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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 18041–18046
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Estimation of displacement derivatives in digital holographic interferometry using a two-dimensional space-frequency distribution

G. Rajshekhar, Sai Siva Gorthi, and Pramod Rastogi  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 18041-18046 (2010)
http://dx.doi.org/10.1364/OE.18.018041


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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

1. Introduction

2. Theory

For deformation analysis using DHI, two holograms are recorded for object states before and after deformation and the corresponding complex amplitudes are obtained using numerical reconstruction [13

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

]. Multiplying the post-deformation complex amplitude with the conjugate of pre-deformation complex amplitude, the reconstructed interference field is obtained which is given as [5

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

, 14

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, (2009). [CrossRef]

]:

I(x,y)=A(x,y)exp[jϕ(x,y)]+η(x,y)
(1)

where A(x,y) is the amplitude term; ϕ(x,y) is the interference phase and η(x,y) represents the noise assumed to be zero mean additive white Gaussian noise (AWGN). Here x and y refer to the pixel values along the N × N size image. The real part of the reconstructed interference field constitutes a fringe pattern. The 2D Wigner-Ville distribution corresponding to I(x,y) is given as [15

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]

]

Γ1(x,y,ω1,ω2)=I(x+u12,y+u22)I*(xu12,yu22)exp[j(ω1u1+ω2u2)]u1u2
(2)

where ‘*’ denotes the complex conjugate. To improve the robustness of the distribution against interference terms or artifacts, a smoothing window is introduced [16

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

] and the resulting so called two-dimensional pseudo Wigner-Ville distribution can be equivalently written as

Γ(x,y,ω1,ω2)=w(τ1,τ2)I(x+τ1,y+τ2)I*(xτ1,yτ2)exp[2j(ω1τ1+ω2τ2)]τ1τ2
(3)

where w(τ 1, τ 2) is a real symmetric window usually having a low-pass behavior. From, Eq. (3), it is clear that 2D-PSWVD maps the reconstructed interference field I(x,y) in spatial domain to Γ(x,y,ω 1,ω 2), thereby providing a joint space-frequency representation. For the simplicity of analysis, the amplitude term is assumed constant within the window region and the noise terms are ignored. Hence from Eq. (1) and Eq. (3), we have

Γ(x,y,ω1,ω2)=A2(x,y)w(τ1,τ2)exp[j(ϕ(x+τ1,y+τ2)ϕ(xτ1,yτ2))]
exp[j(2ω1τ1+2ω2τ2)]τ1τ2
(4)

Assuming that the phase ϕ(x,y) is slowly varying in the neighborhood of (x,y) and using Taylor series expansion up to second order, we have the following approximations

ϕ(x+τ1,y+τ2)ϕ(x,y)+ϕx(x,y)τ1+ϕy(x,y)τ2+12[ϕxx(x,y)τ12+2τ1τ2ϕxy(x,y)+ϕyy(x,y)τ22]
(5)
ϕ(xτ1,yτ2)ϕ(x,y)ϕx(x,y)τ1ϕy(x,y)τ2+12[ϕxx(x,y)τ12+2τ1τ2ϕxy(x,y)+ϕyy(x,y)τ22]
(6)
ϕ(x+τ1,y+τ2)ϕ(xτ1,yτ2)2[ϕx(x,y)τ1+ϕy(x,y)τ2]
(7)

where

ϕx(x,y)=ϕ(x,y)x,ϕxx(x,y)=2ϕ(x,y)x2,ϕxy(x,y)=2ϕ(x,y)xy

and

ϕy(x,y)=ϕ(x,y)y,ϕyy(x,y)=2ϕ(x,y)y2

Using these approximations in Eq. (4), we have

Γ(x,y,ω1,ω2)=A2(x,y)w(τ1,τ2)exp[2j(ϕx(x,y)τ1+ϕy(x,y)τ2)]
exp[2j(ω1τ1+ω2τ2)]τ1τ2
(8)

which is essentially the 2D Fourier transform of w(τ 1,τ 2) modulated by the term exp[2j(ϕx(x,y)τ 1 + ϕy(x,y)τ 2)]. Denoting Ŵ(ω 1,ω 2) as the 2D Fourier transform of w(τ 1,τ 2) and using the frequency shifting property of the Fourier transform, we have

Γ(x,y,ω1,ω2)=A2(x,y)Ŵ(2ω12ϕx(x,y),2ω22ϕy(x,y))
(9)

As window w has a low-pass behavior, Γ(x,y,ω 1,ω 2) is maximum for Ŵ(0,0) i.e. when [ω 1,ω 2] = [ϕx(x,y),ϕy(x,y)]. In other words,

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.
[ϕx(x,y),ϕy(x,y)]=argmaxω1,ω2Γ(x,y,ω1,ω2)
(10)

Hence, the phase derivatives at pixel location (x,y) can be estimated by finding the values of (ω 1,ω 2) at which Γ(x,y,ω 1,ω 2) attains its maximum. Note that the phase derivatives along both directions i.e. x and y can be simultaneously determined using Eq. (10). By repeating the above analysis for all pixels, the phase derivatives for the entire fringe pattern can be estimated. The computational burden for evaluating the 2D Fourier transform can be greatly relieved by using 2D fast Fourier transform (FFT) algorithm.

