Abstract

Recently, a rapid and accurate single-shot phase measurement technique called the sampling moiré method has been developed for small-displacement distribution measurements. In this study, the theoretical phase error of the sampling moiré method caused by linear intensity interpolation in the case of a mismatch between the sampling pitch and the original grating pitch is analyzed. The periodic phase error is proportional to the square of the spatial angular frequency of the moiré fringe. Moreover, an effective phase compensation methodology is developed to reduce the periodic phase error. Single-shot phase analysis can perform accurately even when the sampling pitch is not matched to the original grating pitch exactly. The primary simulation results demonstrate the effectiveness of the proposed phase compensation methodology.

© 2012 Optical Society of America

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References

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  1. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
    [CrossRef]
  2. J. Endo, J. Chen, D. Kobayashi, Y. Wada, and H. Fujita, “Transmission laser microscope using the phase-shifting technique and its application to measurement of optical waveguides,” Appl. Opt. 41, 1308–1314 (2002).
    [CrossRef]
  3. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156–160 (1982).
    [CrossRef]
  4. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977–3982 (1983).
    [CrossRef]
  5. X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
    [CrossRef]
  6. J. Zong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993–4998 (2004).
    [CrossRef]
  7. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
    [CrossRef]
  8. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
    [CrossRef]
  9. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surface and lenses,” Appl. Opt. 13, 2693–2703 (1974).
    [CrossRef]
  10. P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensation phase calculating algorithm,” Appl. Opt. 26, 2504–2506 (1987).
    [CrossRef]
  11. P. D. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
    [CrossRef]
  12. Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 224–231 (1994).
  13. P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
    [CrossRef]
  14. S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
    [CrossRef]
  15. M. Kujawinska and J. Wojciak, “Spatial-carrier phase shifting technique of fringe pattern analysis,” Proc SPIE 1508, 61–67 (1991).
    [CrossRef]
  16. P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
    [CrossRef]
  17. Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
    [CrossRef]
  18. S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
    [CrossRef]
  19. S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method,” Appl. Opt. 45, 6940–6946 (2006).
    [CrossRef]
  20. M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
    [CrossRef]
  21. S. Ri and T. Muramatsu, “A simple technique for measuring thickness distribution of transparent plates from a single image by using the sampling moiré method,” Meas. Sci. Technol. 21, 025305 (2010).
    [CrossRef]
  22. Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
    [CrossRef]
  23. S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
    [CrossRef]
  24. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
    [CrossRef]

2012

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

2011

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

2010

S. Ri and T. Muramatsu, “A simple technique for measuring thickness distribution of transparent plates from a single image by using the sampling moiré method,” Meas. Sci. Technol. 21, 025305 (2010).
[CrossRef]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

2009

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

2007

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

2006

2004

2002

2001

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

2000

P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

1997

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

1995

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
[CrossRef]

P. D. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[CrossRef]

1994

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 224–231 (1994).

1991

M. Kujawinska and J. Wojciak, “Spatial-carrier phase shifting technique of fringe pattern analysis,” Proc SPIE 1508, 61–67 (1991).
[CrossRef]

1987

1984

1983

1982

1974

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Brangaccio, D. J.

Bruning, J. H.

Bryanston-Cross, P. J.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
[CrossRef]

Chan, P. H.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
[CrossRef]

Chen, J.

Chen, W.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

Eiju, T.

Endo, J.

Fujigaki, M.

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method,” Appl. Opt. 45, 6940–6946 (2006).
[CrossRef]

Fujisawa, M.

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 224–231 (1994).

Fujita, H.

Gallagher, J. E.

Groot, P. D.

Halioua, M.

Hariharan, P.

Herriott, D. R.

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Ina, H.

Inuzuka, T.

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

Kobayashi, D.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

J. Endo, J. Chen, D. Kobayashi, Y. Wada, and H. Fujita, “Transmission laser microscope using the phase-shifting technique and its application to measurement of optical waveguides,” Appl. Opt. 41, 1308–1314 (2002).
[CrossRef]

Kobayashi, S.

