Abstract

An approach to computing pixel-by-pixel gradients of optical phase directly from digitally encoded Fourier-transform or phase-stepped fringe patterns is described. This approach can be classified as a phase unwrapping but is really a sine–cosine demodulation technique that finds its roots in the differential cross-multiplier phase-demodulation technique commonly used by the optical fiber sensor community. This technique is algorithmically simple, does not rely on a computation of the arctangent, and therefore is not subject to some of the limitations of the standard phase-unwrapping methodologies. The proposed phase-gradient technique is demonstrated by the calculation of strain fields from moiré interferometric fringe patterns.

© 1994 Optical Society of America

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References

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  1. J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
    [CrossRef]
  2. K. Creath, “Phase-measurement techniques for non-destructive testing,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethal, Conn., 1990), pp. 473–479.
  3. M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry and profilometry: an overview,” Indus. Met. 1, 79–99 (1990).
    [CrossRef]
  4. W. Osten, R. Hofling, “The inverse modulo process in automatic fringe analysis–problems and approaches,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  8. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi-heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119(1989).
  9. D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
    [CrossRef]
  10. V. A. Kovalevsky, “Finite topology as applied to image analysis,” Computer Graphics Image Process. 46, 141–161 (1989).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  13. A. Spik, D. W. Robinson, “Investigation of cellular automated method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
    [CrossRef]
  14. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  15. T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
    [CrossRef]
  16. R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 26–43 (1988).
  17. J. E. Greivenkamp, “Sub-Nyquist interferometry,” Appl. Opt. 26, 5245–5258 (1987).
    [CrossRef] [PubMed]
  18. H. P. Stahl, “Testing large optics: high speed phase-measuring interferometry,” Photon. Spectra 12, 105–112 (1989).
  19. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 358–370 (1982).
    [CrossRef]
  20. K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
    [CrossRef]
  21. R. M. Measures, K. Liu, “Fiber optic sensors focus on smart systems,” Circuits Devices, 8, 37–46 (1992).
    [CrossRef]
  22. D. Post, “Moiré interferometry,” in the Handbook of Experimental Mechanics, A. S. Kobayshi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987).
  23. J. S. Sirkis, H. Singh, “Micro-mechanical analysis of thick composites with embedded optical fibers,” submitted to Exp. Mech.

1992

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

R. M. Measures, K. Liu, “Fiber optic sensors focus on smart systems,” Circuits Devices, 8, 37–46 (1992).
[CrossRef]

T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

1991

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
[CrossRef]

A. Spik, D. W. Robinson, “Investigation of cellular automated method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

1990

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry and profilometry: an overview,” Indus. Met. 1, 79–99 (1990).
[CrossRef]

1989

V. A. Kovalevsky, “Finite topology as applied to image analysis,” Computer Graphics Image Process. 46, 141–161 (1989).
[CrossRef]

H. P. Stahl, “Testing large optics: high speed phase-measuring interferometry,” Photon. Spectra 12, 105–112 (1989).

J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
[CrossRef] [PubMed]

1987

1985

1983

1982

K. Itoh, “Analysis of phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
[CrossRef] [PubMed]

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 358–370 (1982).
[CrossRef]

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
[CrossRef]

Bryanston-Cross, P. J.

T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi-heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119(1989).

Chen, Y. M.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

Cheng, A. Y. H.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement techniques for non-destructive testing,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethal, Conn., 1990), pp. 473–479.

Dandridge, A.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
[CrossRef]

Ghiglia, D. C.

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

Green, R. J.

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 26–43 (1988).

Greivenkamp, J. E.

Hofling, R.

W. Osten, R. Hofling, “The inverse modulo process in automatic fringe analysis–problems and approaches,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

Huntley, J. M.

Itoh, K.

Judge, T. R.

T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi-heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119(1989).

Koo, K. P.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
[CrossRef]

Kovalevsky, V. A.

V. A. Kovalevsky, “Finite topology as applied to image analysis,” Computer Graphics Image Process. 46, 141–161 (1989).
[CrossRef]

Liu, K.

R. M. Measures, K. Liu, “Fiber optic sensors focus on smart systems,” Circuits Devices, 8, 37–46 (1992).
[CrossRef]

Macy, W.

Mastin, G. A.

Measures, R. M.

R. M. Measures, K. Liu, “Fiber optic sensors focus on smart systems,” Circuits Devices, 8, 37–46 (1992).
[CrossRef]

Morgan, C. J.

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 358–370 (1982).
[CrossRef]

Nakadate, S.

Osten, W.

W. Osten, R. Hofling, “The inverse modulo process in automatic fringe analysis–problems and approaches,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

Post, D.

D. Post, “Moiré interferometry,” in the Handbook of Experimental Mechanics, A. S. Kobayshi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987).

Quan, T.

T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Robinson, D. W.

A. Spik, D. W. Robinson, “Investigation of cellular automated method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Romero, L. A.

Saito, H.

Singh, H.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

J. S. Sirkis, H. Singh, “Micro-mechanical analysis of thick composites with embedded optical fibers,” submitted to Exp. Mech.

Sirkis, J. S.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

J. S. Sirkis, H. Singh, “Micro-mechanical analysis of thick composites with embedded optical fibers,” submitted to Exp. Mech.

Spik, A.

