Abstract

This paper presents a non-iterative phase retrieval method from randomly phase-shifted fringe images. By combining the hyperaccurate least squares ellipse fitting method with the subspace method (usually called the principal component analysis), a fast and accurate phase retrieval algorithm is realized. The proposed method is simple, flexible, and accurate. It can be easily coded without iteration, initial guess, or tuning parameter. Its flexibility comes from the fact that totally random phase-shifting steps and any number of fringe images greater than two are acceptable without any specific treatment. Finally, it is accurate because the hyperaccurate least squares method and the modified subspace method enable phase retrieval with a small error as shown by the simulations. A MATLAB code, which is used in the experimental section, is provided within the paper to demonstrate its simplicity and easiness.

© 2016 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2016 (10)

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

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, 105604 (2016).
[Crossref]

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref]

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, 12922–12932 (2016).
[Crossref]

K. Yatabe and Y. Oikawa, “Convex optimization based windowed Fourier filtering with multiple windows for wrapped phase denoising,” Appl. Opt.,  55, 4632–4641 (2016).
[Crossref]

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Compensation of fringe distortion for phase-shifting three-dimensional shape measurement by inverse map estimation,” Appl. Opt. 55, 6017–6024 (2016).
[Crossref]

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, 22881–22891 (2016).
[Crossref]

2015 (5)

2014 (2)

A. Albertazzi Jr., 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. Optomechatron. 8, 340–356 (2014).
[Crossref]

A. Albertazzi Jr., 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. Optomechatron. 8, 340–356 (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]

2013 (5)

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, 626–632 (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 and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51, 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, 1506–1508 (2013).
[Crossref]

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

2011 (5)

2009 (1)

2008 (1)

2007 (1)

2005 (1)

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

2004 (2)

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

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (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, 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, 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, 953–958 (1992).
[Crossref]

1991 (2)

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

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

1988 (1)

1985 (1)

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, 626–632 (2013).
[Crossref]

Albertazzi, A. G.

Albertazzi Jr, A.

A. Albertazzi Jr., 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. Optomechatron. 8, 340–356 (2014).
[Crossref]

Al-Sharadqah, A.

K. Kanatani, A. Al-Sharadqah, N. Chernov, and Y. Sugaya, “Hyper-renormalization: non-minimization approach for geometric estimation,” Inf. Media Technol. 10, 71–87 (2015).
[Crossref]

Awatsuji, Y.

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

Belenguer, T.

Benedet, M. E.

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref]

A. Albertazzi Jr., 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. Optomechatron. 8, 340–356 (2014).
[Crossref]

Bokor, J.

Bruning, J. H.

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

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, 105604 (2016).
[Crossref]

Chen, W.

Cheng, X. C.

Cheng, Y.-Y.

Chernov, N.

K. Kanatani, A. Al-Sharadqah, N. Chernov, and Y. Sugaya, “Hyper-renormalization: non-minimization approach for geometric estimation,” Inf. Media Technol. 10, 71–87 (2015).
[Crossref]

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, 24416 (2016).
[Crossref]

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, 12222–12231 (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, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38, 1506–1508 (2013).
[Crossref]

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. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref]

A. Albertazzi Jr., 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. Optomechatron. 8, 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, 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, 953–958 (1992).
[Crossref]

Gao, P.

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, 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, 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, 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, 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, 20483–20492 (2011).
[Crossref]

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, 626–632 (2013).
[Crossref]

Kanatani, K.

K. Kanatani, A. Al-Sharadqah, N. Chernov, and Y. Sugaya, “Hyper-renormalization: non-minimization approach for geometric estimation,” Inf. Media Technol. 10, 71–87 (2015).
[Crossref]

K. Kanatani and P. Rangarajan, “Hyper least squares fitting of circles and ellipses,” Comput. Statist. Data Anal. 55, 2197–2208 (2011).
[Crossref]

K. Kanatani, P. Rangarajan, Y. Sugaya, and H. Niitsuma, “HyperLS for parameter estimation in geometric fitting,” IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011).
[Crossref]

K. Kanatani, Y. Sugaya, and Y. Kanazawa, “Ellipse fitting for computer vision: implementation and applications,” in Synthesis Lectures on Computer Vision (Morgan & Claypool, 2016).

K. Kanatani, “Statistical optimization for geometric estimation: minimization vs. non-minimization,” in 22nd International Conference on Pattern Recognition (ICPR) (2014), pp. 1–8.

