Abstract

A pointwise least-squares phase-stepping algorithm with an unknown relative phase step is proposed. In phase-stepping interferometry the recorded temporal intensity sequence is a discrete sinusoidal signal biased by a direct-current component. Its value at a certain time can be predicted from its three past samples by use of a recursive formula. Based on this linear prediction property, an unbiased least-squares estimator is deduced to determine the relative phase step from a sequence of intensity values, and the result is used to evaluate the phase value. The validity and performance of this algorithm are verified by computer simulations.

© 2005 Optical Society of America

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References

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  1. K. Okada, A. Sato, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase-shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
    [CrossRef]
  2. G. Lai, T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8, 822–827 (1991).
    [CrossRef]
  3. C. T. Farrell, M. A. Player, “Phase-step measurement and variable-step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
    [CrossRef]
  4. I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
    [CrossRef]
  5. C. Wei, M. Chen, Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999).
    [CrossRef]
  6. K. G. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001).
    [CrossRef] [PubMed]
  7. L. Z. Cai, Q. Liu, X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003).
    [CrossRef] [PubMed]
  8. Z. Wang, B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
    [CrossRef] [PubMed]
  9. M. Chen, H. Guo, C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000).
    [CrossRef]
  10. A. Dobroiu, A. Apostol, V. Nascov, V. Damian, “Tilt-compensation algorithm for phase-shift interferometry,” Appl. Opt. 41, 2435–2439 (2002).
    [CrossRef] [PubMed]
  11. P. Carré, “Installation et utilisation du comparateur photóelectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  12. G. Sotoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Laser Eng. 28, 61–69 (1997).
    [CrossRef]
  13. T. M. Kreis, “Computer aided evaluation of fringe patterns,” Opt. Laser Eng. 19, 221–240 (1993).
    [CrossRef]
  14. X. C. de Lega, P. Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. 35, 5115–5120 (1996).
    [CrossRef]
  15. K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds., (Institute of Physics, 1993), pp. 94–140.
  16. H. M. Ladak, W. F. Decraemer, J. J. J. Dirckx, W. R. J. Funnell, “Systematic errors in small deformations measured by use of shadow-moire topography,” Appl. Opt. 39, 3266–3275 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
    [CrossRef]
  19. H. C. Ho, K. W. Chan, “Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation,” IEEE Trans. Signal Process. 52, 1128–1135 (2004).
    [CrossRef]
  20. R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2001).

2004 (2)

H. C. Ho, K. W. Chan, “Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation,” IEEE Trans. Signal Process. 52, 1128–1135 (2004).
[CrossRef]

Z. Wang, B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
[CrossRef] [PubMed]

2003 (1)

2002 (1)

2001 (1)

2000 (2)

1999 (1)

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

1997 (1)

G. Sotoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Laser Eng. 28, 61–69 (1997).
[CrossRef]

1996 (1)

1995 (1)

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

1993 (1)

T. M. Kreis, “Computer aided evaluation of fringe patterns,” Opt. Laser Eng. 19, 221–240 (1993).
[CrossRef]

1992 (1)

C. T. Farrell, 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, J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase-shifting interferometry,” Opt. Commun. 84, 118–124 (1991).
[CrossRef]

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

1984 (1)

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

1982 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photóelectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Apostol, A.

Cai, L. Z.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photóelectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Chan, K. W.

H. C. Ho, K. W. Chan, “Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation,” IEEE Trans. Signal Process. 52, 1128–1135 (2004).
[CrossRef]

Chen, M.

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

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

Creath, K.

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds., (Institute of Physics, 1993), pp. 94–140.

Damian, V.

de Lega, X. C.

Decraemer, W. F.

Dirckx, J. J. J.

Dobroiu, A.

Dragostinov, T.

G. Sotoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Laser Eng. 28, 61–69 (1997).
[CrossRef]

Farrell, C. T.

C. T. Farrell, M. A. Player, “Phase-step measurement and variable-step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Funnell, W. R. J.

Gonzalez, R. C.

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2001).

Greivenkamp, J. E.

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

Guo, H.

Han, B.

Ho, H. C.

H. C. Ho, K. W. Chan, “Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation,” IEEE Trans. Signal Process. 52, 1128–1135 (2004).
[CrossRef]

Jacquot, P.

Kim, S.-W.

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

Kong, I.-B.

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

Kreis, T. M.

T. M. Kreis, “Computer aided evaluation of fringe patterns,” Opt. Laser Eng. 19, 221–240 (1993).
[CrossRef]

Ladak, H. M.

Lai, G.

Larkin, K. G.

Liu, Q.

Morgan, C. J.

Nascov, V.

Okada, K.

K. Okada, A. Sato, 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, M. A. Player, “Phase-step measurement and variable-step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Sato, A.

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

Sotoilov, G.

G. Sotoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Laser Eng. 28, 61–69 (1997).
[CrossRef]

Tsujiuchi, J.

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

Wang, Z.

Z. Wang, B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29, 1671–1673 (2004).
[CrossRef] [PubMed]

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

Wei, C.

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

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

Woods, R. E.

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2001).

Yang, X. L.

Yatagai, T.