Kondo, H.

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

Kujawinska, M.

M. Kujawinska and J. Wojciak, “Spatial-carrier phase shifting technique of fringe pattern analysis,” Proc SPIE 1508, 61–67 (1991).
[CrossRef]

Liu, H. C.

Masaya, A.

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

Matui, T.

Morimoto, Y.

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method,” Appl. Opt. 45, 6940–6946 (2006).
[CrossRef]

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 224–231 (1994).

Muramatsu, T.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

S. Ri and T. Muramatsu, “A simple technique for measuring thickness distribution of transparent plates from a single image by using the sampling moiré method,” Meas. Sci. Technol. 21, 025305 (2010).
[CrossRef]

Mutoh, K.

Nanbara, K.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Oreb, B. F.

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Parker, S. C.

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
[CrossRef]

Ri, S.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

S. Ri and T. Muramatsu, “A simple technique for measuring thickness distribution of transparent plates from a single image by using the sampling moiré method,” Meas. Sci. Technol. 21, 025305 (2010).
[CrossRef]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method,” Appl. Opt. 45, 6940–6946 (2006).
[CrossRef]

Rosenfeld, D. P.

Saka, M.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Shimo, K.

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Srinivasan, V.

Su, X.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

Takeda, M.

Wada, Y.

Weng, J.

White, A. D.

Wojciak, J.

M. Kujawinska and J. Wojciak, “Spatial-carrier phase shifting technique of fringe pattern analysis,” Proc SPIE 1508, 61–67 (1991).
[CrossRef]

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Zhang, S.

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

Zong, J.

Adv. Mater. Res.

Y. Morimoto, M. Fujigaki, A. Masaya, H. Kondo, and T. Inuzuka, “Accurate displacement measurement for landslide prediction by sampling moiré method,” Adv. Mater. Res. 79–82, 1731–1734 (2009).
[CrossRef]

Appl. Opt.

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surface and lenses,” Appl. Opt. 13, 2693–2703 (1974).
[CrossRef]

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

V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105–3108 (1984).
[CrossRef]

P. D. Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34, 4723–4730 (1995).
[CrossRef]

J. Endo, J. Chen, D. Kobayashi, Y. Wada, and H. Fujita, “Transmission laser microscope using the phase-shifting technique and its application to measurement of optical waveguides,” Appl. Opt. 41, 1308–1314 (2002).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695–2702 (2004).
[CrossRef]

J. Zong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993–4998 (2004).
[CrossRef]

S. Ri, M. Fujigaki, T. Matui, and Y. Morimoto, “Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method,” Appl. Opt. 45, 6940–6946 (2006).
[CrossRef]

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

Exp. Mech.

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, and D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

S. Ri, M. Fujigaki, and Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

J. Mod. Opt.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

J. Opt. Soc. Am.

Meas. Sci. Technol.

S. Ri and T. Muramatsu, “A simple technique for measuring thickness distribution of transparent plates from a single image by using the sampling moiré method,” Meas. Sci. Technol. 21, 025305 (2010).
[CrossRef]

Opt. Eng.

M. Fujigaki, K. Shimo, A. Masaya, and Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50, 101506(2011).
[CrossRef]

Y. Morimoto and M. Fujisawa, “Fringe pattern analysis by a phase-shifting method using Fourier transform,” Opt. Eng. 33, 224–231 (1994).

P. Hariharan, “Phase-shifting interferometry: minimization of systematic errors,” Opt. Eng. 39, 967–969 (2000).
[CrossRef]

Opt. Laser Eng.

