Abstract

We propose a statistical phase-shifting estimation algorithm for temporal phase-shifting interferometry (PSI) based on the continuous wavelet transform (CWT). The proposed algorithm explores spatial information redundancy in the intraframe interferogram dataset using the phase recovery property on the power ridge of the CWT. Despite the errors introduced by the noise of the interferogram, the statistical part of the algorithm is utilized to give a sound estimation of the phase-shifting step. It also introduces the usage of directional statistics as the statistical model, which was validated, so as to offer a better estimation compared with other statistical models. The algorithm is implemented in computer codes, and the validations of the algorithm were performed on numerical simulated signals and actual phase-shifted moiré interferograms. The major advantage of the proposed algorithm is that it imposes weaker conditions on the presumptions in the temporal PSI, which, under most circumstances, requires uniform and precalibrated phase-shifting steps. Compared with other existing deterministic estimation algorithms, the proposed algorithm estimates the phase-shifting step statistically. The proposed algorithm allows the temporal PSI to operate under dynamic loading conditions and arbitrary phase steps and also without precalibration of the phase shifter. The proposed method can serve as a benchmark method for comparing the accuracy of the different phase-step estimation methods.

© 2011 Optical Society of America

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2008 (1)

B. Chen and C. Basaran, “Automatic full strain field Moire interferometry measurement with nano-scale resolution,” Exp. Mech. 48, 665–673 (2008).
[CrossRef]

2007 (4)

C. Han and B. Han, “Phase-shifting in achromatic moiré interferometry system,” Opt. Express 15, 9970–9976 (2007).
[CrossRef] [PubMed]

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
[CrossRef]

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274–280(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 (2)

2005 (1)

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393–403 (2005).
[CrossRef]

2004 (2)

2003 (1)

2002 (1)

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

2001 (1)

1999 (2)

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

1998 (1)

1997 (2)

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

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)

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

1991 (1)

1987 (1)

1985 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

1982 (1)

1974 (1)

Afifi, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Ai, C.

Barnes, T. H.

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Basaran, C.

B. Chen and C. Basaran, “Automatic full strain field Moire interferometry measurement with nano-scale resolution,” Exp. Mech. 48, 665–673 (2008).
[CrossRef]

H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef] [PubMed]

Batschelet, E.

E. Batschelet, Circular Statistics in Biology, Mathematics in Biology (Academic, 1981).

Bizuet, R.

Bowman, K. O.

L. R. Shenton and K. O. Bowman, Maximum Likelihood Estimation in Small Samples, Griffin’s Statistical Monographs & Courses (Macmillan, 1977).

Brangaccio, D. J.

Bruning, J. H.

Cai, L. Z.

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

Cartwright, A. N.

Chen, B.

B. Chen and C. Basaran, “Automatic full strain field Moire interferometry measurement with nano-scale resolution,” Exp. Mech. 48, 665–673 (2008).
[CrossRef]

Chen, M.

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

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

Cheng, Y.-Y.

Delprat, N.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Dong, G. Y.

Edwards, A. W. F.

A. W. F. Edwards, Likelihood: An Account of the Statistical Concept of Likelihood and Its Application to Scientific Inference (Cambridge University, 1972).
[PubMed]

Escudie, B.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[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]

Fassi-Fihri, A.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Gallagher, J. E.

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Guillemain, P.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Guo, H.

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

Gutmann, B.

Han, B.

C. Han and B. Han, “Phase-shifting in achromatic moiré interferometry system,” Opt. Express 15, 9970–9976 (2007).
[CrossRef] [PubMed]

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274–280(2007).
[CrossRef]

Han, C.

Herriott, D. R.

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

Jammalamadaka, S. R. S. A.

S. R. S. A. Jammalamadaka, Topics in Circular Statistics, Series on Multivariate Analysis (World Scientific, 2001).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing, Prentice Hall Signal Processing Series (Prentice-Hall, 1993).

