## Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform

Optics Express, Vol. 11, Issue 6, pp. 617-623 (2003)

http://dx.doi.org/10.1364/OE.11.000617

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### Abstract

In this study, we propose the Hilbert transform (HT) method for phase analysis of a Dynamic ESPI signal. The data processing is performed in the temporal domain, using the temporal history of the interference signal at every single pixel. The final results give a temporal development of the two-dimensional deformation field. To reduce the influence of the fluctuations of bias intensity on the calculated phase, it was removed prior to performing the HT. This method was demonstrated for defects distinction and the determination of the sign change in the deformation field in two different experiments. The range of measurement lies between submicrons and tens of microns and the spatial resolution is better when compared to the fringe analysis method and the spatial carrier method.

© 2003 Optical Society of America

## 1. Introduction

5. V. Madjarova, S. Toyooka, R. Widiastuti, and H. Kadono, “Dynamic ESPI with subtraction-addition method for obtaining the phase,” Opt. Commun. **212**, 35 (2002) [CrossRef]

6. Xavier Colonna de Lega and Pierre Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. **35**, 5115–5121 (1994). [CrossRef]

11. C. Joenathan., P. Haible, and H. Tiziani, “Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera,” Appl. Opt. **38**,1169–1178 (1999). [CrossRef]

12. D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of Hilbert transform in ECG signal analysis,” Computers in Biology and Medicine **31**, 399–406 (2001) [CrossRef] [PubMed]

## 2. Data-analysis method

*I*

_{0}(

*x,y,t*

_{i}) and

*I*(

_{m}*x,y,t*) are the bias and the modulation intensities, respectively. ϕ(

_{i}*x,y,t*)=θ(

_{i}*x,y,t*)+φ(

_{i}*x,y,t*) is the signal phase, where θ(

_{i}*x,y,t*) is the random phase of the speckle field, and φ(

_{i}*x,y,t*) is the phase introduced by the deformation of the object under study, which usually varies slowly in the space and the time domains.

_{i}*t*is the time the

_{i}*i-th*frame of speckle interference pattern is taken. We can treat each point in space domain having an independent time signal of interference intensity. To obtain the phase, the temporal history of the development of the interference signal at each point in the image is processed, applying signal-processing technique. One advantage of working in time domain in case of ESPI is the significantly low presence of noise. Some other advantages are significantly simpler algorithms for one-dimensional signals, and the straightforwardness in the unwrapping procedure. In addition, since the interference signal that we are using is three dimensional, we can receive the full spatio-temporal information of a given process.

*u*(

*t*), we may associate a complex wave function ψ(

*t*)=

*u*(

*t*)+

*iv*(

*t*) where the imaginary part is the HT of the real signal. The HT of

*u*(

*t*) is defined by [16]

*Hi*{

*u*(

*t*)} is a linear functional of

*u*(

*t*). In fact, it is obtained from

*u*(

*t*) by convolution with (-π

*t*)

^{-1}expressed by

*t*. The HT of the signal

*f*(ϕ(

*t*))=cos(ϕ(

*t*)) gives sin(ϕ(

*t*)). Using this characteristic, we can determine the phase of the signal through the equation given by

5. V. Madjarova, S. Toyooka, R. Widiastuti, and H. Kadono, “Dynamic ESPI with subtraction-addition method for obtaining the phase,” Opt. Commun. **212**, 35 (2002) [CrossRef]

22. The Math Works, MATLAB Signal Processing Toolbox: User’s guide (December 1996), www.mathworks.com

*p*represents the reference frame. The speckle phase value will change for a long term of experiments. For this reason, the reference value is renewed after a certain interval. The unwrapped phase values are recorded as two-dimensional images, which represent the space development of the deformation. Because some pixels have low modulation intensity, the phase in these pixels could not be determined correctly. These pixels give spiky noise in the final results. To remove this noise, a median filter is applied in space domain.