Kanazawa, Y.

K. Kanatani, Y. Sugaya, and Y. Kanazawa, “Ellipse fitting for computer vision: implementation and applications,” in Synthesis Lectures on Computer Vision (Morgan & Claypool, 2016).

Kim, S.-W.

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

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85, 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, 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, 10794–10807 (2015).
[Crossref]

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, 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, 1506–1508 (2013).
[Crossref]

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]

Malacara, D.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).

Malacara, Z.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).

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, 13589–13604 (2015).
[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, 626–632 (2013).
[Crossref]

Meng, X. F.

Niitsuma, H.

K. Kanatani, P. Rangarajan, Y. Sugaya, and H. Niitsuma, “HyperLS for parameter estimation in geometric fitting,” IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011).
[Crossref]

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, 118–124 (1991).
[Crossref]

Onuma, T.

Patil, A.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 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, 105604 (2016).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 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, 953–958 (1992).
[Crossref]

Qin, J.

Quiroga, J. A.

Rangarajan, P.

K. Kanatani, P. Rangarajan, Y. Sugaya, and H. Niitsuma, “HyperLS for parameter estimation in geometric fitting,” IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011).
[Crossref]

K. Kanatani and P. Rangarajan, “Hyper least squares fitting of circles and ellipses,” Comput. Statist. Data Anal. 55, 2197–2208 (2011).
[Crossref]

Rastogi, P.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 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, 626–632 (2013).
[Crossref]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85, 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, 118–124 (1991).
[Crossref]

Schreiber, H.

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

Schwider, J.

Servín, M.

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).

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, 637–641 (2013).
[Crossref]

Streibl, N.

Sugaya, Y.

K. Kanatani, A. Al-Sharadqah, N. Chernov, and Y. Sugaya, “Hyper-renormalization: non-minimization approach for geometric estimation,” Inf. Media Technol. 10, 71–87 (2015).
[Crossref]

K. Kanatani, P. Rangarajan, Y. Sugaya, and H. Niitsuma, “HyperLS for parameter estimation in geometric fitting,” IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011).
[Crossref]

K. Kanatani, Y. Sugaya, and Y. Kanazawa, “Ellipse fitting for computer vision: implementation and applications,” in Synthesis Lectures on Computer Vision (Morgan & Claypool, 2016).

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, 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, 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, 24416 (2016).
[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 and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51, 637–641 (2013).
[Crossref]

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

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36, 2215–2217 (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, 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, 1506–1508 (2013).
[Crossref]

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, 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, 24416 (2016).
[Crossref]

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, 12222–12231 (2015).
[Crossref]

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, 025704 (2015).
[Crossref]

Wang, Y. R.

Wang, Z.

Willemann, D. P.

A. V. Fantin, D. P. Willemann, M. E. Benedet, and A. G. Albertazzi, “Robust method to improve the quality of shearographic phase maps obtained in harsh environments,” Appl. Opt. 55, 1318–1323 (2016).
[Crossref]

A. Albertazzi Jr., 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. Optomechatron. 8, 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, 24416 (2016).
[Crossref]

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, 12222–12231 (2015).
[Crossref]

Wu, 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, 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, 10794–10807 (2015).
[Crossref]

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, 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, 10794–10807 (2015).
[Crossref]

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, 1506–1508 (2013).
[Crossref]

Appl. Opt. (6)

Appl. Phys. Lett. (1)

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

Comput. Statist. Data Anal. (1)

K. Kanatani and P. Rangarajan, “Hyper least squares fitting of circles and ellipses,” Comput. Statist. Data Anal. 55, 2197–2208 (2011).
[Crossref]

Inf. Media Technol. (1)

K. Kanatani, A. Al-Sharadqah, N. Chernov, and Y. Sugaya, “Hyper-renormalization: non-minimization approach for geometric estimation,” Inf. Media Technol. 10, 71–87 (2015).
[Crossref]

Int. J. Optomechatron. (1)

A. Albertazzi Jr., 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. Optomechatron. 8, 340–356 (2014).
[Crossref]

IPSJ Trans. Comput. Vis. Appl. (1)

K. Kanatani, P. Rangarajan, Y. Sugaya, and H. Niitsuma, “HyperLS for parameter estimation in geometric fitting,” IPSJ Trans. Comput. Vis. Appl. 3, 80–94 (2011).
[Crossref]

J. Opt. (2)

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, 105604 (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, 025704 (2015).
[Crossref]

J. Opt. Soc. Am. A (2)

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, 953–958 (1992).
[Crossref]

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5, 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]

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

Opt. Eng. (1)

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

Opt. Express (6)

Opt. Lasers Eng. (5)

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

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43, 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, 626–632 (2013).
[Crossref]

Opt. Lett. (8)

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, 24416 (2016).
[Crossref]

Other (4)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis For Optical Testing, 2nd ed. (CRC, 2005).

