Abstract

This paper presents a simple algorithm for estimating phase shifts from only three interferograms. In it, the fringe pattern differences are computed first in order to remove the background component, and then the variances and further the standard deviations (SDs) of fringe pattern differences are calculated. The phase shifts are estimated, by using the law of cosines, from a triangle whose lengths of sides are the SDs just calculated. This algorithm offers several advantages over others, e.g., being efficient, easy to implement, accurate, and less sensitive to noise. Numerical simulations and an experiment are performed to demonstrate its validity.

© 2013 Optical Society of America

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    [CrossRef]
  2. 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]
  3. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
    [CrossRef]
  4. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
    [CrossRef]
  5. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
    [CrossRef]
  6. 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]
  7. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
    [CrossRef]
  8. L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
    [CrossRef]
  9. 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, 1966–1968 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  17. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 25–33 (2007).
    [CrossRef]
  18. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19, 7807–7815 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2013

2011

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

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19, 7807–7815 (2011).
[CrossRef]

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

E. Hack and J. Burke, “Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[CrossRef]

2010

2009

2007

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 25–33 (2007).
[CrossRef]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
[CrossRef]

2006

2004

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
[CrossRef]

K. Qian, S. H. Soon, and A. Asundi, “Calibration of phase shift from two fringe patterns,” Meas. Sci. Technol. 15, 2142–2144 (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]

2003

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).
[CrossRef]

2001

2000

1995

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]

1992

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

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

1985

1982

Asundi, A.

K. Qian, S. H. Soon, and A. Asundi, “Calibration of phase shift from two fringe patterns,” Meas. Sci. Technol. 15, 2142–2144 (2004).
[CrossRef]

Belenguer, T.

Bokor, J.

Brohinsky, W. R.

Burke, J.

E. Hack and J. Burke, “Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[CrossRef]

Cai, L. Z.

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, 1966–1968 (2006).
[CrossRef]

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
[CrossRef]

Chai, L.

Chen, M.

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 25–33 (2007).
[CrossRef]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
[CrossRef]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).
[CrossRef]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
[CrossRef]

Chen, X.

Chin, K. C.

Deng, J.

Dong, G. Y.

Fan, J.

Farrell, C. T.

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.

Goodman, J. W.

Gramaglia, M.

Guo, H.

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19, 7807–7815 (2011).
[CrossRef]

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 25–33 (2007).
[CrossRef]

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
[CrossRef]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).
[CrossRef]

M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
[CrossRef]

Hack, E.

E. Hack and J. Burke, “Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[CrossRef]

Han, B.

Hao, Q.

Harder, I.

Hu, Y.

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]

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]

Larkin, K. G.

Li, Y.

Lindlein, N.

Liu, Q.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
[CrossRef]

Lu, X.

Mantel, K.

Meng, X. F.

Okada, K.

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

Player, M. A.

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]

Qian, K.

K. Qian, S. H. Soon, and A. Asundi, “Calibration of phase shift from two fringe patterns,” Meas. Sci. Technol. 15, 2142–2144 (2004).
[CrossRef]

Quiroga, J. A.

Sato, A.

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

Shen, X. X.

Soon, S. H.

K. Qian, S. H. Soon, and A. Asundi, “Calibration of phase shift from two fringe patterns,” Meas. Sci. Technol. 15, 2142–2144 (2004).
[CrossRef]

Stetson, K. A.

Tsujiuchi, J.

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

Tur, M.

Vargas, J.

Wang, H.

Wang, Z.

Wei, C.

Xu, J.

Xu, Q.

Xu, X. F.

Yang, X. L.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
[CrossRef]

Yao, B.

Yeazell, J. A.

Yu, Y.

Zhang, D.

Zhao, Z.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
[CrossRef]

Zhong, L.

Zhu, Q.

Appl. Opt.

IEEE Signal Process. Mag.

