Abstract

The paper introduces a two-dimensional space-frequency distribution based method to directly obtain the unwrapped estimate of the phase derivative which corresponds to strain in digital holographic interferometry. In the proposed method, a two-dimensional pseudo Wigner-Ville distribution of the reconstructed interference field is evaluated and the peak of the distribution provides information about the phase derivative. The presence of a two-dimensional window provides high robustness against noise and enables simultaneous measurement of phase derivatives along both spatial directions. Simulation and experimental results are presented to demonstrate the method’s applicability for phase derivative estimation.

© 2010 Optical Society of America

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  1. R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
    [CrossRef]
  2. G. K. Bhat, “A Fourier transform technique to obtain phase derivatives in interferometry,” Opt. Commun. 110, 279–286 (1994).
    [CrossRef]
  3. Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. 111, 427–432 (1994).
    [CrossRef]
  4. 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]
  5. C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443–3446 (2003).
    [CrossRef]
  6. C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282, 809–815 (2009).
    [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. C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
    [CrossRef]
  9. C. A. Sciammarella, and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003).
    [CrossRef]
  10. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
    [CrossRef] [PubMed]
  11. 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, 17784–17791 (2009).
    [CrossRef] [PubMed]
  12. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
    [CrossRef] [PubMed]
  13. U. Schnars, and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  14. S. S. Gorthi, and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
    [CrossRef]
  15. L. Debnath, and B. Rao, “On new two-dimensional Wigner-Ville nonlinear integral transforms and their basic properties,” Integr. Transf. Spec. F 21, 165–174 (2010).
    [CrossRef]
  16. L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).
  17. A. Choudry, “Digital holographic interferometry of convective heat transport,” Appl. Opt. 20, 1240–1244 (1981).
    [CrossRef] [PubMed]
  18. N. Pandey, and B. Hennelly, “Fixed-point numerical reconstruction for digital holographic microscopy,” Opt. Lett. 35, 1076–1078 (2010).
    [CrossRef] [PubMed]
  19. L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
    [CrossRef]

2010 (3)

2009 (6)

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

S. S. Gorthi, and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

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, 17784–17791 (2009).
[CrossRef] [PubMed]

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

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

2008 (1)

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[CrossRef]

2003 (3)

C. A. Sciammarella, and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (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]

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

2002 (1)

U. Schnars, and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

1994 (3)

G. K. Bhat, “A Fourier transform technique to obtain phase derivatives in interferometry,” Opt. Commun. 110, 279–286 (1994).
[CrossRef]

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

1981 (1)

Ahrenberg, L.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Asundi, A.

Badulescu, C.

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

Bhat, G. K.

G. K. Bhat, “A Fourier transform technique to obtain phase derivatives in interferometry,” Opt. Commun. 110, 279–286 (1994).
[CrossRef]

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]

Choudry, A.

Cordero, R. R.

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[CrossRef]

Debnath, L.

L. Debnath, and B. Rao, “On new two-dimensional Wigner-Ville nonlinear integral transforms and their basic properties,” Integr. Transf. Spec. F 21, 165–174 (2010).
[CrossRef]

Gorthi, S. S.

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi, and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

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, 17784–17791 (2009).
[CrossRef] [PubMed]

Gr’ediac, M.

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

Hennelly, B.

Hennelly, B. M.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Juptner, W. P. O.

U. Schnars, and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 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]

Labbe, F.

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[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]

Martinez, A.

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[CrossRef]

Mathias, J. D.

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

McDonald, J. B.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Molimard, J.

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[CrossRef]

Naughton, T. J.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Page, A. J.

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Pandey, N.

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]

Rajshekhar, G.

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
[CrossRef] [PubMed]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Rao, B.

L. Debnath, and B. Rao, “On new two-dimensional Wigner-Ville nonlinear integral transforms and their basic properties,” Integr. Transf. Spec. F 21, 165–174 (2010).
[CrossRef]

Rastogi, P.

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18, 560–565 (2010).
[CrossRef] [PubMed]

S. S. Gorthi, and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. 20, 075307 (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, 17784–17791 (2009).
[CrossRef] [PubMed]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Roux, D.

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

Schnars, U.

U. Schnars, and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 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. (2)

Exp. Mech. (1)

C. Badulescu, M. Gr’ediac, J. D. Mathias, and D. Roux, “A procedure for accurate one-dimensional strain measurement using the grid method,” Exp. Mech. 49, 841–854 (2009).
[CrossRef]

Integr. Transf. Spec. F (1)

L. Debnath, and B. Rao, “On new two-dimensional Wigner-Ville nonlinear integral transforms and their basic properties,” Integr. Transf. Spec. F 21, 165–174 (2010).
[CrossRef]

J. Disp. Technol. (1)

L. Ahrenberg, A. J. Page, B. M. Hennelly, J. B. McDonald, and T. J. Naughton, “Using commodity graphics hardware for real-time digital hologram view reconstruction,” J. Disp. Technol. 5, 111–119 (2009).
[CrossRef]

