Abstract

A rapid and precise phase-retrieval method based on Lissajous ellipse fitting and ellipse standardization is demonstrated. It only requires two interferograms without pre-filtering, which reduces its complexity and shortens the processing time. The elliptic coefficients obtained by ellipse fitting are used for ellipse standardization. After compensating phase-shift errors by ellipse standardization, the phase distribution is extracted with high precision. It is suitable for fluctuation, noise, tilt-shift, simple and complex fringes. This method is effective for the number of fringes less than 1. The reliability of the method is verified by simulations and experiments, indicating high accuracy and less time consumption.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  15. Y. Zhang, X.B. Tian, and R.G Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15072 (2018).
    [Crossref]
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    [Crossref]

2019 (3)

2018 (2)

2017 (1)

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

2015 (3)

2014 (1)

2013 (2)

Q. Liu, Y. Wang, F. Ji, and J. He, “A three-step least-squares iterative method for tilt phase-shift interferometry,” Opt. Express 21(24), 29505–29515 (2013).
[Crossref]

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

2008 (1)

2002 (1)

W. Wei, J. Guo, and J.H. Guo, “Phase-shift interferometry surface plasmon microscopy,” N. Opt and N. St 4923, 26–30 (2002).
[Crossref]

2000 (1)

1997 (2)

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]

Albertazzi, A.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Benedet, M. E.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Bob, F.O.

Carazo, J.M.

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

Cha, L.Q.

Chen, M.Y.

David, I.F

Estrada, J.C.

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

Fan, C.

Fantin, A. V.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

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]

Guo, H.W.

Guo, J.

W. Wei, J. Guo, and J.H. Guo, “Phase-shift interferometry surface plasmon microscopy,” N. Opt and N. St 4923, 26–30 (2002).
[Crossref]

Guo, J.H.

W. Wei, J. Guo, and J.H. Guo, “Phase-shift interferometry surface plasmon microscopy,” N. Opt and N. St 4923, 26–30 (2002).
[Crossref]

He, J.

Hibino, K.

Ji, F.

Kieran, G.L.

Lara-Cortes, F. A.

Lei, H.B.

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

Liang, R.G

Liang, R.G.

Liu, F.W.

Liu, H.P.

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

Liu, Q.

Lu, X.X.

Luo, C.S.

Ma, S.Z.

Maia, A. F.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Malacara, D.

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007).

Meneses-Fabian, C.

Patorski, K.

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]

Sorzano, C.O.S.

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

Sunderland, Z.

Tian, X.B.

Vargas, J.

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

Viotti, M.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Wang, H.L.

Wang, Y.

Wei, C.L.

Wei, W.

W. Wei, J. Guo, and J.H. Guo, “Phase-shift interferometry surface plasmon microscopy,” N. Opt and N. St 4923, 26–30 (2002).
[Crossref]

Wielgus, M.

Willemann, D. P.

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

Wu, F.

Wu, Y.Q.

Xu, J.C.

Xu, Q.

Yamaguchi, I.

Yang, Y.F

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

Yao, Y.

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

Zhang, H.Y.

Zhang, T.

Zhang, Y.

Zhao, H.

Zhao, Z.X.

Zhong, L.Y.

Zhuang, Y.Y.

AOPC (1)

H.B. Lei, Y. Yao, H.P. Liu, and Y.F Yang, “Efficient phase extraction from three random interferograms based on Lissajous figure and ellipse fitting method,” AOPC 10458, 5 (2017).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

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]

N. Opt and N. St (1)

W. Wei, J. Guo, and J.H. Guo, “Phase-shift interferometry surface plasmon microscopy,” N. Opt and N. St 4923, 26–30 (2002).
[Crossref]

Opt. Commun. (1)

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

Opt. Express (9)

H.L. Wang, C.S. Luo, L.Y. Zhong, S.Z. Ma, and X.X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref]

Y. Zhang, X.B. Tian, and R.G. Liang, “Three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization,” Opt. Express 26(22), 29170–29182 (2018).
[Crossref]