H. Guo, “A simple algorithm for fitting a Gaussian function,” IEEE Signal Process. Mag. 28(5), 134–137 (2011).
[CrossRef]

J. Opt. Soc. Am. A

Meas. Sci. Technol.

K. Qian, S. H. Soon, and A. Asundi, “Calibration of phase shift from two fringe patterns,” Meas. Sci. Technol. 15, 2142–2144 (2004).
[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]

Opt. Commun.

L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233, 21–26 (2004).
[CrossRef]

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

Opt. Eng.

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

Opt. Lasers Eng.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45, 281–292 (2007).
[CrossRef]

Opt. Lett.

Proc. SPIE

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437–444 (2003).
[CrossRef]

Rev. Sci. Instrum.

E. Hack and J. Burke, “Measurement uncertainty of linear phase-stepping algorithms,” Rev. Sci. Instrum. 82, 061101 (2011).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

(a) Vectors of complex amplitudes in the complex plane and (b) triangle with the side lengths being the standard deviations of pattern differences.

Fig. 2.
Fig. 2.

(a) Simulated interferogram with 1.5 fringes where the SD of Gaussian noise is 0.04 and (b) its phase map recovered using the phase shifts estimated with the proposed algorithm, where the phase range is from π to π rad. (c) and (d) are similar to (a) and (b), but the interferogram contains four fringes.

Fig. 3.
Fig. 3.

RMS errors of recovered phase maps, depending on the levels of Gaussian noise, are functions of the number of fringes appearing in each interferogram.

Fig. 4.
Fig. 4.

(a) Simulated interferogram with 1.5 fringes contaminated by speckle noise (SD is 0.2) and (b) its phase map recovered using the phase shifts estimated with the proposed algorithm, where the phase range is from π to π rad. (c) and (d) are similar to (a) and (b), but the interferogram contains four fringes.

Fig. 5.
Fig. 5.

RMS errors of recovered phase maps, depending on the levels of speckle noise, are functions of the number of fringes appearing in each interferogram.

Fig. 6.
Fig. 6.

(a) Simulated interferogram with circular fringes, in which the SDs of Gaussian noise and speckle noise are 0.01 and 0.1 and (b) its phase map recovered using the phase shifts estimated with the proposed algorithm, where the phase range is from π to π rad.

Fig. 7.
Fig. 7.

Experimental result. (a) Interferogram of grating and (b) its phase map reconstructed with the phase shifts estimated using the proposed algorithm, where the phase range is from π to π rad.

Tables (2)

Tables Icon

Table 1. Simulation Results in the Presence of Additive Gaussian Noise with Interferograms Containing 1.5 and 4 Fringes

Tables Icon

Table 2. Simulation Results in the Presence of Speckle Noise with Interferograms Containing 1.5 and 4 Fringes

Equations (13)

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

Ik(x,y)=a(x,y)+b(x,y)cos[φ(x,y)+δk],
Δk,l=IlIk=b{cos[φ+δl]cos[φ+δk]}=2bsin[(δkδl)/2]sin[φ+(δk+δl)/2],
Ak,l=2bsin[(δkδl)/2]exp(iδk+δl2),
Al,m=Ak,mAk,l.
δmδl2=arccos(|Ak,m|2+|Ak,l|2|Al,m|22|Ak,m||Ak,l|).
σkl2=σlk2=1Mx,y[Il(x,y)Ik(x,y)]2,
σkl=σlk=1Mx,y[Il(x,y)Ik(x,y)]2.
σkl2=σlk2=2sin2[(δkδl)/2]M×{x,yb2(x,y)x,yb2(x,y)cos[2φ(x,y)+(δk+δl)]}.
σkl=σlk2|sin[(δkδl)/2]|1Mx,yb2(x,y).
σlmσkmσkl=|Al,m||Ak,m||Ak,l|.
δmδl2=arccos(σkm2+σkl2σlm22σkmσkl).
δ1=2arccos(σ212+σ202σ1022σ21σ20).
δ2=2π2arccos(σ212+σ102σ2022σ21σ10).

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