Abstract

A three-frame phase-shifting algorithm with a constant but unknown phase shift is proposed. The algorithm is based on background-intensity removal prior to phase retrieval to eliminate an undetermined factor in a fringe pattern. The proposed method is validated on three-dimensional profilometry by fringe projection and on deformation measurement by means of digital speckle shearing interferometry. For a fringe pattern with slow-varying background intensity, the background removal is achieved in the frequency domain. For a speckle pattern, a background removal technique is integrated with the three-frame algorithm. In this process, manual intervention is minimal, and high computational speed is achieved. In addition, high-frequency phase signals would not be removed in the noise-reduction process as is the case in the bandpass-filtering technique. Accuracy of the method is discussed.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
  7. X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
    [CrossRef]
  8. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  13. F. Mendoza-Santoyo, D. Kerr, J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).
  14. S. Almazan-Cuellar, D. Malacara-Hemandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3524–3531 (2003).
    [CrossRef]
  15. M. Lehmann, “Speckle statistic in the context of digital speckle,” in Digital Speckle Pattern Interferometry and Related Techniques,P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001).
  16. D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
    [CrossRef]
  17. C. C. Kao, G. B. Yeh, S. S. Lee, C. K. Lee, C. S. Yang, K. C. Wu, “Phase-shifting algorithms for electronic speckle pattern interferometry,” Appl. Opt. 41, 46–54 (2002).
    [CrossRef] [PubMed]
  18. D. R. Schmitt, R. W. Hunt, “Optimization of fringe pattern calculation with direct correlations in speckle interferometry,” Appl. Opt. 36, 8848–8857 (1997).
    [CrossRef]
  19. T. W. Ng, F. S. Chau, “Automated analysis in digital speckle shearing interferometry using an object step-loading method,” Opt. Commun. 108, 214–218 (1994).
    [CrossRef]
  20. H. A. Aebischer, S. Waldner, “A simple and effective method of filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
    [CrossRef]

2003 (2)

J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik 114, 63–68 (2003).
[CrossRef]

S. Almazan-Cuellar, D. Malacara-Hemandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3524–3531 (2003).
[CrossRef]

2002 (1)

2001 (2)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

1999 (1)

H. A. Aebischer, S. Waldner, “A simple and effective method of filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

1998 (1)

1997 (1)

1994 (1)

T. W. Ng, F. S. Chau, “Automated analysis in digital speckle shearing interferometry using an object step-loading method,” Opt. Commun. 108, 214–218 (1994).
[CrossRef]

1989 (1)

D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
[CrossRef]

1988 (1)

1987 (1)

1986 (1)

1985 (2)

1984 (1)

1983 (1)

1982 (1)

1966 (1)

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Aebischer, H. A.

H. A. Aebischer, S. Waldner, “A simple and effective method of filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Almazan-Cuellar, S.

S. Almazan-Cuellar, D. Malacara-Hemandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3524–3531 (2003).
[CrossRef]

Carré, P.

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Chau, F. S.

T. W. Ng, F. S. Chau, “Automated analysis in digital speckle shearing interferometry using an object step-loading method,” Opt. Commun. 108, 214–218 (1994).
[CrossRef]

Cheng, Y. Y.

Creath, K.

Eiju, T.

Franze, B.

Haible, P.

Hariharan, P.

He, X. Y.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Hunt, R. W.

Joenathan, C.

Kao, C. C.

Kerr, D.

D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
[CrossRef]

F. Mendoza-Santoyo, D. Kerr, J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Kreis, T.

Le, X.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Lee, C. K.

Lee, S. S.

Lehmann, M.

M. Lehmann, “Speckle statistic in the context of digital speckle,” in Digital Speckle Pattern Interferometry and Related Techniques,P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001).

Malacara-Hemandez, D.

S. Almazan-Cuellar, D. Malacara-Hemandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3524–3531 (2003).
[CrossRef]

Mendoza-Santoyo, F.

D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
[CrossRef]

F. Mendoza-Santoyo, D. Kerr, J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Morgan, C. J.

Mutoh, K.

Ng, T. W.

T. W. Ng, F. S. Chau, “Automated analysis in digital speckle shearing interferometry using an object step-loading method,” Opt. Commun. 108, 214–218 (1994).
[CrossRef]

Novak, J.

J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik 114, 63–68 (2003).
[CrossRef]

Oreb, B. F.

Quan, C.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Schmitt, D. R.

Shang, H. M.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Takeda, M.

Tao, G.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Tay, C. J.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Tiziani, H. J.

Tyrer, J. R.

D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
[CrossRef]

F. Mendoza-Santoyo, D. Kerr, J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

Waldner, S.

H. A. Aebischer, S. Waldner, “A simple and effective method of filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Wang, C. F.

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

Wu, K. C.

Wyant, J. C.

Yang, C. S.

Yang, Y.

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Yeh, G. B.

Appl. Opt. (9)

M. Takeda, K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
[CrossRef] [PubMed]

J. C. Wyant, B. F. Oreb, P. Hariharan, “Testing aspherics using two-wavelength holography: use of digital electronic techniques,” Appl. Opt. 23, 4020–4023 (1984).
[CrossRef] [PubMed]

P. Hariharan, B. F. Oreb, T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. 26, 2504–2506 (1987).
[CrossRef] [PubMed]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985).
[CrossRef] [PubMed]

Y. Y. Cheng, J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. 24, 3049–3052 (1985).
[CrossRef] [PubMed]

F. Mendoza-Santoyo, D. Kerr, J. R. Tyrer, “Interferometric fringe analysis using a single phase step technique,” Appl. Opt. 27, 4362–4364 (1988).