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]

Kim, T.

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393–403 (2005).
[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]

Kovacevic, J.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995).

Kronland-Martinet, R.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Lai, G.

Larkin, K.

Liu, H.

H. Liu, A. N. Cartwright, and C. Basaran, “Moire interferogram phase extraction: a ridge detection algorithm for continuous wavelet transforms,” Appl. Opt. 43, 850–857 (2004).
[CrossRef] [PubMed]

H. Liu, “Phase reconstruction of phase shifted moire interferograms using continuous wavelet transforms,” Ph.D. dissertation (State University of New York at Buffalo, 2003).

Liu, Q.

Malacara, D.

D. Malacara, S. Manuel, and M. Zacarias, Interferogram Analysis for Optical Testing, 2nd ed., Optical Engineering and Science (Taylor & Francis, 2005).
[CrossRef]

Manuel, S.

D. Malacara, S. Manuel, and M. Zacarias, Interferogram Analysis for Optical Testing, 2nd ed., Optical Engineering and Science (Taylor & Francis, 2005).
[CrossRef]

Marjane, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Martinez, A.

Meng, X. F.

Morgan, C. J.

Nassim, K.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[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]

Rachafi, S.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Rayas, J. A.

Rivera, M.

Rosenfeld, D. P.

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393–403 (2005).
[CrossRef]

Shen, X. X.

Shenton, L. R.

L. R. Shenton and K. O. Bowman, Maximum Likelihood Estimation in Small Samples, Griffin’s Statistical Monographs & Courses (Macmillan, 1977).

Sidki, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Tan, S. M.

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Tchamitchian, P.

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Torresani, B.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

van den Bos, A.

A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley-Interscience, 2007).
[CrossRef]

Vetterli, M.

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995).

Wang, Z.

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274–280(2007).
[CrossRef]

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

Watkins, L. R.

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Weber, H.

Wei, C.

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

White, A. D.

Wyant, J. C.

Xu, X. F.

Yang, X. L.

Yatagai, T.

Zacarias, M.

D. Malacara, S. Manuel, and M. Zacarias, Interferogram Analysis for Optical Testing, 2nd ed., Optical Engineering and Science (Taylor & Francis, 2005).
[CrossRef]

Zhao, Z.

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

Appl. Opt. (5)

Electron. Lett. (1)

L. R. Watkins, S. M. Tan, and T. H. Barnes, “Interferometer profile extraction using continuous wavelet transform,” Electron. Lett. 33, 2116–2117 (1997).
[CrossRef]

Exp. Mech. (2)

B. Chen and C. Basaran, “Automatic full strain field Moire interferometry measurement with nano-scale resolution,” Exp. Mech. 48, 665–673 (2008).
[CrossRef]

C. A. Sciammarella and T. Kim, “Frequency modulation interpretation of fringes and computation of strains,” Exp. Mech. 45, 393–403 (2005).
[CrossRef]

IEEE Trans. Inf. Theory (1)

N. Delprat, B. Escudie, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, and B. Torresani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
[CrossRef]

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

Meas. Sci. Technol. (1)

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. (1)

M. Afifi, A. Fassi-Fihri, M. Marjane, K. Nassim, M. Sidki, and S. Rachafi, “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47–51 (2002).
[CrossRef]

Opt. Eng. (3)

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
[CrossRef]

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]

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

Opt. Express (3)

Opt. Lasers Eng. (3)

Z. Wang and B. Han, “Advanced iterative algorithm for randomly phase-shifted interferograms with intra- and inter-frame intensity variations,” Opt. Lasers Eng. 45, 274–280(2007).
[CrossRef]

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

L. R. Watkins, “Phase recovery from fringe patterns using the continuous wavelet transform,” Opt. Lasers Eng. 45, 298–303 (2007).
[CrossRef]

Opt. Lett. (5)

Other (9)

D. Malacara, S. Manuel, and M. Zacarias, Interferogram Analysis for Optical Testing, 2nd ed., Optical Engineering and Science (Taylor & Francis, 2005).
[CrossRef]

H. Liu, “Phase reconstruction of phase shifted moire interferograms using continuous wavelet transforms,” Ph.D. dissertation (State University of New York at Buffalo, 2003).

