Abstract

We develop an error-free nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness (RH), and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter and, as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to “visually demonstrate” that their new nPSA works better than its competitors. This often leads to biased/favorable fringe pattern selections which “visually demonstrate” the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2017 (1)

2016 (7)

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

K. Ishikawa, K. Yatabe, N. Chitanont, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “High-speed imaging of sound using parallel phase-shifting interferometry,” Opt. Express 24(12), 12922–12932 (2016).
[Crossref] [PubMed]

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (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]

2015 (4)

2014 (3)

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]

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

2013 (5)

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

2011 (3)

2009 (1)

2008 (1)

2007 (1)

2005 (1)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

2004 (2)

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]

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

1997 (1)

1996 (1)

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

1992 (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

1991 (2)

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[Crossref]

1988 (1)

1985 (1)

1901 (1)

K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
[Crossref]

Aguilar, L. A.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Albertazzi, A.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Belenguer, T.

Benedet, M. E.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Bokor, J.

Cai, L. Z.

Carazo, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chai, L.

Chen, M.

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, W.

Cheng, X. C.

Cheng, Y.-Y.

Chitanont, N.

Deng, J.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[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. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Dong, G. Y.

Estrada, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Fan, J.

Fantin, A. V.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Gao, P.

Garnica, G.

Geist, E.

Goldberg, K. A.

Guerrero-Sánchez, F.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Guo, H.

Han, B.

Han, H.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Hao, J.

Harder, I.

He, J.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[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]

Ikeda, Y.

Ishikawa, K.

Ji, F.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Ji, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Jin, W.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

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]

Juarez-Salazar, R.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kimbrough, B.

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

Kinnstaetter, K.

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Lai, G.

Lara-Cortes, F. A.

Li, C.

Lindlein, N.

Liu, F.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (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]

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, Q.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Lohmann, A. W.

Lu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[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, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Lv, X.

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.

Medecki, H.

Meneses-Fabian, C.

C. Meneses-Fabian and F. A. Lara-Cortes, “Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps,” Opt. Express 23(10), 13589–13604 (2015).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Meng, X. F.

Niwa, H.

Oikawa, Y.

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[Crossref]

Onuma, T.

Padilla, M.

Paez, G.

Patil, A.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Patorski, K.

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]

Pearson, K.

K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rastogi, P.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Robledo-Sánchez, C.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[Crossref]

Schwider, J.

Servin, M.

Shen, X. X.

Shou, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Song, W.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

Sorzano, C.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Streibl, N.

Sun, W. J.

Tejnil, E.

Tian, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Trusiak, M.

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]

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[Crossref]

Vargas, J.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

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

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

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.

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]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Wang, J.

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, K.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

Wang, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Wang, Y. R.

Wang, Z.

Willemann, D. P.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Wu, D.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[Crossref] [PubMed]

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, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (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, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (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]

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]

Wyant, J. C.

Xu, J.

Xu, Q.

Xu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Xu, X. F.

Xu, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Yamaguchi, I.

Yao, B.

Yatabe, K.

Yatagai, T.

Yoshii, M.

Yu, Y.

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]

Zhang, H.

Zhang, T.

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, Z.

Zheng, D.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Zhong, L.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[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, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Int. J. Optomechatronics (2)

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

J. Opt. (2)

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[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]

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (2)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

Opt. Commun. (4)

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[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. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991).
[Crossref]

Opt. Eng. (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Opt. Express (6)

Opt. Lasers Eng. (5)

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Opt. Lett. (8)

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[Crossref] [PubMed]

P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
[Crossref] [PubMed]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[Crossref] [PubMed]

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]

H. Medecki, E. Tejnil, K. A. Goldberg, and J. Bokor, “Phase-shifting point diffraction interferometer,” Opt. Lett. 21(19), 1526–1528 (1996).
[Crossref] [PubMed]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[Crossref] [PubMed]

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

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

Philos. Mag. (1)

K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
[Crossref]

Sci. Rep. (1)

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

Other (2)

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, (WILEY-VCH, 2014).

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing (Wiley, 2006), pp. 547–666.

