Abstract

Measurement of strain is an important application of digital holographic interferometry. As strain relates to the displacement derivative, it depends on the derivative of the interference phase corresponding to the reconstructed interference field. The paper proposes an elegant method for direct measurement of unwrapped phase derivative. The proposed method relies on approximating the interference phase as a piecewise cubic polynomial and subsequently evaluating the polynomial coefficients using cubic phase function algorithm. The phase derivative is constructed using the evaluated polynomial coefficients. The method’s performance is demonstrated using simulation and experimental results.

© 2010 Optical Society of America

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  1. U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Tech. 13, R85-R101 (2002).
    [CrossRef]
  2. Y. Zou, G. Pedrini, and H. Tiziani, "Derivatives obtained directly from displacement data," Opt. Commun. 111, 427-432 (1994).
    [CrossRef]
  3. U. Schnars and W. P. O. Juptner, "Digital recording and reconstruction of holograms in hologram interferometry and shearography," Appl. Opt. 33, 4373-4377 (1994).
    [CrossRef] [PubMed]
  4. C. Liu, "Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method," Opt. Eng. 42, 3443-3446 (2003).
    [CrossRef]
  5. C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
    [CrossRef]
  6. C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
    [CrossRef]
  7. K. Qian, S. H. Soon, and A. Asundi, "Phase-shifting windowed Fourier ridges for determination of phase derivatives," Opt. Lett. 28, 1657-1659 (2003).
    [CrossRef] [PubMed]
  8. S. S. Gorthi and P. Rastogi, "Simultaneous measurement of displacement, strain and curvature in digital holographic interferometry using high-order instantaneous moments," Opt. Express 17, 17,784-17,791 (2009).
    [CrossRef]
  9. S. S. Gorthi and P. Rastogi, "Windowed high-order ambiguity function method for fringe analysis," Rev. Sci. Inst. 80, 073,109 (2009).
    [CrossRef]
  10. B. Porat, Digital Processing of Random Signals (Prentice hall, Englewood Cliffs, NJ, 1994).
  11. P. O’Shea, "A fast algorithm for estimating the parameters of a quadratic FM signal," IEEE Trans. Sig. Proc. 52, 385-393 (2004).
    [CrossRef]
  12. E. Aboutanios and B. Mulgrew, "Iterative frequency estimation by interpolation on Fourier coefficients," IEEE. Trans. Sig. Proc. 53, 1237-1242 (2005).
    [CrossRef]

2009

C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, "Simultaneous measurement of displacement, strain and curvature in digital holographic interferometry using high-order instantaneous moments," Opt. Express 17, 17,784-17,791 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, "Windowed high-order ambiguity function method for fringe analysis," Rev. Sci. Inst. 80, 073,109 (2009).
[CrossRef]

2005

E. Aboutanios and B. Mulgrew, "Iterative frequency estimation by interpolation on Fourier coefficients," IEEE. Trans. Sig. Proc. 53, 1237-1242 (2005).
[CrossRef]

2004

P. O’Shea, "A fast algorithm for estimating the parameters of a quadratic FM signal," IEEE Trans. Sig. Proc. 52, 385-393 (2004).
[CrossRef]

2003

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

C. Liu, "Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method," Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

K. Qian, S. H. Soon, and A. Asundi, "Phase-shifting windowed Fourier ridges for determination of phase derivatives," Opt. Lett. 28, 1657-1659 (2003).
[CrossRef] [PubMed]

2002

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Tech. 13, R85-R101 (2002).
[CrossRef]

1994

Aboutanios, E.

E. Aboutanios and B. Mulgrew, "Iterative frequency estimation by interpolation on Fourier coefficients," IEEE. Trans. Sig. Proc. 53, 1237-1242 (2005).
[CrossRef]

Asundi, A.

Chen, W.

C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
[CrossRef]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, "Windowed high-order ambiguity function method for fringe analysis," Rev. Sci. Inst. 80, 073,109 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, "Simultaneous measurement of displacement, strain and curvature in digital holographic interferometry using high-order instantaneous moments," Opt. Express 17, 17,784-17,791 (2009).
[CrossRef]

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Tech. 13, R85-R101 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, "Digital recording and reconstruction of holograms in hologram interferometry and shearography," Appl. Opt. 33, 4373-4377 (1994).
[CrossRef] [PubMed]

Kim, T.

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Liu, C.

C. Liu, "Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method," Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

Mulgrew, B.

E. Aboutanios and B. Mulgrew, "Iterative frequency estimation by interpolation on Fourier coefficients," IEEE. Trans. Sig. Proc. 53, 1237-1242 (2005).
[CrossRef]

O’Shea, P.

P. O’Shea, "A fast algorithm for estimating the parameters of a quadratic FM signal," IEEE Trans. Sig. Proc. 52, 385-393 (2004).
[CrossRef]

Pedrini, G.

Y. Zou, G. Pedrini, and H. Tiziani, "Derivatives obtained directly from displacement data," Opt. Commun. 111, 427-432 (1994).
[CrossRef]

Qian, K.