C. C. Kao, G. B. Yeh, S. S. Lee, C. K. Lee, C. S. Yang, K. C. Wu, “Phase-shifting algorithms for electronic speckle pattern interferometry,” Appl. Opt. 41, 46–54 (2002).
[CrossRef] [PubMed]

D. R. Schmitt, R. W. Hunt, “Optimization of fringe pattern calculation with direct correlations in speckle interferometry,” Appl. Opt. 36, 8848–8857 (1997).
[CrossRef]

J. Mod. Opt. (1)

D. Kerr, F. Mendoza-Santoyo, J. R. Tyrer, “Manipulation of the Fourier components of speckle fringe patterns as part of an interferometric analysis process,” J. Mod. Opt. 36, 195–302 (1989).
[CrossRef]

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

Metrologia (1)

P. Carré, “Installation et utilization du compateur photoelectrique et interferential du Bureau International des Poids et Mesures,” Metrologia 2, 13–20 (1966).
[CrossRef]

Opt. Commun. (3)

C. Quan, X. Y. He, C. F. Wang, C. J. Tay, H. M. Shang, “Shape measurement of small objects using LCD fringe projection with phase shifting,” Opt. Commun. 189, 21–29 (2001).
[CrossRef]

T. W. Ng, F. S. Chau, “Automated analysis in digital speckle shearing interferometry using an object step-loading method,” Opt. Commun. 108, 214–218 (1994).
[CrossRef]

H. A. Aebischer, S. Waldner, “A simple and effective method of filtering speckle-interferometric phase fringe patterns,” Opt. Commun. 162, 205–210 (1999).
[CrossRef]

Opt. Eng. (1)

S. Almazan-Cuellar, D. Malacara-Hemandez, “Two-step phase-shifting algorithm,” Opt. Eng. 42, 3524–3531 (2003).
[CrossRef]

Opt. Laser Technol. (1)

X. Le, G. Tao, Y. Yang, “Continual deformation analysis with scanning phase method and time sequence phase method in temporal speckle pattern interferometry,” Opt. Laser Technol. 33, 53–59 (2001).
[CrossRef]

Opt. Lett. (1)

Optik (1)

J. Novak, “Five-step phase-shifting algorithms with unknown values of phase shift,” Optik 114, 63–68 (2003).
[CrossRef]

Other (1)

M. Lehmann, “Speckle statistic in the context of digital speckle,” in Digital Speckle Pattern Interferometry and Related Techniques,P. K. Rastogi, ed. (Wiley, Chichester, UK, 2001).

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Figures (13)

Fig. 1
Fig. 1

(a) |I1 + I3| > |2I2|. Small |I2| is sensitive to the system nonlinearity. (b) |I1 + I3| < |2I2|. Larger |I2| has higher tolerance to the system nonlinearity.

Fig. 2
Fig. 2

Experimental setup for shape measurement.

Fig. 3
Fig. 3

Experimental setup for slope measurement.

Fig. 4
Fig. 4

Projected fringe pattern on an object.

Fig. 5
Fig. 5

Differences in background intensity between the Fourier transform and phase shifting.

Fig. 6
Fig. 6

(a) Wrapped phase map and (b) 3-D plot of the surface profile.

Fig. 7
Fig. 7

Speckle fringe pattern obtained with DSSI (small load).

Fig. 8
Fig. 8

(a) Smoothened fringe pattern by use of bandpass filtering and (b) wrapped phase map extracted from phase-shifted fringe patterns.

Fig. 9
Fig. 9

(a) Wrapped phase map obtained with the proposed technique and (b) phase map of Fig. 9(a) processed by a sine–cosine filter.

Fig. 10
Fig. 10

Speckle fringe pattern obtained with DSSI (large load).

Fig. 11
Fig. 11

(a) Smoothened fringe pattern with bandpass filtering and (b) wrapped phase map extracted from phase-shifted fringe patterns.

Fig. 12
Fig. 12

(a) Wrapped phase map obtained with the proposed technique (processed by the sine–cosine filter) and (b) 3-D plot of the slope of the deformation pattern.

Fig. 13
Fig. 13

(a) Calculated and theoretical phase shifts and (b) mean difference between calculated and theoretical deformation phase maps.

Equations (13)

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

I i = a + b cos ( ϕ + δ i ) ,
I 1 = b cos ( ϕ - δ ) ,
I 2 = b cos ϕ ,
I 3 = b cos ( ϕ + δ ) ,
ϕ = arctan { I 1 - I 3 [ 4 I 2 2 - ( I 1 + I 3 ) 2 ] 1 / 2 } ,
sign ( sin ϕ ) = sign ( I 1 - I 3 ) ,
sign ( cos ϕ ) = sign ( I 2 ) ,
δ = arccos ( I 1 + I 3 2 I 2 ) .
I B = I o + I r + 2 I o I r cos θ ,
I A = I o + I r + 2 I o I r cos ( θ + ϕ ) ,
I 1 = I o + I r + 2 I o I r cos ( θ - δ ) , I 2 = I o + I r + 2 I o I r cos θ , I 3 = I o + I r + 2 I o I r cos ( θ + δ ) ,
I 4 = I o + I r + 2 I o I r cos ( θ + ϕ - δ ) , I 5 = I o + I r + 2 I o I r cos ( θ + ϕ ) , I 6 = I o + I r + 2 I o I r cos ( θ + ϕ + δ ) .
1 6 i = 1 6 I i = I o + I r + 2 I o I r 3 ( 1 + 2 cos δ ) cos ( θ + ϕ 2 ) × cos ϕ 2 .

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