## Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts |

Optics Express, Vol. 19, Issue 15, pp. 14464-14472 (2011)

http://dx.doi.org/10.1364/OE.19.014464

Acrobat PDF (1020 KB)

### Abstract

A non-iterative method based on principal component analysis (PCA) is presented to directly extract the phase from multiple-beam Fizeau interferograms with random phase shifts. The PCA method is the approach that decomposes the multiple-beam Fizeau interferograms into many uncorrelated quadrature signals and then applies principal component analysis algorithm to extract the measured phase without any prior guess about the phase shifts. Some factors that affect the performance of the proposed method are analyzed and discussed. Numerical simulations and experiments demonstrate that the proposed method extracts phase fast and exhibits high precision. The method can be applied in high precision interferometry.

© 2011 OSA

## 1. Introduction

5. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. **26**(13), 2506–2507 (1987). [CrossRef] [PubMed]

6. C. Ai and J. C. Wyant, “Effect of retroreflection on a Fizeau phase-shifting interferometer,” Appl. Opt. **32**(19), 3470–3478 (1993). [CrossRef] [PubMed]

8. B. V. Dorrío, J. Blanco-García, C. López, A. F. Doval, R. Soto, J. L. Fernández, and M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. **35**(1), 61–64 (1996). [CrossRef] [PubMed]

9. P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. **44**(11), 899–902 (1967). [CrossRef]

10. P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. **37**(10), 2751–2753 (1998). [CrossRef]

5. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. **26**(13), 2506–2507 (1987). [CrossRef] [PubMed]

11. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A **12**(4), 761–768 (1995). [CrossRef]

12. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. **35**(1), 51–60 (1996). [CrossRef] [PubMed]

*j*th-order harmonic can be minimized with phase steps of

13. J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. **22**(21), 3421–3432 (1983). [CrossRef] [PubMed]

14. P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. **5**(2), 167–194 (1996). [CrossRef]

15. A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express **13**(11), 4070–4084 (2005). [CrossRef] [PubMed]

16. R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. **30**(24), 3326–3328 (2005). [CrossRef] [PubMed]

17. R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise,” Opt. Lett. **31**(8), 1058–1060 (2006). [CrossRef] [PubMed]

18. 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]

20. X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. **31**(13), 1966–1968 (2006). [CrossRef] [PubMed]

21. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. **40**(17), 2886–2894 (2001). [CrossRef] [PubMed]

25. J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express **18**(23), 24368–24378 (2010). [CrossRef] [PubMed]

5. P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. **26**(13), 2506–2507 (1987). [CrossRef] [PubMed]

11. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A **12**(4), 761–768 (1995). [CrossRef]

25. J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express **18**(23), 24368–24378 (2010). [CrossRef] [PubMed]

27. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. **36**(8), 1326–1328 (2011). [CrossRef] [PubMed]

## 2. Multiple-beam Fizeau interferograms

*A*is the local mean intensity, and

*B*and

*C*are constants that depend on the reflectivity coefficients of reference surface

*r*

_{1}and test surface

*r*

_{2}

*φ*,

8. B. V. Dorrío, J. Blanco-García, C. López, A. F. Doval, R. Soto, J. L. Fernández, and M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. **35**(1), 61–64 (1996). [CrossRef] [PubMed]

26. J. Xu, Q. Xu, L. Chai, and H. Peng, “Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts,” Opt. Express **16**(23), 18922–18932 (2008). [CrossRef] [PubMed]

*j*th harmonic decreases with increasing of

*j*. Moreover, the rate of amplitude between consecutive harmonics is equal to

## 3. Principal component analysis

### 3.1 Calculating the covariance matrix

*N*images of multiple-beam Fizeau interferograms and each image is reshaped into one column with size of

*M*, (here

*N*images can be expressed in a matrix form as

*n*th image with size of

*M*, and

*I*is

*I*is

*N*). The size of

*p*=

*q*. The covariance matrix of

### 3.2 Diagonalization process

*Φ*consisting of

*Λ*is a diagonal matrix consisting of

*Φ*. In fact, Eq. (12) is the diagonalization process of the symmetric matrix

### 3.3 Principal Component Transform

*I*. The forward transform [27

27. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. **36**(8), 1326–1328 (2011). [CrossRef] [PubMed]