## 3. Experimental results and discussion

## 4. Conclusions

5. V. Madjarova, S. Toyooka, R. Widiastuti, and H. Kadono, “Dynamic ESPI with subtraction-addition method for obtaining the phase,” Opt. Commun. **212**, 35 (2002) [CrossRef]

## Acknowledgements

## References and Links

1. | P. K. Rastogi (Ed.), |

2. | P.K. Rastogi, |

3. | D. W. Robinson and C. R. Raed (Ed.), |

4. | Wolfgang Osten, “Digital Processing and Evaluation of Fringe Patterns in Optical Metrology and Non-Destructive Testing, D-28359 (Bremer Institut fur Angenwandte Strahltechnik, Bremen, 1998). |

5. | V. Madjarova, S. Toyooka, R. Widiastuti, and H. Kadono, “Dynamic ESPI with subtraction-addition method for obtaining the phase,” Opt. Commun. |

6. | Xavier Colonna de Lega and Pierre Jacquot, “Deformation measurement with object-induced dynamic phase shifting,” Appl. Opt. |

7. | X. Colonna de Lega, “Continuous deformation measurement using dynamic phase shifting and wavelet transforms,” |

8. | J. M. Huntly, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1kHz,” Appl. Opt. |

9. | T. E. Carlsson and A. Wei, “Phase Evaluation of Speckle Patterns During Continuous Deformation by use of Phase-Shifting Speckle Interferometry,” Appl. Opt. |

10. | C. Joenathan, B. Franze, P. Haible, and H.J. Tiziani, “Large in-plane displacement in dual-beam speckle interferometry using temporal phase measurement,” J. mod. Opt. |

11. | C. Joenathan., P. Haible, and H. Tiziani, “Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera,” Appl. Opt. |

12. | D. Benitez, P. A. Gaydecki, A. Zaidi, and A. P. Fitzpatrick, “The use of Hilbert transform in ECG signal analysis,” Computers in Biology and Medicine |

13. | Yuuki Watanabe and Ichirou Yamaguchi, “Digital Hilbert transformation for separation measurement of thickness and refractive indices of layered objects by use of a wavelength-scanning heterodyne interference confocal microscope,” Appl. Opt. |

14. | V. A. Grechikhin and B.S. Rinkevichius, “Hilbert transform for processing of laser Doppler vibrometer signals,” Opt. Laser Eng. |

15. | Yanghua Zhao, Zhongping Chen, Christopher Saxer, Shaohua Xiang, Johannes F. de Boer, and J. Stuart Nelson, “Phase resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,” Opt. Lett. |

16. | Stefan L. Hahn, |

17. | G. Cloude, Speckle interferometry made simple and cheap, |

18. | Ronald N. Bracewell, |

19. | Max Born and Emil Wolf, |

20. | E.C. Titchmarsh, |

21. | Sanjit K. Mitra, |

22. | The Math Works, MATLAB Signal Processing Toolbox: User’s guide (December 1996), www.mathworks.com |

23. | Dennis C. Ghiglia and Mark D Pritt, |

**OCIS Codes**

(100.2650) Image processing : Fringe analysis

(120.6160) Instrumentation, measurement, and metrology : Speckle interferometry

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 3, 2003

Revised Manuscript: March 17, 2003

Published: March 24, 2003

**Citation**

Violeta Madjarova, Hirofumi Kadono, and Satoru Toyooka, "Dynamic electronic speckle pattern interferometry (DESPI) phase analyses with temporal Hilbert transform," Opt. Express **11**, 617-623 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-6-617