3. Simulation and Experimental Results

A complex reconstructed interference field signal was simulated at signal to noise ratio (SNR) of 10 dB using ‘AWGN’ function of MATLAB 7.8.0(R2009a) and the corresponding fringe pattern (size 128 × 128) is shown in Fig. 1(a). To apply the 2D-PSWVD for phase derivative estimation, a 2D Gaussian window of the following form was used:

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

table-icon
View This Table
w(τ1,τ2)=12πσxσyexp[(τ122σx2+τ222σy2)]τ1[σx2,σx2],τ2[σy2,σy2]
(11)

with σx = σy = 16. Applying the proposed method, the obtained phase derivative estimate along x in radians/pixel and its wrapped form are shown in Fig. 1(b) and Fig. 1(c). The phase derivative estimate along y in radians/pixel and its wrapped form are shown in Fig. 1(e) and Fig. 1(f). Note that the proposed method directly provides unwrapped estimates and the wrapped forms are shown here for the sake of illustration only. The errors between original and estimated phase derivatives along x and y are shown in Fig. 1(d) and Fig. 1(g) respectively. The root mean square errors (RMSE) for phase derivative estimation along x and y are 0.0048 and 0.0036 radians/pixel respectively. The pixels near the borders of the fringe pattern were neglected in the analysis to ignore the errors at the boundaries.

In DHI measurements, noise is an important consideration and its various sources are discussed in [17

17. A. Choudry, “Digital holographic interferometry of convective heat transport,” Appl. Opt. 20, 1240–1244 (1981). [CrossRef] [PubMed]

]. The major advantage of the proposed method is the improved robustness against noise as compared to the phase derivative estimation method used in [10

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]

]. This is because a 2D window captures data samples along both x and y for the evaluation of the PSWVD in the proposed method and consequently provides smoothing in both directions effectively increasing the method’s immunity against noise. The increased noise robustness comes at the cost of increased computational burden due to the need of evaluating 2D Fourier transforms at each pixel location. On the contrary, the method in [10

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]

] uses a window along only one direction and though it is quite computationally efficient due to the use of 1D Fourier transform, it is more error-prone under high noise. To compare the performance of both methods with respect to noise-immunity, a reconstructed interference field signal similar to the one used for Fig. 1 was simulated for different SNRs. The RMSEs (in radians/pixel) for phase derivative estimation obtained by the proposed method and the method in [10

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]

] at various SNRs (in dB) are shown in Table 1. From the table, it is clear that the proposed method exhibits superior performance and provides phase derivative estimates with reasonable accuracy even for very low SNR such as 2 dB. The computational times required for the proposed method and the other method are about 8 minutes and 3 seconds on a 2.66 GHz Intel Core 2 Quad Processor machine with 3.23 GB random access memory. Though the computational cost of the proposed method is high, it can be significantly improved using techniques like fixed-point computations [18

18. N. Pandey and B. Hennelly, “Fixed-point numercial reconstruction for digital holographic microscopy,” Opt. Lett. 35, 1076–1078 (2010). [CrossRef] [PubMed]

] and graphics processing units [19

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]

].

To test the practical applicability of the proposed method, a DHI experiment was conducted by subjecting a circularly clamped object to central load and recording two holograms before and after deformation using a Coherent Verdi laser (532 nm). The complex amplitudes of the object wave before and after deformation were obtained by numerical reconstruction performed using discrete Fresnel transform [13

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

]. The reconstructed interference field was obtained as discussed earlier in the ‘Theory’ section. The experimental fringe pattern is shown in Fig. 2(a). The phase derivative estimate along x in radians/pixel and the corresponding wrapped form obtained using the proposed method are shown in Fig. 2(b) and Fig. 2(c). Similarly, the phase derivative estimate along y in radians/pixel and the corresponding wrapped form are shown in Fig. 2(d) and Fig. 2(e). For comparison, the digital shearing approach [5

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

] was also applied for phase derivative estimation along y and the obtained wrapped estimate is shown in Fig. 2(f). It is clear from Fig. 2(e) and Fig. 2(f) that the proposed method shows superior performance. Moreover, since the obtained estimates are wrapped, unwrapping operation is required in digital shearing approach.

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.

4. Conclusions

A 2D-PSWVD based method is proposed in the paper for the direct estimation of the phase derivative in DHI. The method provides phase derivative estimates without requiring any unwrapping or filtering operations. The method also has high robustness against noise and enables simultaneous retrieval of phase derivatives along both spatial directions. The authors believe that the method could prove to be a potential choice for DHI applications involving measurement of displacement derivatives.

Acknowledgments

This work is funded by Swiss National Science Foundation under Grant 200020-121555.

References and links

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. Grédiac, 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, (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 numercial 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]

OCIS Codes
(090.2880) Holography : Holographic interferometry
(120.2880) Instrumentation, measurement, and metrology : Holographic interferometry
(090.1995) Holography : Digital holography

ToC Category:
Holography

History
Original Manuscript: June 4, 2010
Revised Manuscript: August 2, 2010
Manuscript Accepted: August 3, 2010
Published: August 6, 2010

Citation
G. Rajshekhar, Sai Siva Gorthi, and Pramod Rastogi, "Estimation of displacement derivatives in digital holographic interferometry using a two-dimensional space-frequency distribution," Opt. Express 18, 18041-18046 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-18041


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References

  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]

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