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

K. Kanatani, Y. Sugaya, and Y. Kanazawa, “Ellipse fitting for computer vision: implementation and applications,” in Synthesis Lectures on Computer Vision (Morgan & Claypool, 2016).

K. Kanatani, “Statistical optimization for geometric estimation: minimization vs. non-minimization,” in 22nd International Conference on Pattern Recognition (ICPR) (2014), pp. 1–8.

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Figures (6)

Fig. 1.
Fig. 1. Schematic illustration of the subspace method ( M = 3 ) . Data vectors are ideally distributed as a circle as (a), which is shown in Eq. (4). By subtracting the mean vector as in (b), the data vectors lie in a linear subspace as in (c), which can be detected by SVD of the corresponding data matrix. The red and green arrows represent the two leading vectors of U . The 2D representation of data vectors is obtained by extracting the subspace as in (d). In this example, distribution of the extracted data in (d) is a circle because the two conditions in Section 2.D are satisfied.
Fig. 2.
Fig. 2. Schematic illustration of the ellipse fitting method. When the assumptions of phase reconstruction are not satisfied, observed data form an elliptic distribution in the 2D space as in (a). By fitting an ellipse to the data (b), the elliptic distribution can be corrected as in (c) and (d). Since phase is the angle between the horizontal axis and a data vector that is depicted by the dashed lines, the deformation and translation of the data distribution results in an error of the phase.
Fig. 3.
Fig. 3. MATLAB code for steps 5–7 in the proposed method. The input is an M × N data matrix D introduced in Section 2, where M is the number of fringe images, and N is the total number of pixels to be considered. The extended L M × N data matrix, which incorporates L adjacent pixels in each data vector as explained in Section 2.C, is also acceptable by the same code. In this example, the functions are divided only for demonstration purposes. Obviously, combining all functions into a single function can reduce the number of total lines.
Fig. 4.
Fig. 4. Example of data distributions obtained by the phase retrieval algorithms ( L = 1 ). The phase φ = x 1 2 + x 2 2 shown in (a) was observed as three fringe images as in (b). (c–h) show the distribution of the data vectors, where each blue dot represents a 3D data vector for an ( i , j ) -th pixel, d i , j = [ I i , j [ 1 ] , I i , j [ 2 ] , I i , j [ 3 ] ] T , projected to the 2D subspace by SVD. (f) and (h) were obtained by converting the ellipses of (e) and (g) back into the circles, respectively (Fig. 2). The black cross marks illustrate the origin of the coordinate, while the red ones show the centers of the fitted ellipses. In this case, only HEFS achieved a comparable RMSE to AIA with the best initial guess whose RMSE was 0.0097 rad.
Fig. 5.
Fig. 5. RMSE of each method for the equally phase-shifted data with different noise levels. Each line corresponds to the different σ { 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } as in the legend. The horizontal axis represents the number of fringe images M used for the phase extraction. The vertical axis denotes RMSE, where every axis is illustrated by the same scale.
Fig. 6.
Fig. 6. Mean RMSE of each method for the randomly phase-shifted data with different noise levels. Each line corresponds to different σ { 0.1 , 0.2 , 0.3 , 0.4 , 0.5 } as in the legend. The horizontal axis represents the number of fringe images M used for the phase extraction. The vertical axis denotes the mean RMSE, where every axis is illustrated by the same scale.