X. Su and W. Chen, “Fourier transform profilometry: a review,” Opt. Laser Eng. 35, 263–284 (2001).
[CrossRef]

Opt. Lasers Eng.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications, and implementations,” Opt. Lasers Eng. 45, 304–317 (2007).
[CrossRef]

P. H. Chan, P. J. Bryanston-Cross, and S. C. Parker, “Spatial phase stepping method of fringe-pattern analysis,” Opt. Lasers Eng. 23, 343–354 (1995).
[CrossRef]

S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48, 141–148 (2010).
[CrossRef]

Proc SPIE

M. Kujawinska and J. Wojciak, “Spatial-carrier phase shifting technique of fringe pattern analysis,” Proc SPIE 1508, 61–67 (1991).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Fundamental principle behind the sampling moiré method. (a) The original grating with a regular pitch P. (b) The intensity recorded by the CCD camera (only one horizontal line in the CCD plane is shown). (c) Four phase-shifted coarse moiré fringes obtained by down-sampling with a pitch of T=4 pixels. (d) Four smooth moiré fringe obtained by intensity interpolation.

Fig. 2.
Fig. 2.

Example of single-shot phase analysis by the sampling moiré method. (a) Original grating image f(x,y). (b) 8 phase-shifted moiré fringes fm(x,y;k) obtained by down-sampling with a sampling pitch T=8 pixels and linear intensity interpolation. (c) Phase distribution φm(x,y) of the moiré fringe. (d) Phase distribution φ(x,y) of the original grating (a).

Fig. 3.
Fig. 3.

Single-shot phase analysis of Fig. 2(a) with different sampling pitches. (a) The phase distribution φm(x,y) of the moiré fringe. (b) The phase distribution φ(x,y) of the original grating with a sampling pitch T=7 pixels. (c) The phase distribution φm(x,y) of the moiré fringe. (d) The phase distribution φ(x,y) of the original grating with a sampling pitch T=9 pixels.

Fig. 4.
Fig. 4.

Flow chart of single-shot phase analysis using the sampling moiré method and symbols used in theoretical analysis of the phase error.

Fig. 5.
Fig. 5.

Procedure of down-sampling and linear intensity interpolation in the sampling moiré method.

Fig. 6.
Fig. 6.

Relationship between the intensity of the original grating and the phase error before and after compensation in the case of (a) P=10 and T=8 pixels, and (b) P=10 and T=9 pixels by simulation. The phase errors were reduced by factors of 5.9 and 12 in the case of T=8 and 9 pixels, respectively.

Fig. 7.
Fig. 7.

Relationship between the sampling pitch T and the RMS phase error E before and after compensation in the case of (a) without noise, and (b) with 1.0% random noise by simulation.

Equations (43)