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 (2015).
[Crossref]

F.W. Liu, Y.Q. 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]

Q. Liu, Y. Wang, F. Ji, and J. He, “A three-step least-squares iterative method for tilt phase-shift interferometry,” Opt. Express 21(24), 29505–29515 (2013).
[Crossref]

Y. Zhang, X.B. Tian, and R.G Liang, “Two-step random phase retrieval approach based on Gram-Schmidt orthonormalization and Lissajous ellipse fitting method,” Opt. Express 27(3), 2575–2588 (2019).
[Crossref]

H.Y. Zhang, H. Zhao, Z.X. Zhao, Y.Y. Zhuang, and C. Fan, “Two-frame fringe pattern phase demodulation using Gram-Schmidt orthonormalization with least squares method,” Opt. Express 27(8), 10495–10508 (2019).
[Crossref]

Y. Zhang, X.B. Tian, and R.G Liang, “Accurate and fast two-step phase shifting algorithm based on principle component analysis and Lissajous ellipse fitting with random phase shift and no pre-filtering,” Opt. Express 27(14), 20047–20063 (2019).
[Crossref]

Y. Zhang, X.B. Tian, and R.G Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15072 (2018).
[Crossref]

Opt. Lett. (2)

Other (2)

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc., 2007).

A. Albertazzi, A. V. Fantin, A. F. Maia, D. P. Willemann, M. E. Benedet, and M. Viotti, “Use of generalized Ndimensional Lissajous figures for phase retrieval from sequences of interferometric images with unknown phase shifts,” Fringe 2013, 191–196 (2014).

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Figures (12)

Fig. 1.
Fig. 1. Lissajous figures. (a) The oblique ellipse (with errors), (b) the standard ellipse (without errors).
Fig. 2.
Fig. 2. Simulated phase distribution and phase-shift distributions. (a) The simulated phase distribution, (b) the simulated linear tilt-shift distribution, (c) the simulated nonlinear tilt-shift distribution.
Fig. 3.
Fig. 3. The comparison between two algorithms with linear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.
Fig. 4.
Fig. 4. The comparison between the two algorithms with nonlinear tilt-shift. (a) The unwrapped phase calculated by the proposed algorithm, (b) the phase error distribution calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by Wielgus’s algorithm.
Fig. 5.
Fig. 5. The RMS phase error and processing time with the two algorithms. (a) RMS phase error with different levels of noise, (b) consuming time with different levels of noise.
Fig. 6.
Fig. 6. Performance comparison between the two algorithms (a) RMS phase error of different methods with different levels of fluctuation (b) RMS phase errors of proposed algorithm with different levels of fluctuation.
Fig. 7.
Fig. 7. The RMS phase errors of the two algorithms with different fringes.
Fig. 8.
Fig. 8. The phase distribution and phase error distribution with highly non-uniformity. (a) the unwrapped phase with linear tilt-shift, (b) the phase error distribution with linear tilt-shift, (c) the unwrapped phase with non-linear tilt-shift, (d) the phase error distribution with non-linear tilt-shift.
Fig. 9.
Fig. 9. Simulated result of complex fringes. (a) The simulated phase distribution, (b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm, (d) the phase error distribution calculated by the proposed algorithm, (e) the phase error distribution calculated by Wielgus’s algorithm.
Fig. 10.
Fig. 10. Experimental results of straight fringe interferograms with the number of fringes more than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.
Fig. 11.
Fig. 11. Experimental results of straight fringe interferograms with the number of fringes less than 1. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.
Fig. 12.
Fig. 12. Experimental results of circular fringe interferograms. (a) One of the phase shifted interferograms, b) the unwrapped phase calculated by the proposed algorithm, (c) the unwrapped phase calculated by Wielgus’s algorithm.

