Abstract

A method combining the principal component analysis (PCA) and the least squares method (LSM) is proposed to extract the phase from interferograms with random phase shifts. The method estimates the initial phase by PCA, and then determines the correct global phase sign and reduces the residual phase error by LSM. Some factors that may influence the performance of the proposed method are analyzed and discussed, such as the number of frames used, the number of fringes in interferogram and the amplitude of random phase shifts. Numerical simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for randomly phase-shifted interferograms.

© 2011 OSA

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References

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2011

2010

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]

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

2009

2008

2007

2006

2004

2001

2000

1998

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

1997

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

1995

1994

1993

1992

1991

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]

1990

1987

1985

Apostol, D.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Belenguer, T.

Bokor, J.

Cai, L. Z.

Chai, L.

Chen, M.

Chen, X.

Cheng, X. C.

Cheng, Y.-Y.

Dai-Lin, L.

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

Damian, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Deck, L. L.

Dobroiu, A.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Dong, G. Y.

Eiju, T.

Farrant, D. I.

Gao, P.

Geist, E.

Goldberg, K. A.

Gramaglia, M.

Guo, H.

Guo, J. P.

Han, B.

Han, G.-S.

Harder, I.

Hariharan, P.

Hibino, K.

Ishii, Y.

Kim, S.-W.

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]

G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. 33(31), 7321–7325 (1994).
[CrossRef] [PubMed]

Kong, B.

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]

Langoju, R.

Larkin, K. G.

Li, A. M.

Li, Y.

Lindlein, N.

Liu, Q.

Logofatu, P. C.

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

Lu-Zhong, C.

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

Mantel, K.

Meng, X. F.

Nascov, V.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

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]

Onodera, R.

Oreb, B. F.

Patil, A.

Peng, X.

Quiroga, J. A.

Rastogi, P.

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.

Shen, X. X.

Sun, L.

Sun, W. J.

Surrel, Y.

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.

Wang, H.

Wang, Y. R.

Wang, Z.

Wizinowich, P. L.

Wyant, J. C.

Xian-Feng, X.

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

Xu, J.

Xu, Q.

Xu, X. F.

Yang, X. L.

Yao, B.

Yeazell, J. A.

Yu, Y.

Yu-Rong, W.

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

Zhang, H.

Appl. Opt.

Chin. Phys. Lett.

X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. 27(2), 024215 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. 8(7), 738–745 (1997).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. 9(5), 744–750 (1998).
[CrossRef]

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. 9(9), 1451–1455 (1998).
[CrossRef]

Opt. Commun.

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.

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

Opt. Lett.

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]

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]

Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. 20(18), 1883–1885 (1995).
[CrossRef] [PubMed]

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

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. 34(8), 1210–1212 (2009).
[CrossRef] [PubMed]

L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. 29(2), 183–185 (2004).
[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]

R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise,” Opt. Lett. 31(8), 1058–1060 (2006).
[CrossRef] [PubMed]

X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006).
[CrossRef] [PubMed]

Other

http://en.wikipedia.org/wiki/Principal_component_analysis .

http://en.wikipedia.org/wiki/Singular_value_decomposition .

http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem .

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Figures (6)

Fig. 1
Fig. 1

Simulation results: (a) the fringe; (b) the given measured phase; (c) and (d) the extracted phase and the residual phase error of the PCA method; and (e) and (f) the extracted phase and the residual phase error of the “PCA+LSM” method.

Fig. 2
Fig. 2

The influence of the number of frames used on (a) the phase-shift extraction error and (b) the phase error.

Fig. 3
Fig. 3

The influence of the number of fringes in each interferogram on (a) the phase-shift extraction error and (b) the phase error.

Fig. 4
Fig. 4

The influence of the amplitude of random phase shifts on the residual phase error for (a) N = 3 and (b) N = 5.

Fig. 5
Fig. 5

Comparison of the performance of phase shift extraction between our method and Gao’s method.

Fig. 6
Fig. 6

Experimental results. (a) the interferogram, (b) the extracted phase by AIA, (c) the extracted phase by PCA, (d) the extracted phase by PCA + LSM, (e) the phase error of PCA, and (f) the phase error of PCA + LSM.

Equations (13)

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

I= [ I 1 , I 2 ,, I n ,, I N ] T
[I μ I ]( B 1 cos( φ 1 + θ 1 ) B m cos( φ m + θ 1 ) B M cos( φ M + θ 1 ) B 1 cos( φ 1 + θ N ) B m cos( φ m + θ N ) B M cos( φ M + θ N ) )
[I μ I ] mn = B m cos( φ m + θ n )= a n u m + b n v m
m=1 M u m v m = 1 2 m=1 M B m 2 sin(2 φ m ) << m=1 M u m u m = m=1 M B m 2 cos 2 ( φ m )
m=1 M u m v m = 1 2 m=1 M B m 2 sin(2 φ m ) << m=1 M v m v m = m=1 M B m 2 sin 2 ( φ m )
C=[I μ I ] [I μ I ] T
Φ T CΦ=D
Y= Φ T (I μ I )
φ=ta n 1 (v/u )=±ta n 1 ( y 2 / y 1 )
[ a n b n c n ]= [ M m=1 M cos φ m m=1 M sin φ m m=1 M cos φ m m=1 M cos 2 φ m m=1 M sin φ m cos φ m m=1 M sin φ m m=1 M sin φ m cos φ m m=1 M sin 2 φ m ] 1 [ m=1 M I nm m=1 M I nm cos φ m m=1 M I nm sin φ m ]
θ n ~ =ta n 1 ( c n / b n )
[ a m b m c m ]= [ N n=1 N cos θ n n=1 N sin θ n n=1 N cos θ n n=1 N cos 2 θ n n=1 N sin θ n cos θ n n=1 N sin θ n n=1 N sin θ n cos θ n n=1 N sin 2 θ n ] 1 [ n=1 N I nm n=1 N I nm cos θ n n=1 N I nm sin θ n ]
φ m ~ =ta n 1 ( c m / b m )

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