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

f(x)=a(x)cos(2πxP+φ0)+b(x)=a(x)cos[φ(x)]+b(x),
fm(x;k)=a(x)cos[2π(1P1T)x+2πkT+φ0]+b(x)=a(x)cos[φ(x)φs(x)+2πkT]+b(x)=a(x)cos[φm(x)+2πkT]+b(x).
φ(x)=φm(x)+2πxT.
f(x,y)=255{0.5+0.3cos[2πx/P+1.5×peaks(x,y)]}.
peaks(x,y)=3×(1x)2×exp[x2(y+1)2]10×(x/5x3y5)×exp(x2y2)1/3exp[(x+1)2y2].
f^(x,y)=acos[2πxP+φ0]+b=acos[φ^(x,y)]+b,
fm(x,y;k)=1T[(Tl)·f(xl,y)+l·f(xl+T,y)],(k=0,1,,T1),
fm(x,y;k)=TlTacos{φ^(x,y)+Δφ1}+lTacos{φ^(x,y)+Δφ2}+b,
Δφ1=2πl(1P1T)=lωm,
Δφ2=2π(Tl)(1P1T)=(Tl)ωm,
ωm=2π(1P1T).
fm(x,y;k)fm(x,y;k)|D1+Δφ1fmΔφ1|D1+Δφ2fmΔφ2|D1+(Δφ1)222fm(Δφ1)2|D1+(Δφ2)222fm(Δφ2)2|D1,
fm(x,y;k)am(x,y;k)cos[φ^m(x,y)+2πkT]+bam(x,y;k)cos[φ^(x,y)]+b,
φ^m(x,y)=2πx(1P1T)+φ0=φ(x,y)φs(x,y),
am(x,y;k)=a+Δam(x,y;k)=a[1l(Tl)2ωm2].
Fm(x,y)=k=0T1fm(x,y;k)exp(j2πkT).
Fm(x,y)=F^m(x,y)+ΔFm(x,y).
F^m(x,y)=Fm(x,y)|D2=Ta2exp[jφ^m(x,y)],
ΔFm(x,y)Δam(x,y;0)Fm(x,y)am(x,y;0)|D2+Δam(x,y;1)Fm(x,y)am(x,y;1)|D2++Δam(x,y;T1)Fm(x,y)am(x,y;T1)|D2
ΔFm(x,y)Ta2ωm2(αexp{j[φ^m(x,y)+4πxT]}βexp[jφ^m(x,y)]),
α=12Tk=0T1k(kT)cos(4πkT),
β=T2112.
φm(x,y)=arctan[Fi(x,y)Fr(x,y)]φ^m(x,y)+Δφm(x,y).
Δφm(x,y)ΔFrφm(x,y)Fr(x,y)|D3+ΔFiφm(x,y)Fi(x,y)|D3
Δφm(x,y)=αωm2sin2[φ^m(x,y)+2πxT]=αωm2sin[2φ^(x,y)].
φ^m(x,y)x=2π(1P1T)=ωm.
φm(x,y)x=φm(x+1,y)φm(x1,y)2ωm(x,y).
I(x)=cos(2πxP)+n(x),
E=1Nx=0N1[Δφm(x)]2[%]
Δφm(x)=100·φm(x)φ^m(x)2π[%]
fmΔφ1|D1=TlTasin[φ^m(x,y)+2πkT],
fmΔφ2|D1=lTasin[φ^m(x,y)+2πkT],
2fm(Δφ1)2|D1=TlTacos[φ^m(x,y)+2πkT],
2fm(Δφ2)2|D1=lTacos[φ^m(x,y)+2πkT].
fm(x,y;k)acos{φ^m(x,y)+2πkT}+baT[(Tl)Δφ1+lΔφ2]sin{φ^m(x,y)+2πkT}a2T[(Tl)(Δφ1)2+l(Δφ2)2]cos{φ^m(x,y)+2πkT}
fm(x,y;k)a[1l(Tl)2ωm2]cos{φ^m(x,y)+2πkT}+b.
Fm(x,y)am(x,y;k)|D2=cos[φ^m(x,y)+2πkT]exp(j2πkT)=12{exp[jφ^m(x,y)]exp(j4πkT)+exp[jφ^m(x,y)]}.
ΔFm(x,y)=k=0T1[al(Tl)2ωm2]12{exp[jφ^m(x,y)]exp(j4πkT)+exp[jφ^m(x,y)]}=Ta2ωm2(αexp[jφ^m(x,y)]βexp[jφ^m(x,y)]),
α=12Tk=0T1[l(Tl)]exp(j4πkT)=αexp(j4πxT)
β=12Tk=0T1l(Tl)=T2112.
φm(x,y)Fr(x,y)=Fi(x,y)[Fm(x,y)]2,φm(x,y)Fi(x,y)=Fr(x,y)[Fm(x,y)]2.
φm(x,y)Fr(x,y)|D3=2Tasinφ^m(x,y),φm(x,y)Fi(x,y)|D3=2Tacosφ^m(x,y).
Δφm(x,y)=ωm2{αsin[φ^m(x,y)+4πkT]βsinφ^m(x,y)}cosφ^m(x,y)ωm2{αcos[φ^m(x,y)+4πkT]βcosφ^m(x,y)}sinφ^m(x,y)=αωm2sin2[φ^m(x,y)+2πxT].

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