**can be seen as the coordinates in the orthogonal base. We can reconstruct the original data vector**

*Y***by**

*Y**Y*. This minimizes the mean-square error between the data and this representation with given number of eigenvectors. Instead of using all the eigenvectors of the covariance matrix, we may represent the data in terms of only a few basis vectors of the orthogonal basis. For the analysis of multiple beam Fizeau interferograms expressed by Eq. (1), it has only five unknows (A, B, C,

*φ*and

### 3.4 Calculating the measured phase

## 4. Numerical simulation and discussion

*π*, and

*φ*, the phase error of our algorithm is obtained and shown in Fig. 1 (d). The peak-to-valley (PV) and root-mean-square (RMS) values of the phase error are 0.670 and 0.103 rad, respectively. Figure 1 shows that the proposed algorithm can accurately extract the phase from seven multiple-beam Fizeau interferograms with random phase shifts.

*N*) in our simulation is 7 and each frame has signal-to-noise ratio (

*SNR*) of 30 dB. We let

*r*

_{1}= 0.2 and

*r*

_{2}vary from 0.1 to 1. Then we calculate the RMS value of phase error for different

*r*

_{2}and the result is shown in Fig. 2 (a) . For

*r*

_{2}when

*r*

_{2}, the contrast of the multiple-beam fringe decreases, the relative amplitude of high-order harmonics increases, and the influence of noise becomes serious. In addition, the covariance of

*r*

_{2}when

*r*

_{2}approximates to 1.

*r*

_{1}= 0.2,

*r*

_{2}= 0.7, and

*N*= 7. Then we perform simulation at different

*SNRs*to compute the RMS value of phase error, and the result is shown in Fig. 2(b). Figure 2(b) shows that the RMS value of phase error decreases with the increasing of

*SNR*. When SNR becomes larger than 40, the effect of noise on phase error becomes insignificant, so the RMS value of phase error keeps nearly constant of 0.025rad.

*r*

_{1}= 0.2,

*r*

_{2}= 0.7, and

*SNR*= 30dB. Then we calculate the RMS value of phase error when we use different numbers of multiple-beam fringes, and the result is shown in Fig. 2(c). Figure 2(c) shows that the RMS value of phase error decreases as the used number of multiple-beam fringes increases. From Figs. 2 (a-c) we can conclude that the RMS value of phase error is less than 0.1rad when

## 5. Experiments

26. J. Xu, Q. Xu, L. Chai, and H. Peng, “Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts,” Opt. Express **16**(23), 18922–18932 (2008). [CrossRef] [PubMed]

10. P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. **37**(10), 2751–2753 (1998). [CrossRef]

## 6. Conclusion

## Acknowledgments

## References and links

1. | K. Creath, “Temporal phase measurement method, ” in |

2. | D. Malacara, M. Servín, and Z. Malacara, |

3. | B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. |

4. | Y. Surrel, “Fringe analysis,” in |

5. | P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. |

6. | C. Ai and J. C. Wyant, “Effect of retroreflection on a Fizeau phase-shifting interferometer,” Appl. Opt. |

7. | G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) |

8. | B. V. Dorrío, J. Blanco-García, C. López, A. F. Doval, R. Soto, J. L. Fernández, and M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. |

9. | P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. |

10. | P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. |

11. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A |

12. | Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. |

13. | J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. |

14. | P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. |

15. | A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express |

16. | R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. |

17. | R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise,” Opt. Lett. |

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

19. | H. Guo and M. Chen, “Least-squares algorithm for phase-stepping interferometry with an unknown relative step,” Appl. Opt. |

20. | X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. |

21. | K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. |

22. | 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. |

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

24. | T. E. Zander, V. Madyastha, A. Patil, P. Rastogi, and L. M. Reindl, “Phase-step estimation in interferometry via an unscented Kalman filter,” Opt. Lett. |

25. | J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express |

26. | J. Xu, Q. Xu, L. Chai, and H. Peng, “Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts,” Opt. Express |

27. | J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. |

28. | R. C. Gonzalez and R. E. Woods, |

29. | B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. |

**OCIS Codes**

(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 23, 2011

Revised Manuscript: June 23, 2011

Manuscript Accepted: June 24, 2011

Published: July 13, 2011

**Citation**

Jiancheng Xu, Lili Sun, Yanli Li, and Yong Li, "Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts," Opt. Express **19**, 14464-14472 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14464

