett. 39(18), 5375–5378 (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]

Zhuo, Y.

Appl. Opt. (6)

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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]

Chin. Phys. Lett. (1)

X.-F. Xu, L.-Z. Cai, Y.-R. Wang, and D.-L. Li, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[Crossref]

J. Glob. Optim. (1)

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]

J. Opt. (4)

J. Li, Y. Wang, X. Meng, X. Yang, and Q. Wang, “An evaluation method for phase shift extraction algorithms in generalized phase-shifting interferometry,” J. Opt. 15(10), 105408 (2013).
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[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
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C. Tian and S. Liu, “Demodulation of two-shot fringe patterns with random phase shifts by use of orthogonal polynomials and global optimization,” Opt. Express 24(4), 3202–3215 (2016).
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F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[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).
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M. Trusiak, Ł. Służewski, and K. Patorski, “Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis,” Opt. Express 24(4), 4221–4238 (2016).
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C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
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J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express 18(23), 24368–24378 (2010).
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C. Tian, X. Chen, and S. Liu, “Modal wavefront reconstruction in radial shearing interferometry with general aperture shapes,” Opt. Express 24(4), 3572–3583 (2016).
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[Crossref]

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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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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Figures (8)

Fig. 1
Fig. 1 Simulated two-shot fringe patterns [(a) and (b)] with a phase shift δ = 0.01π rad and the continuous (c) and wrapped (d) phase. The three red disks on (a) and (b) are three masks used to evaluate the phase shift. Colorbar unit: rad.
Fig. 2
Fig. 2 Phase shift evaluation using fringe data in different masks. First column: fringe patterns in mask 1 (first row), 2 (second row), and 3 (third row). Second column: evaluation process showing that the value of the cost function decreases with evolution generations of the solution vector (X). Third column: optimized fit coefficients. Fourth column: computed phase using Eq. (9). Last column: computed intensity map using Eq. (7). The extracted phase shifts for mask 1, 2, and 3 are (0.0095 ± 0.0002)π, (0.0103 ± 0.0003)π, and (0.01 ± 0.0003)π rad, respectively.
Fig. 3
Fig. 3 Comparison of retrieved phase using different algorithms. From left to right: retrieved phase maps and error maps by the proposed method, the parametric method [37], the Kreis method [27], the OF method [19], and the GS method [28]. Colorbar unit: rad.
Fig. 4
Fig. 4 Specific example showing the relation between extracted phase shift δ and mask radius. When the mask has a moderate size of 15%-60% of the whole image, the extracted phase shift is most accurate (δ true value 0.01π). The results, shown in mean ± standard deviation, are calculated from five independent runs. The insets are snapshots of the images in the mask at different radii.
Fig. 5
Fig. 5 Influences of background (a), modulation (b), and additive noise (c) on the accuracy of phase retrieval results.
Fig. 6
Fig. 6 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with closed fringes. (c) Reference phase map obtained using the AIA method (13 frames) [43]. (d) Fringe pattern in the mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.58π rad in this case. (e) – (h) Recovered phase maps using the proposed method, the parametric method, the Kreis method, and the GS method, respectively. (i) – (l) Corresponding error maps with respect to the AIA result. Colorbar unit: rad.
Fig. 7
Fig. 7 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with open fringes. (c) Fringe pattern in the selected mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.49π rad in this case. (d), (e), and (f) Retrieved phase maps using the proposed method, the AIA method (four frames), and their difference, respectively. Colorbar unit: rad.
Fig. 8
Fig. 8 Phase retrieval from two-shot experimental fringe patterns [(a) and (b)] with background modulation. (c) Fringe pattern in the selected mask [red disk in (a)] for phase shift evaluation. The extracted phase shift is 0.48π rad in this case. (d), (e), and (f) Retrieved phase maps using the proposed method, the AIA method (13 frames), and their difference, respectively. Colorbar unit: rad.

Equations (14)

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I 1 (x,y)=a(x,y)+b(x,y)cos[ϕ(x,y)]+n(x,y),
I 2 (x,y)=a(x,y)+b(x,y)cos[ϕ(x,y)+δ]+n(x,y),
I 1 hp (x,y)=b(x,y)cos[ϕ(x,y)],
I 2 hp (x,y)=b(x,y)cos[ϕ(x,y)+δ].
Q 1 hp (x,y)=b(x,y)sinϕ(x,y)= I 1 hp (x,y)cosδ I 2 hp (x,y) sinδ .
ϕ(x,y)=arctan{ Q 1 hp I 1 hp }=arctan{ I 1 hp (x,y)cosδ I 2 hp (x,y) I 1 hp sinδ }.
I 1 n (x,y)=cos[ ϕ mask (x,y)],(x,y)mask,
I 2 n (x,y)=cos[ ϕ mask (x,y)+δ],(x,y)mask,
ϕ mask (x,y) j c j Z j (x,y) ,(x,y)mask,
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 ,
f(X)= (x,y)mask ( { I 1 n (x,y)cos[ ϕ mask (x,y)] } 2 + { I 2 n (x,y)cos[ ϕ mask (x,y)+δ] } 2 ) .
X= argmin X f(X).
ϕ(x,y)=0.5{ ( x 2 + y 2 )10[ cos(2πx)+cos(2πy) ] }π,

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