A. van den Bos, Parameter Estimation for Scientists and Engineers (Wiley-Interscience, 2007).
[CrossRef]

A. W. F. Edwards, Likelihood: An Account of the Statistical Concept of Likelihood and Its Application to Scientific Inference (Cambridge University, 1972).
[PubMed]

L. R. Shenton and K. O. Bowman, Maximum Likelihood Estimation in Small Samples, Griffin’s Statistical Monographs & Courses (Macmillan, 1977).

S. M. Kay, Fundamentals of Statistical Signal Processing, Prentice Hall Signal Processing Series (Prentice-Hall, 1993).

E. Batschelet, Circular Statistics in Biology, Mathematics in Biology (Academic, 1981).

S. R. S. A. Jammalamadaka, Topics in Circular Statistics, Series on Multivariate Analysis (World Scientific, 2001).
[CrossRef]

M. Vetterli and J. Kovacevic, Wavelets and Subband Coding (Prentice-Hall, 1995).

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Figures (6)

Fig. 1
Fig. 1

Flow chart of the proposed phase-shifting step estimation algorithm.

Fig. 2
Fig. 2

Input signal at δ 1 = 0 and δ 3 = 4 / 5 π .

Fig. 3
Fig. 3

CWT estimation of the phase difference for δ 3 = 4 / 5 π compared with the true value.

Fig. 4
Fig. 4

Probability distribution of the CWT estimation of the phase-shift step in the numerical simulation for δ 3 = 4 / 5 π .

Fig. 5
Fig. 5

Moiré interferometry data at (a)  δ 1 and (b)  δ 3 .

Fig. 6
Fig. 6

Probability of the CWT estimation of the phase-shifting step in the moiré interferometry experimental data for δ 3 .

Tables (3)

Tables Icon

Table 1 Probability Distribution Candidates for the Parametric Parameter Estimation

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Table 2 Parameter Estimate for the Numerical Simulation

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Table 3 Parameter Estimate for the Moiré Interferometry Experimental Data

Equations (14)

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I k ( i , j ) = a ( i , j ) + b ( i , j ) cos [ ϕ ( i , j ) + Δ ϕ k ] ,
I k ( i , j ) = a ( i , j ) + b ( i , j ) cos ( Δ ϕ k ) cos [ ϕ ( i , j ) ] b ( i , j ) sin ( Δ ϕ k ) sin [ ϕ ( i , j ) ] .
MSE = Σ i , j [ I e ( i , j ) I k ( i , j ) ] 2 ,
2 Σ i , j I k ( i , k ) c n I k ( i , j ) = 0 , n = 1 , 2 , 3 , ,
c ( i , j ) = [ a ( i , j ) b ( i , j ) cos ϕ ( i , j ) b ( i , j ) sin ϕ ( i , j ) ] ,
Δ ϕ = 4 π δ Λ ,
W f ( a , b ) = + f ( t ) ψ a , b * ( t ) d t ,
ψ a , b ( x ) = 1 a ψ ( x b a ) .
ψ ( t ) = 1 2 π exp ( j ω 0 t ) exp ( t 2 2 ) ,
arg W f ( a max , b ) = ϕ ( b ) ,
Δ ϕ k ( i , j ) = δ k + ε k ( i , j ) ,
n θ ln ( p n ( ω n ; θ ) ) = 0.
F ( θ ) = E ( [ θ ln ( p ( ω ; θ ) ) ] [ θ ln ( p ( ω ; θ ) ) ] T | θ ) ,
s [ n ] = 1 2 [ 1 + cos ( 2 π · 10 · n L + δ k ) ] ,

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