Quan, C.

C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
[CrossRef]

Rastogi, P.

S. S. Gorthi and P. Rastogi, "Windowed high-order ambiguity function method for fringe analysis," Rev. Sci. Inst. 80, 073,109 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, "Simultaneous measurement of displacement, strain and curvature in digital holographic interferometry using high-order instantaneous moments," Opt. Express 17, 17,784-17,791 (2009).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Tech. 13, R85-R101 (2002).
[CrossRef]

U. Schnars and W. P. O. Juptner, "Digital recording and reconstruction of holograms in hologram interferometry and shearography," Appl. Opt. 33, 4373-4377 (1994).
[CrossRef] [PubMed]

Sciammarella, C. A.

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Soon, S. H.

Tay, C. J.

C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
[CrossRef]

Tiziani, H.

Y. Zou, G. Pedrini, and H. Tiziani, "Derivatives obtained directly from displacement data," Opt. Commun. 111, 427-432 (1994).
[CrossRef]

Zou, Y.

Y. Zou, G. Pedrini, and H. Tiziani, "Derivatives obtained directly from displacement data," Opt. Commun. 111, 427-432 (1994).
[CrossRef]

Appl. Opt.

IEEE Trans. Sig. Proc.

P. O’Shea, "A fast algorithm for estimating the parameters of a quadratic FM signal," IEEE Trans. Sig. Proc. 52, 385-393 (2004).
[CrossRef]

IEEE. Trans. Sig. Proc.

E. Aboutanios and B. Mulgrew, "Iterative frequency estimation by interpolation on Fourier coefficients," IEEE. Trans. Sig. Proc. 53, 1237-1242 (2005).
[CrossRef]

Meas. Sci. Tech.

U. Schnars and W. P. O. Juptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Tech. 13, R85-R101 (2002).
[CrossRef]

Opt. Commun.

Y. Zou, G. Pedrini, and H. Tiziani, "Derivatives obtained directly from displacement data," Opt. Commun. 111, 427-432 (1994).
[CrossRef]

C. Quan, C. J. Tay, and W. Chen, "Determination of displacement derivative in digital holographic interferometry," Opt. Commun. 282, 809-815 (2009).
[CrossRef]

Opt. Eng.

C. A. Sciammarella and T. Kim, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

C. Liu, "Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method," Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

Opt. Express

S. S. Gorthi and P. Rastogi, "Simultaneous measurement of displacement, strain and curvature in digital holographic interferometry using high-order instantaneous moments," Opt. Express 17, 17,784-17,791 (2009).
[CrossRef]

Opt. Lett.

Rev. Sci. Inst.

S. S. Gorthi and P. Rastogi, "Windowed high-order ambiguity function method for fringe analysis," Rev. Sci. Inst. 80, 073,109 (2009).
[CrossRef]

Other

B. Porat, Digital Processing of Random Signals (Prentice hall, Englewood Cliffs, NJ, 1994).

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Figures (3)

Fig. 1.
Fig. 1.

Original vs estimated phase derivative in radians/pixel at SNR of (a) 5 dB, (b) 10 dB, (c) 15 dB. (d) Absolute error for phase derivative estimation

Fig. 2.
Fig. 2.

(a) Simulated fringe pattern. (b) Original phase derivative in radians/pixel. (c) Estimated phase derivative in radians/pixel. (d) Wrapped form of the estimated phase derivative

Fig. 3.
Fig. 3.

(a) Fringe pattern obtained in a DHI experiment. (b) Estimated phase derivative in radians/pixel. (c) Wrapped estimated phase derivative. (d) Wrapped phase derivative estimate using digital shearing method.

Equations (13)

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

I ( x , y ) = A ( x , y ) exp [ ( x , y ) ] + η ( x , y )
I k ( y ) = A k ( y ) exp [ j ϕ k ( y ) ] + η k ( y )
ϕ k ( y ) = a 0 k + a 1 k y + a 2 k y 2 + a 3 k y 3
ϕ k ( y ) y = a 1 k + 2 a 2 k y + 3 a 3 k y 2
2 ϕ k ( y ) y 2 = 2 ( a 2 k + 3 a 3 k y )
CPF k ( y , Ω ) = τ = 0 ( N s 1 ) / 2 I k ( y + τ ) I k ( y τ ) exp ( τ 2 )
U k ( y ) = arg max Ω CPF k ( y , Ω )
U k ( y ) = 2 ( a 2 k + 3 a 3 k y )
U k ( y 1 ) = 2 ( a 2 k + 3 a 3 k y 1 )
U k ( y 2 ) = 2 ( a 2 k + 3 a 3 k y 2 )
I k ' ( y ) = I k ( y ) exp [ j ( a ̂ 2 k y 2 + a ̂ 3 k y 3 ) ]
G k ( ω ) = FT [ I k ' ( y ) ]
a ̂ 1 k = arg max ω G k ( ω )

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