## Estimation of displacement derivatives in digital holographic interferometry using a two-dimensional space-frequency distribution |

Optics Express, Vol. 18, Issue 17, pp. 18041-18046 (2010)

http://dx.doi.org/10.1364/OE.18.018041

Acrobat PDF (781 KB)

### 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

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]

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]

## 2. Theory

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

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

*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]

*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

*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

*w*(

*τ*

_{1},

*τ*

_{2}) modulated by the term exp[2

*j*(

*ϕ*(

_{x}*x,y*)

*τ*

_{1}+

*ϕ*(

_{y}*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

*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}] = [

*ϕ*(

_{x}*x,y*),

*ϕ*(

_{y}*x,y*)]. In other words,

*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

*σ*=

_{x}*σ*= 16. Applying the proposed method, the obtained phase derivative estimate along

_{y}*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.

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

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]

*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]

**80**, 093107 (2009). [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]

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

*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]

*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.

## 4. Conclusions

## Acknowledgments

## 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. |

2. | G. K. Bhat, “A Fourier transform technique to obtain phase derivatives in interferometry,” Opt. Commun. |

3. | Y. Zou, G. Pedrini, and H. Tiziani, “Derivatives obtained directly from displacement data,” Opt. Commun. |

4. | U. Schnars and W. P. O. Juptner, “Digital recording and reconstruction of holograms in hologram interferometry and shearography,” Appl. Opt. |

5. | C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. |

6. | C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. |

7. | K. Qian, S. H. Soon, and A. Asundi, “Phase-shifting windowed fourier ridges for determination of phase derivatives,” Opt. Lett. |

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. |

9. | C. A. Sciammarella and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. |

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. |

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 |

12. | S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express |

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

14. | S. S. Gorthi and P. Rastogi, “Analysis of reconstructed interference fields in digital holographic interferometry using the polynomial phase transform,” Meas. Sci. Technol. |

15. | L. Debnath and B. Rao, “On new two-dimensional Wigner-Ville nonlinear integral transforms and their basic properties,” Integr. Transf. Spec. F |

16. | L. Cohen, |

17. | A. Choudry, “Digital holographic interferometry of convective heat transport,” Appl. Opt. |

18. | N. Pandey and B. Hennelly, “Fixed-point numercial reconstruction for digital holographic microscopy,” Opt. Lett. |

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. |

**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

- 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]
- 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]
- C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443–3446 (2003). [CrossRef]
- C. Quan, C. J. Tay, and W. Chen, “Determination of displacement derivative in digital holographic interferometry,” Opt. Commun. 282, 809–815 (2009). [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. 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]
- C. A. Sciammarella, and T. Kim, “Determination of strains from fringe patterns using space-frequency representations,” Opt. Eng. 42, 3182–3193 (2003). [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]
- 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]
- 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]
- 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]
- L. Cohen, Time Frequency Analysis (Prentice Hall, 1995).
- A. Choudry, “Digital holographic interferometry of convective heat transport,” Appl. Opt. 20, 1240–1244 (1981). [CrossRef] [PubMed]
- N. Pandey, and B. Hennelly, “Fixed-point numerical reconstruction for digital holographic microscopy,” Opt. Lett. 35, 1076–1078 (2010). [CrossRef] [PubMed]
- 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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