Appl. Opt. (4)

IEEE Trans. Signal Process. (1)

H. C. Ho, K. W. Chan, “Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation,” IEEE Trans. Signal Process. 52, 1128–1135 (2004).
[CrossRef]

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

Meas. Sci. Technol. (1)

C. T. Farrell, M. A. Player, “Phase-step measurement and variable-step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953–958 (1992).
[CrossRef]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photóelectrique et intertérentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Commun. (1)

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

Opt. Eng. (3)

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

I.-B. Kong, S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–187 (1995).
[CrossRef]

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

Opt. Express (1)

Opt. Laser Eng. (2)

G. Sotoilov, T. Dragostinov, “Phase-stepping interferometry: five-frame algorithm with an arbitrary step,” Opt. Laser Eng. 28, 61–69 (1997).
[CrossRef]

T. M. Kreis, “Computer aided evaluation of fringe patterns,” Opt. Laser Eng. 19, 221–240 (1993).
[CrossRef]

Opt. Lett. (3)

Other (2)

R. C. Gonzalez, R. E. Woods, Digital Image Processing, 2nd ed. (Prentice-Hall, 2001).

K. Creath, “Temporal phase measurement method,” in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson, G. Reid, eds., (Institute of Physics, 1993), pp. 94–140.

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Figures (4)

Fig. 1
Fig. 1

Measurement results with the proposed algorithm, where the noise SD is 0.01 and the phase-step number is 8. The unit on the vertical axes is radians, and the horizontal axes denote the pixel position. (a) Predefined phase, (b) predefined relative phase step, (c) recovered phase, (d) estimated relative phase step, (e) difference in (c) and (a), (f) difference in (d) and (b).

Fig. 2
Fig. 2

Root-mean-square errors in recovered phases with different algorithms, where the noise SD is 0.01.

Fig. 3
Fig. 3

Statistics of the measurement errors at a general point. The unit on the vertical axes is in radians, and the horizontal axes denote the number of phase steps. Noise SDs: dashed line, 0; solid curve, 0.01; dotted curve, 0.02. (a) Mean errors of the relative phase steps, (b) SDs of the step errors, (c) mean errors of the phase, (d) SDs of the phase errors with the suggested algorithm, (e)–(h) parallel sequence, but the relative phase step is estimated by using Eq. (4).

Fig. 4
Fig. 4

Computational time: solid line, suggested algorithm; dotted curve, iterative algorithm.

Equations (15)

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I n ( i , j ) = a ( i , j ) + b ( i , j ) cos { ϕ ( i , j ) + [ n - ( N - 1 ) / 2 ] δ ( i , j ) } ,
I n = ( 2 cos δ + 1 ) ( I n - 1 - I n - 2 ) + I n - 3 .
e n = I n - ( 2 cos δ + 1 ) ( I n - 1 - I n - 2 ) - I n - 3 .
δ = arccos [ n = 3 N - 1 ( I n - I n - 3 ) ( I n - 1 - I n - 2 ) 2 n = 3 N - 1 ( I n - 1 - I n - 2 ) 2 - 1 2 ] .
I ^ n = I n + η n ,             n = 0 , 1 , , N - 1 ,
E { e n 2 } = E { [ I ^ n - ( 2 cos δ + 1 ) ( I ^ n - 1 - I ^ n - 2 ) - I ^ n - 3 ] 2 } = E ( { ( I n + η n ) - ( 2 cos δ + 1 ) [ ( I n - 1 + η n - 1 ) - ( I n - 2 + η n - 2 ) ] - ( I n - 3 + η n - 3 ) } 2 ) = E ( { [ I n - ( 2 cos δ + 1 ) ( I n - 1 - I n - 2 ) - I n - 3 ] + [ η n - ( 2 cos δ + 1 ) ( η n - 1 - η n - 2 ) - η n - 3 ] } 2 ) = E { [ I n - ( 2 cos δ + 1 ) ( η n - 1 - I n - 2 ) - I n - 3 ] 2 } + E [ η n 2 + ( 2 cos δ + 1 ) ( η n - 1 2 + η n - 2 2 ) + η n - 3 2 ] = [ I n - ( 2 cos δ + 1 ) ( I n - 1 - I n - 2 ) - I n - 3 ] 2 + [ 2 + 2 ( 2 cos δ + 1 ) 2 ] σ η 2 .
ɛ n = e n [ 2 + 2 ( 2 cos δ + 1 ) 2 ] 1 / 2 .
A y 2 - B y - A = 0 ,
y = 2 cos δ + 1 ,
A = n = 3 N - 1 [ ( I n - 2 - I n - 1 ) ( I n - I n - 3 ) ] ,
B = n = 3 N - 1 [ ( I n - 2 - I n - 1 ) 2 - ( I n - I n - 3 ) 2 ] .
δ = arccos [ B - ( B 2 + 4 A 2 ) 1 / 2 4 A - 1 2 ] .
I n = c 0 + c 1 cos [ n - ( N - 1 ) / 2 ] δ + c 2 sin [ n - ( N - 1 ) / 2 ] δ , n = 0 , 1 , , N - 1 ,
[ N n = 0 N - 1 cos [ n - ( N - 1 ) / 2 ] δ 0 n = 0 N - 1 cos [ n - ( N - 1 ) / 2 ] δ n = 0 N - 1 cos 2 [ n - ( N - 1 ) / 2 ] δ 0 0 0 n = 0 N - 1 sin 2 [ n - ( N - 1 ) / 2 ] δ ] [ c 0 c 1 c 2 ] = [ n = 0 N - 1 I n n = 0 N - 1 I n cos [ n - ( N - 1 ) / 2 ] δ n = 0 N - 1 I n sin [ n - ( N - 1 ) / 2 ] δ ] .
ϕ = arctan ( - c 2 c 1 ) .

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