Tables (1)

Tables Icon

Table 1. Computational Time of the Phase Extraction from Five Fringe Images ( M = 5 ) Whose Size Was 300 × 300 , Where Constructions of the Data Matrices Were Included

Equations (40)

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

I i , j [ m ] = B i , j + A i , j cos ( φ i , j + δ [ m ] ) ,
d i , j = [ I i , j [ 1 ] , I i , j [ 2 ] , , I i , j [ M ] ] T ,
I i , j [ m ] = B i , j + A i , j [ cos ( φ i , j ) cos ( δ [ m ] ) sin ( φ i , j ) sin ( δ [ m ] ) ] ,
d i , j = B i , j o + C i , j c + S i , j s ,
c = [ cos ( δ [ 1 ] ) , cos ( δ [ 2 ] ) , , cos ( δ [ M ] ) ] T ,
s = [ sin ( δ [ 1 ] ) , sin ( δ [ 2 ] ) , , sin ( δ [ M ] ) ] T .
d i , j d ¯ i , j = d i , j d ¯ i , j o = d i , j o M m = 1 M I i , j [ m ] = d i , j o o T o 2 d i , j ,
φ i , j = Arg [ C i , j ι S i , j ] = Arg [ A i , j { cos ( φ i , j ) + ι sin ( φ i , j ) } ] ,
D = [ d 1 , 1 , d 2 , 1 , , d N v , 1 , d 1 , 2 , d 2 , 2 , d N v , N h ] ,
D = U Σ V T ,
D D T = U Σ V T V Σ U T = U Σ 2 U T .
d i , j ( 2 ) = [ d i , j T , d i + 1 , j T ] T ,
d i , j ( 5 ) = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T ] T ,
d i , j ( 9 ) = [ d i , j T , d i + 1 , j T , d i 1 , j T , d i , j + 1 T , d i , j 1 T , d i + 1 , j + 1 T , d i + 1 , j 1 T , d i 1 , j + 1 T , d i 1 , j 1 T ] T .
α 1 x 2 + 2 α 2 x y + α 3 y 2 + 2 β ( α 4 x + α 5 y ) + β 2 α 6 = 0 ,
α T χ = 0 ,
α = [ α 1 , α 2 , α 3 , α 4 , α 5 , α 6 ] T ,
χ = [ x 2 , 2 x y , y 2 , 2 β x , 2 β y , β 2 ] T .
1 N n = 1 N ( α T χ n ) 2 ,
χ n = [ x n 2 , 2 x n y n , y n 2 , 2 β x n , 2 β y n , β 2 ] T ,
α arg min α 1 N n = 1 N ( α T χ n ) 2 s.t. α 2 = 1 ,
α arg min α α T X α α T α ,
X = 1 N n = 1 N χ n χ n T ,
1 N n = 1 N ( α T χ n ) 2 = 1 N n = 1 N α T χ n χ n T α = α T [ 1 N n = 1 N χ n χ n T ] α .
X α = λ α ,
φ = Arg [ α 1 + α 3 + κ ( y ˜ + τ x ˜ ) + ι α 1 + α 3 κ ( x ˜ τ y ˜ ) ] ,
x ˜ = x β ( α 3 α 4 α 2 α 5 ) / ( α 2 2 α 1 α 3 ) ,
y ˜ = y β ( α 1 α 5 α 2 α 4 ) / ( α 2 2 α 1 α 3 ) ,
κ = 4 α 2 2 + ( α 1 α 3 ) 2 ,
τ = ( α 1 α 3 + κ ) / ( 2 α 2 ) .
W = [ 6 x ¯ 2 6 x y ¯ x ¯ 2 + y ¯ 2 6 β x ¯ 2 β y ¯ β 2 6 x y ¯ 4 ( x ¯ 2 + y ¯ 2 ) 6 x y ¯ 4 β y ¯ 4 β x ¯ 0 x ¯ 2 + y ¯ 2 6 x y ¯ 6 y ¯ 2 2 β x ¯ 6 β y ¯ β 2 6 β x ¯ 4 β y ¯ 2 β x ¯ 4 β 2 0 0 2 β y ¯ 4 β x ¯ 6 β y ¯ 0 4 β 2 0 β 2 0 β 2 0 0 0 ] ,
x ¯ 2 = 1 N n = 1 N x n 2 ,
y ¯ 2 = 1 N n = 1 N y n 2 ,
x y ¯ = 1 N n = 1 N x n y n .
α arg min α α T X α α T W α ,
X α = λ W α .
W α = λ X α ,
I [ m ] ( x ) = B ( x ) + A ( x ) cos ( φ ( x ) + δ [ m ] ) + σ ε ( x ) ,
δ [ m ] = 2 π m / M ,
φ = Arg [ ( m = 1 M I [ m ] cos ( 2 π m / M ) ) ι ( m = 1 M I [ m ] sin ( 2 π m / M ) ) ] ,

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