## Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method |

Optics Express, Vol. 21, Issue 14, pp. 17234-17248 (2013)

http://dx.doi.org/10.1364/OE.21.017234

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### Abstract

A novel method is presented to extract phase distribution from phase-shifted interferograms with unknown tilt phase shifts. The proposed method can estimate the tilt phase shift between two temporal phase-shifted interferograms with high accuracy, by extending the regularized optical flow method with the spatial image processing and frequency estimation technology. With all the estimated tilt phase shifts, the phase component encoded in the interferograms can be extracted by the least-squares method. Both simulation and experimental results have fully proved the feasibility of the proposed method. Particularly, a flat-based diffractive optical element with quasi-continuous surface is tested by the proposed method with introduction of considerably large tilt phase shift amounts (i.e., the highest estimated tilt phase shift amount between two consecutive frame reaches 6.18λ). The phase extraction result is in good agreement with that of Zygo’s MetroPro software under steady-state testing conditions, and the residual difference between them is discussed. In comparison with the previous methods, the proposed method not only has relatively little restrictions on the amounts or orientations of the tilt phase shifts, but also works well with interferograms including open and closed fringes in any combination.

© 2013 OSA

## 1. Introduction

1. J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express **19**(2), 584–596 (2011). [CrossRef] [PubMed]

3. R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. **51**(5), 626–632 (2013). [CrossRef]

1. J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express **19**(2), 584–596 (2011). [CrossRef] [PubMed]

8. J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. **51**(5), 637–641 (2013). [CrossRef]

## 2. Principle

### 2.1 The regularized optical flow method for the two-step interferometry

10. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. **36**(17), 3485–3487 (2011). [CrossRef] [PubMed]

### 2.2 Computation of the wrapped phase shift distribution between two interferograms

### 2.3 Unwrap the phase shift distribution

### 2.5 Elimination of the sign ambiguity among the estimated phase-shifting planes

### 2.6 Phase extraction with the least-squares method

## 3. Simulation results

*x-y*coordinate system with

*PV*and

*RMS*measures are shown in Table 1. As can be seen from Table 1, all the residual

*PV*errors and

*RMS*errors are within

## 4. Experimental results

10. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. **36**(17), 3485–3487 (2011). [CrossRef] [PubMed]

4. Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. **29**(14), 1671–1673 (2004). [CrossRef] [PubMed]

*PV*measure,

*RMS*measure [see Fig. 3(e)], and

*PV*measure,

*RMS*measure [see Fig. 3(f)], respectively. Particularly, we think the residual errors shown in Fig. 3(e) can be taken as an indirect measure of the non-ideality of the testing environment. Then we can conclude that in this experiment the proposed method has similar accuracy to the AIA method, as the residual errors shown in Fig. 3(e) and Fig. 3(f) are comparable in

*PV*and

*RMS*measures. The typical “double-frequency” fringe error shown in Fig. 3(f) is due to the discrepancies in phase shift estimations between the AIA method and the proposed method, i.e., as to the AIA method, the phase shifts are estimated in an iterative way and the information of different interferograms will be coupled in the least-squares sense during the iteration process, while for the proposed method, the estimation of phase shift between two interferograms is independent of the information encoded in other interferograms.

## 5. Conclusion

10. J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. **36**(17), 3485–3487 (2011). [CrossRef] [PubMed]

1. J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express **19**(2), 584–596 (2011). [CrossRef] [PubMed]

11. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**(22), 3894–3898 (2000). [CrossRef] [PubMed]

15. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. **10**(7), 075011 (2008). [CrossRef]

11. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. **39**(22), 3894–3898 (2000). [CrossRef] [PubMed]

12. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. **41**(13), 2435–2439 (2002). [CrossRef] [PubMed]

14. J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. **47**(3), 480–485 (2008). [CrossRef] [PubMed]

13. O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express **13**(10), 3743–3753 (2005). [CrossRef] [PubMed]

**19**(2), 584–596 (2011). [CrossRef] [PubMed]

15. J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. **10**(7), 075011 (2008). [CrossRef]

## Acknowledgments

## References and links

1. | J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express |

2. | D. Malacara, |

3. | R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. |

4. | Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. |

5. | X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. |

6. | P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. |

7. | B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett. |

8. | J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng. |

9. | H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express |

10. | J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett. |

11. | M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. |

12. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. |

13. | O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express |

14. | J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt. |

15. | J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt. |

16. | S. Ye and E. Aboutanios, “Two dimensional frequency estimation by interpolation on Fourier coefficients,” in Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 3353–3356 (2012). [CrossRef] |

17. | R. C. Gonzalez, R. E. Woods, and S. L. Eddins, |

18. | L. G. Shapiro and G. C. Stockman, |

**OCIS Codes**

(050.5080) Diffraction and gratings : Phase shift

(120.2650) Instrumentation, measurement, and metrology : Fringe analysis

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(120.5050) Instrumentation, measurement, and metrology : Phase measurement

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 13, 2013

Revised Manuscript: July 3, 2013

Manuscript Accepted: July 3, 2013

Published: July 11, 2013

**Citation**

Fa Zeng, Qiaofeng Tan, Huarong Gu, and Guofan Jin, "Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method," Opt. Express **21**, 17234-17248 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-17234

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### References

- J. Vargas, J. A. Quiroga, A. Álvarez-Herrero, and T. Belenguer, “Phase-shifting interferometry based on induced vibrations,” Opt. Express19(2), 584–596 (2011). [CrossRef] [PubMed]
- D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 14, pp. 547–666.
- R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. M. Arévalo Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng.51(5), 626–632 (2013). [CrossRef]
- Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett.29(14), 1671–1673 (2004). [CrossRef] [PubMed]
- X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett.33(8), 776–778 (2008). [CrossRef] [PubMed]
- P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett.34(22), 3553–3555 (2009). [CrossRef] [PubMed]
- B. Li, L. Chen, W. Tuya, S. Ma, and R. Zhu, “Carrier squeezing interferometry: suppressing phase errors from the inaccurate phase shift,” Opt. Lett.36(6), 996–998 (2011). [CrossRef] [PubMed]
- J. Vargas and C. O. S. Sorzano, “Quadrature Component Analysis for interferometry,” Opt. Lasers Eng.51(5), 637–641 (2013). [CrossRef]
- H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express19(8), 7807–7815 (2011). [CrossRef] [PubMed]
- J. Vargas, J. A. Quiroga, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Two-step interferometry by a regularized optical flow algorithm,” Opt. Lett.36(17), 3485–3487 (2011). [CrossRef] [PubMed]
- M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt.39(22), 3894–3898 (2000). [CrossRef] [PubMed]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt.41(13), 2435–2439 (2002). [CrossRef] [PubMed]
- O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express13(10), 3743–3753 (2005). [CrossRef] [PubMed]
- J. Xu, Q. Xu, and L. Chai, “Iterative algorithm for phase extraction from interferograms with random and spatially nonuniform phase shifts,” Appl. Opt.47(3), 480–485 (2008). [CrossRef] [PubMed]
- J. Xu, Q. Xu, and L. Chai, “Tilt-shift determination and compensation in phase-shifting interferometry,” J. Opt. A, Pure Appl. Opt.10(7), 075011 (2008). [CrossRef]
- S. Ye and E. Aboutanios, “Two dimensional frequency estimation by interpolation on Fourier coefficients,” in Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing, 3353–3356 (2012). [CrossRef]
- R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MATLAB (Prentice Hall, 2004).
- L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, Upper Saddle River, New Jersey, USA, 2001).

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