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### References

- P. K. Rastogi (Ed.), Photomechanics (Springer-Verlag, Berlin, 1999).
- P.K. Rastogi, Digital Speckle Pattern Interferometry and Related Techniques (John and Wiley and Sons Ltd., Chichester, 2001).
- D. W. Robinson, C. R. Raed (Ed.), Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics Publishing, Bristol, 1993).
- Wolfgang Osten, �??Digital Processing and Evaluation of Fringe Patterns in Optical Metrology and Non-Destructive Testing,�?? D-28359 (Bremer Institut fur Angenwandte Strahltechnik, Bremen, 1998).
- V. Madjarova, S. Toyooka, R. Widiastuti, H. Kadono, �??Dynamic ESPI with subtraction-addition method for obtaining the phase,�?? Opt. Commun. 212, 35 (2002) [CrossRef]
- Xavier Colonna de Lega, Pierre Jacquot, �??Deformation measurement with object-induced dynamic phase shifting,�?? Appl. Opt. 35, 5115-5121 (1994). [CrossRef]
- X. Colonna de Lega, �??Continuous deformation measurement using dynamic phase shifting and wavelet transforms,�?? Applied Optics and Optoelectronics, ed. K.T.V. Gratten (Institute of Physics Publishing 1996)
- J. M. Huntly, G. H. Kaufmann, D. Kerr, �??Phase-shifted dynamic speckle pattern interferometry at 1kHz,�?? Appl. Opt. 38, 6556-6563 (1999). [CrossRef]
- T. E. Carlsson, A. Wei, �??Phase Evaluation of Speckle Patterns During Continuous Deformation by use of Phase-Shifting Speckle Interferometry,�?? Appl. Opt. 39, 2628-2637 (2000). [CrossRef]
- C. Joenathan, B. Franze, P. Haible, H.J. Tiziani, �??Large in-plane displacement in dual-beam speckle interferometry using temporal phase measurement,�?? J. Mod. Opt. 44, 1975-1984 (1998). [CrossRef]
- C. Joenathan., P. Haible, H. Tiziani, �??Speckle interferometry with temporal phase evaluation: influence of decorrelation, speckle size, and nonlinearity of the camera,�?? Appl. Opt. 38, 1169-1178 (1999). [CrossRef]
- D. Benitez, P. A. Gaydecki, A. Zaidi, A. P. Fitzpatrick, �??The use of Hilbert transform in ECG signal analysis,�?? Computers in Biology and Medicine 31, 399-406 (2001) [CrossRef] [PubMed]
- Yuuki Watanabe, Ichirou Yamaguchi, �??Digital Hilbert transformation for separation measurement of thickness and refractive indices of layered objects by use of a wavelength-scanning heterodyne interference confocal microscope,�?? Appl. Opt. 41, 4497-4502 (2002) [CrossRef] [PubMed]
- V. A. Grechikhin, B.S. Rinkevichius, �??Hilbert transform for processing of laser Doppler vibrometer signals,�?? Opt. Laser Eng. 30, 151-161 (1998). [CrossRef]
- Yanghua Zhao, Zhongping Chen, Christopher Saxer, Shaohua Xiang, Johannes F. de Boer, J. Stuart Nelson, �??Phase resolved optical coherence tomography and optical Doppler tomography for imaging blood flow in human skin with fast scanning speed and high velocity sensitivity,�?? Opt. Lett. 25, 14-16 (2000) [CrossRef]
- Stefan L. Hahn, Hilbert Transforms in Signal Processing (Artech House: Boston, 1996)
- G. Cloude, Speckle interferometry made simple and cheap, Proc. Int. Conf. Theoretical, Experimental and Computational Mechanics 2000, 796.
- Ronald N. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Companies, Inc., 2000)
- Max Born, Emil Wolf, Principle of Optics: Electromagnetic theory of propagation, Interference and Diffraction of light (Cambridge University Press, Edinburg, 1999: 7th extended edition)
- E.C. Titchmarsh, Introduction to the theory of Fourier integrals (Chelsea Publishing Company, New York, 1986)
- Sanjit K. Mitra, Digital Signal Processing: A computer based approach (McGraw-Hill Companies, Inc., 2002)
- The Math Works, MATLAB Signal Processing Toolbox: User�??s guide (December 1996), <a href="http://www.mathworks.com">www.mathworks.com</a>
- Dennis C. Ghiglia, Mark D Pritt, Two-Dimensional Phase Unwrapping (John Wiley and Sons, Inc., New York, 1998)

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