A. Spik, D. W. Robinson, “Investigation of cellular automated method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Stahl, H. P.

H. P. Stahl, “Testing large optics: high speed phase-measuring interferometry,” Photon. Spectra 12, 105–112 (1989).

Takeda, M.

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry and profilometry: an overview,” Indus. Met. 1, 79–99 (1990).
[CrossRef]

Towers, D. P.

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi-heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119(1989).

Tveten, A. B.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
[CrossRef]

Walker, J. G.

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 26–43 (1988).

Appl. Opt.

Appl. Phys. Lett.

K. P. Koo, A. B. Tveten, A. Dandridge, “Passive stabilization scheme for fiber interferometers using (3 × 3) fiber directional couplers,” Appl. Phys. Lett. 4, 616–618 (1982).
[CrossRef]

Circuits Devices

R. M. Measures, K. Liu, “Fiber optic sensors focus on smart systems,” Circuits Devices, 8, 37–46 (1992).
[CrossRef]

Computer Graphics Image Process.

V. A. Kovalevsky, “Finite topology as applied to image analysis,” Computer Graphics Image Process. 46, 141–161 (1989).
[CrossRef]

Indus. Met.

M. Takeda, “Spatial carrier fringe pattern analysis and its application to precision interferometry and profilometry: an overview,” Indus. Met. 1, 79–99 (1990).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Eng.

J. S. Sirkis, Y. M. Chen, H. Singh, A. Y. H. Cheng, “Computerized optical fringe pattern analysis in photomechanics: a review,” Opt. Eng. 31, 304–314(1992).
[CrossRef]

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “Analysis of holographic fringe data using the dual reference approach,” Opt. Eng. 30, 452–460 (1991).
[CrossRef]

T. R. Judge, T. Quan, P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533–543 (1992).
[CrossRef]

Opt. Laser Eng.

A. Spik, D. W. Robinson, “Investigation of cellular automated method for phase unwrapping and its implementation on an array processor,” Opt. Laser Eng. 14, 25–37 (1991).
[CrossRef]

Opt. Lett.

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 358–370 (1982).
[CrossRef]

Photon. Spectra

H. P. Stahl, “Testing large optics: high speed phase-measuring interferometry,” Photon. Spectra 12, 105–112 (1989).

Other

K. Creath, “Phase-measurement techniques for non-destructive testing,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethal, Conn., 1990), pp. 473–479.

W. Osten, R. Hofling, “The inverse modulo process in automatic fringe analysis–problems and approaches,” in Proceedings of 1990 Fall Conference (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 301–309.

J. J. Gierloff, “Phase unwrapping by regions,” in Current Developments in Optical Engineering II, R. E. Fischer, W. J. Smith, eds., Proc. Soc. Photo-Opt. Instrum. Eng.818, 2–9 (1987).

D. P. Towers, T. R. Judge, P. J. Bryanston-Cross, “A quasi-heterodyne holographic technique and automatic algorithms for phase unwrapping,” in Fringe Pattern Analysis, G. T. Reid, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1163, 95–119(1989).

R. J. Green, J. G. Walker, “Phase unwrapping using a priori knowledge about the band limits of a function,” in Industrial Inspection, D. W. Braggins, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1010, 26–43 (1988).

D. Post, “Moiré interferometry,” in the Handbook of Experimental Mechanics, A. S. Kobayshi, ed. (Prentice-Hall, Englewood Cliffs, N.J., 1987).

J. S. Sirkis, H. Singh, “Micro-mechanical analysis of thick composites with embedded optical fibers,” submitted to Exp. Mech.

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Figures (3)

Fig. 1
Fig. 1

Moiré fringe patterns: (a) horizontal field (U field), (b) vertical field (V field).

Fig. 2
Fig. 2

Strain fields found with the DCM algorithm: (a) transverse normal strain (ɛ xx ), (b) longitudinal normal strain (ɛ yy ), (c) shear strain.

Fig. 3
Fig. 3

Strain distributions calculated with the column line-scan phase-unwrapping method: (a) transverse normal strain (ɛ xx ), (b) longitudinal normal strain (ɛ yy ), (c) shear strain.

Equations (17)

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

I 1 = cos ( ϕ ) ,
I 2 = sin ( ϕ ) .
Z = I 1 d I 2 d ψ - I 2 d I 1 d ψ ,
d ϕ d ψ = Z ,
I = n = 0 b n cos ( ϕ + 2 π n f ˜ c · r ˜ ) ,
I sb = 1 2 b 1 exp ( i ϕ ) .
cos ( ϕ ) = 2 b 1 Re ( I sb ) ,
sin ( ϕ ) = 2 b 1 Im ( I sb ) .
I n = a + b cos ( ϕ + α n ) ,
I n = a + M cos ( α n ) + N sin ( α n ) ,
M = b cos ( ϕ ) = n = 1 m I n cos ( α n ) ,
N = - b sin ( ϕ ) = n = 1 m I m sin ( α n ) ,
cos ( ϕ ) = ( 1 b ) n = 1 m I n cos ( α n ) ,
sin ( ϕ ) = - ( 1 b ) n = 1 m I n sin ( α n ) .
ɛ x x = P 2 π ϕ x x ,
ɛ y y = P 2 π ϕ y y ,
ɛ x y = P 4 π ( ϕ x y + ϕ y x ) ,

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