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Figures (8)

Fig. 1
Fig. 1 Panel (a) shows 9 nonuniform temporal samples of a continuous-time fringe. Panel (b) shows the Fourier spectrum of the continuous-time fringe.
Fig. 2
Fig. 2 Three nonlinear phase-shifted samples taken at θn = {0, 1.49, 5.13} radians.
Fig. 3
Fig. 3 The 3 nonuniform phase-shifted, harmonics free, fringe images.
Fig. 4
Fig. 4 Here we show plain PCA-nPSA demodulated phase, and its phase-error. This figure uses the standard 256 gray-level linear-map from [-π,π) to [0,255). The amplitude of the two phase-error images was multiplied by 2 for illustrative purposes.
Fig. 5
Fig. 5 Correcting 3-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA. Panel (b) shows the corrected FTF along its analytic signal and Lissajous circle. We also show the SNR-gain slight degradation due to FTF correction.
Fig. 6
Fig. 6 Here we show 4 out of 9 nonlinear phase-shifted, harmonics free, fringe patterns.
Fig. 7
Fig. 7 The red dots plot represents 9 nonlinear/nonuniform phase steps.
Fig. 8
Fig. 8 Correction of 9-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA-nPSA applied to Fig. 7 data. Panel (b) shows the corrected FTF, analytic signal, and Lissajous circle. Note that the SNR-gain decreases due to FTF correction.

Equations (27)

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

I( x,y,t )=a(x,y)+b(x,y)cos[ φ(x,y)+ ω 0 t ];t.
F[ I(t) ]=aδ(ω)+ b 2 e iφ δ( ω+1 )+ b 2 e iφ δ( ω1 ).
I n = [ a+bcos( φ+ ω 0 t ) ]δ(t t n ) dt;n{0,1,...,N1}.
I n =a+bcos( φ+ θ n );(x,y)L×L.
I n ( x,y )=a(x,y)+b(x,y)cos[ φ(x,y)+ θ n ];n={0,1,...,N1}.
a ^ (x,y)= 1 N n=0 N1 I n (x,y) .
[ C ] m,n = 1 L 2 x=0 L1 y=0 L1 [ I m (x,y) a ^ (x,y) ][ I n (x,y) a ^ (x,y) ] .
C v n = λ n v n ;n{0,1,...,N1}.
A(x,y)= n=0 N1 ( [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
r[ φ(x,y) ]=Re[ A(x,y) ]i+Im[ A(x,y) ]j.
ρ= x=0 L1 y=0 L1 | Im[ A(x,y) ] | x=0 L1 y=0 L1 | Re[ A(x,y) ] | ;( 0<ρ1.0 ).
A 2 (x,y)= n=0 N1 ( ρ [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
A(x,y)= n=0 N1 c ¯ n I n (x,y) = [ h ¯ (t)I(x,y,t) ] t=N1 , h(t)= n=0 N1 c n δ( t θ n 1.0 ) ; c n =ρ [ v 0 ] n +i [ v 1 ] n ; I(x,y,t)= n=0 N1 I(x,y, θ n )δ( t θ n 1.0 ) .
H(ω)=F[ n=0 N1 c n δ( t θ n 1.0 ) ]= n=0 N1 c n e i θ n ω .
H(1)=0;H(0)=0;H(+1)0.
A(x,y)= b(x,y) 2 H(1) e iφ(x,y) .
A error = b 2 [ e iφ H(1)+ e iφ H(1) ]= b 2 H(1) e iφ [ 1+ H(1) H(1) e 2iφ ];(x,y)LxL.
SNR= QuadratureSignalPower FilteredAWGNPower = ( b 2 /4) (η/2) | H(1.0) | 2 n=0 N1 | c n | 2 .
G SNR = | H(1.0) | 2 n=0 N1 | c n | 2 ;(0< G SNR N).
I(φ, θ n )=a+bcos[ φ+ θ n ]+ k=2 b k cos[ kφ+k θ n ] ;(x,y)LxL.
A(x,y)= b 2 e iφ H(1)+ k=2 ( b k 2 )[ e ikφ H(k)+ e ikφ H(k) ] .
R H = QuadratureSignalPower TotalFringeHarmonicPower = | H(1.0) | 2 k=2 ( 1 k 2 )[ | H(k) | 2 + | H(k) | 2 ] .
[ C n,m ]=[ 3.89 3.73 0.16 3.73 17.02 13.3 0.16 13.3 13.46 ]( λ 0 λ 1 λ 2 )=( 28.95 5.42 0 )
A(x,y)= n=0 2 ( [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
H(ω)= n=0 2 ( [ v 0 ] n +i [ v 1 ] n ) e i θ n ω .
A= b 2 H(1) e iφ [ 1+ H(1) H(1) e 2iφ ]= b 2 e iφ [ 1+0.31 e 2iφ ];(x,y).
ρ= x=0 L1 y=0 L1 | Im[ A(x,y) ] | x=0 L1 y=0 L1 | Re[ A(x,y) ] | =0.432.

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