Meas. Sci. Technol. (2)

U. Schnars, and W. P. O. Juptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

S. S. Gorthi, and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. 20, 075307 (2009).
[CrossRef]

Opt. Commun. (4)

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

R. R. Cordero, J. Molimard, F. Labbe, and A. Martinez, “Strain maps obtained by phase-shifting interferometry: an uncertainty analysis,” Opt. Commun. 281, 2195–2206 (2008).
[CrossRef]

G. K. Bhat, “A Fourier transform technique to obtain phase derivatives in interferometry,” Opt. Commun. 110, 279–286 (1994).
[CrossRef]

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

Opt. Eng. (2)

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

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

Opt. Express (2)

Opt. Lett. (2)

Rev. Sci. Instrum. (1)

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Strain, curvature, and twist measurements in digital holographic interferometry using pseudo Wigner-Ville distribution based method,” Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef] [PubMed]

Other (1)

L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).

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Figures (2)

Fig. 1.
Fig. 1.

(a) Simulated fringe pattern. (b) Estimated phase derivative along x in radians/pixel. (c) Wrapped phase derivative along x. (d) Error between original and estimated phase derivatives along x in radians/pixel. (e) Estimated phase derivative along y in radians/pixel. (f) Wrapped phase derivative along y. (g) Error between original and estimated phase derivatives along y in radians/pixel.

Fig. 2.
Fig. 2.

(a) Experimental fringe pattern. (b) Estimated phase derivative along x in radians/pixel. (c) Wrapped form of phase derivative along x. (d) Estimated phase derivative along y in radians/pixel. (e) Wrapped form of phase derivative along y. (f) Wrapped estimate along y using digital shearing approach.

Tables (1)

Tables Icon

Table 1. Comparison of RMSEs (in radians/pixel) for phase derivative estimation along x at various SNRs (in dB)

Equations (13)

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I ( x , y ) = A ( x , y ) exp [ j ϕ ( x , y ) ] + η ( x , y )
Γ 1 ( x , y , ω 1 , ω 2 ) = I ( x + u 1 2 , y + u 2 2 ) I * ( x u 1 2 , y u 2 2 ) exp [ j ( ω 1 u 1 + ω 2 u 2 ) ] u 1 u 2
Γ ( x , y , ω 1 , ω 2 ) = w ( τ 1 , τ 2 ) I ( x + τ 1 , y + τ 2 ) I * ( x τ 1 , y τ 2 ) exp [ 2 j ( ω 1 τ 1 + ω 2 τ 2 ) ] τ 1 τ 2
Γ ( x , y , ω 1 , ω 2 ) = A 2 ( x , y ) w ( τ 1 , τ 2 ) exp [ j ( ϕ ( x + τ 1 , y + τ 2 ) ϕ ( x τ 1 , y τ 2 ) ) ]
exp [ j ( 2 ω 1 τ 1 + 2 ω 2 τ 2 ) ] τ 1 τ 2
ϕ ( x + τ 1 , y + τ 2 ) ϕ ( x , y ) + ϕ x ( x , y ) τ 1 + ϕ y ( x , y ) τ 2 + 1 2 [ ϕ x x ( x , y ) τ 1 2 + 2 τ 1 τ 2 ϕ x y ( x , y ) + ϕ y y ( x , y ) τ 2 2 ]
ϕ ( x τ 1 , y τ 2 ) ϕ ( x , y ) ϕ x ( x , y ) τ 1 ϕ y ( x , y ) τ 2 + 1 2 [ ϕ x x ( x , y ) τ 1 2 + 2 τ 1 τ 2 ϕ x y ( x , y ) + ϕ y y ( x , y ) τ 2 2 ]
ϕ ( x + τ 1 , y + τ 2 ) ϕ ( x τ 1 , y τ 2 ) 2 [ ϕ x ( x , y ) τ 1 + ϕ y ( x , y ) τ 2 ]
Γ ( x , y , ω 1 , ω 2 ) = A 2 ( x , y ) w ( τ 1 , τ 2 ) exp [ 2 j ( ϕ x ( x , y ) τ 1 + ϕ y ( x , y ) τ 2 ) ]
exp [ 2 j ( ω 1 τ 1 + ω 2 τ 2 ) ] τ 1 τ 2
Γ ( x , y , ω 1 , ω 2 ) = A 2 ( x , y ) W ̂ ( 2 ω 1 2 ϕ x ( x , y ) , 2 ω 2 2 ϕ y ( x , y ) )
[ ϕ x ( x , y ) , ϕ y ( x , y ) ] = arg max ω 1 , ω 2 Γ ( x , y , ω 1 , ω 2 )
w ( τ 1 , τ 2 ) = 1 2 π σ x σ y exp [ ( τ 1 2 2 σ x 2 + τ 2 2 2 σ y 2 ) ] τ 1 [ σ x 2 , σ x 2 ] , τ 2 [ σ y 2 , σ y 2 ]

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