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### References

- K. Creath, “Temporal phase measurement method, ” in Interferogram Analysis, D. W. Robinson and G. T. Reid, eds. (Institute of Physics, Bristol, UK, 1993), pp. 94–140.
- D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, New York, 1998).
- B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999). [CrossRef]
- Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed., Vol. 77 of Topics in Applied Physics (Springer, 2000), pp. 55–102.
- P. Hariharan, “Digital phase-stepping interferometry: effects of multiply reflected beams,” Appl. Opt. 26(13), 2506–2507 (1987). [CrossRef] [PubMed]
- C. Ai and J. C. Wyant, “Effect of retroreflection on a Fizeau phase-shifting interferometer,” Appl. Opt. 32(19), 3470–3478 (1993). [CrossRef] [PubMed]
- G. Bonsch and H. Bohme, “Phase-determination of Fizeau interferences by phase-shifting interferometry,” Optik (Stuttg.) 82, 161–164 (1989).
- B. V. Dorrío, J. Blanco-García, C. López, A. F. Doval, R. Soto, J. L. Fernández, and M. Pérez-Amor, “Phase error calculation in a Fizeau interferometer by Fourier expansion of the intensity profile,” Appl. Opt. 35(1), 61–64 (1996). [CrossRef] [PubMed]
- P. B. Clapham and G. D. Dew, “Surface-coated reference flats for testing fully aluminized surfaces by means of the Fizeau interferometer,” J. Sci. Instrum. 44(11), 899–902 (1967). [CrossRef]
- P. Hariharan, “Interferometric measurements of small-scale irregularities: highly reflecting surfaces,” Opt. Eng. 37(10), 2751–2753 (1998). [CrossRef]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12(4), 761–768 (1995). [CrossRef]
- Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef] [PubMed]
- J. Schwider, R. Burow, K. E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital wave-front measuring interferometry: some systematic error sources,” Appl. Opt. 22(21), 3421–3432 (1983). [CrossRef] [PubMed]
- P. Picart, R. Mercier, and M. Lamare, “Influence of multiple-beam interferences in a phase-shifting Fizeau interferometer and error-reduced algorithms,” Pure Appl. Opt. 5(2), 167–194 (1996). [CrossRef]
- A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13(11), 4070–4084 (2005). [CrossRef] [PubMed]
- R. Langoju, A. Patil, and P. Rastogi, “Resolution-enhanced Fourier transform method for the estimation of multiple phases in interferometry,” Opt. Lett. 30(24), 3326–3328 (2005). [CrossRef] [PubMed]
- R. Langoju, A. Patil, and P. Rastogi, “Phase-shifting interferometry in the presence of nonlinear phase steps, harmonics, and noise,” Opt. Lett. 31(8), 1058–1060 (2006). [CrossRef] [PubMed]
- 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]
- H. Guo and M. Chen, “Least-squares algorithm for phase-stepping interferometry with an unknown relative step,” Appl. Opt. 44(23), 4854–4859 (2005). [CrossRef] [PubMed]
- X. F. Xu, L. Z. Cai, X. F. Meng, G. Y. Dong, and X. X. Shen, “Fast blind extraction of arbitrary unknown phase shifts by an iterative tangent approach in generalized phase-shifting interferometry,” Opt. Lett. 31(13), 1966–1968 (2006). [CrossRef] [PubMed]
- K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [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, J. Schwider, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009). [CrossRef] [PubMed]
- T. E. Zander, V. Madyastha, A. Patil, P. Rastogi, and L. M. Reindl, “Phase-step estimation in interferometry via an unscented Kalman filter,” Opt. Lett. 34(9), 1396–1398 (2009). [CrossRef] [PubMed]
- J. Xu, Y. Li, H. Wang, L. Chai, and Q. Xu, “Phase-shift extraction for phase-shifting interferometry by histogram of phase difference,” Opt. Express 18(23), 24368–24378 (2010). [CrossRef] [PubMed]
- J. Xu, Q. Xu, L. Chai, and H. Peng, “Algorithm for multiple-beam Fizeau interferograms with arbitrary phase shifts,” Opt. Express 16(23), 18922–18932 (2008). [CrossRef] [PubMed]
- J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011). [CrossRef] [PubMed]
- R. C. Gonzalez and R. E. Woods, Digital Image Processing, 3rd ed. (Prentice-Hall, 2007).
- B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009). [CrossRef] [PubMed]

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