Tables (3)

Tables Icon

Table 1. The RMS phase errors and processing time with different levels of noise

Tables Icon

Table 2. RMS phase errors with different levels of fluctuation using different methods

Tables Icon

Table 3. RMS phase errors of different methods with different number of fringes

Equations (17)

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

I 1 ( x , y ) = a 1 ( x , y ) + b 1 ( x , y ) cos [ ϕ ( x , y ) ]
I 2 ( x , y ) = a 2 ( x , y ) + b 2 ( x , y ) cos [ ϕ ( x , y ) + δ ( x , y ) ]
I 1 = a 1 + b 1 cos ϕ
I 2 = a 2 + b 2 cos ( ϕ + δ ) = a 2 + b 2 cos ϕ cos δ b 2 sin ϕ sin δ
cos ϕ = ( I 1 a 1 ) / b 1
sin ϕ = a 2 b 2 sin δ a 1 cos δ b 1 sin δ + cos δ b 1 sin δ I 1 1 b 2 sin δ I 2
1 b 1 sin 2 δ I 1 2 2 cos δ b 1 b 2 sin 2 δ I 1 I 2 + 1 b 2 2 sin 2 δ I 2 2  + ( 2 a 2 cos δ b 1 b 2 sin 2 δ 2 a 1 cos 2 δ b 1 2 sin 2 δ 2 a 1 b 1 2 ) I 1 + ( 2 a 1 cos δ b 1 b 2 sin 2 δ 2 a 2 b 2 2 sin 2 δ ) I 2 + ( a 2 b 2 sin δ a 1 cos δ b 1 sin δ ) 2 + a 1 2 b 1 2 1 = 0
a I 1 2 + b I 1 I 2 + c I 2 2 + d I 1 + f I 2 + g = 0
( ( I 1 x 0 ) cos θ + ( I 2 y 0 ) sin θ ) 2 a x 2 + ( ( I 2 y 0 ) cos θ ( I 1 x 0 ) sin θ ) 2 a y 2 1 = 0
x 0 = b f 2 c d 4 a c b 2 , y 0 = b d 2 a f 4 a c b 2 a x = 2 a f 2 + c d 2 + g b 2 b d f 4 a c g ( b 2 4 a c ) ( ( a c ) 2 + b 2 ( a + c ) ) a y = 2 a f 2 + c d 2 + g b 2 b d f 4 a c g ( b 2 4 a c ) ( ( a c ) 2 + b 2 ( a + c ) ) θ = { 0 f o r b = 0 a n d a < c π 2 f o r b = 0 a n d a > c 1 2 arctan ( b a c ) f o r b 0 a n d a < c π 2 + 1 2 arctan ( b a c ) f o r b 0 a n d a > c
{ sin ϕ = X a x cos ϕ = Y a y
ϕ = arctan ( r X Y )  =  arctan ( r ( I 1 x 0 ) cos θ + ( I 2 y 0 ) sin θ ( I 2 y 0 ) cos θ ( I 1 x 0 ) sin θ )
a 1 ( x , y ) = 1 + 0.1 e x p ( 0.02 ( x 2 + y 2 ) ) ,   a 2 ( x , y ) = 0.9 + 0.21 e x p ( 0.02 ( x 2 + y 2 ) ) b 1 ( x , y ) = 1 + 0.1 e x p ( 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 0.9 + 0.21 e x p ( 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ( 0.02 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ( 0.02 ( x 2 + y 2 ) ) a 2 ( x , y ) = ( 1 + k ) e x p ( 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = ( 1 + k ) e x p ( 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ( 0.02 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ( 0.02 ( x 2 + y 2 ) ) a 2 ( x , y ) = 2 e x p ( 0.02 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 2 e x p ( 0.02 ( x 2 + y 2 ) )
a 1 ( x , y ) = e x p ( 0.2 ( x 2 + y 2 ) ) ,   b 1 ( x , y ) = e x p ( 0.2 ( x 2 + y 2 ) ) a 2 ( x , y ) = 2 e x p ( 0.2 ( x 2 + y 2 ) ) ,   b 2 ( x , y ) = 2 e x p ( 0.2 ( x 2 + y 2 ) )
ϕ ( x , y ) = 5 π ( 3 x 2 + y 2 ) + 4 p e a k s ( 512 )

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