## Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method |

Optics Express, Vol. 19, Issue 21, pp. 20483-20492 (2011)

http://dx.doi.org/10.1364/OE.19.020483

Acrobat PDF (2120 KB)

### Abstract

A method combining the principal component analysis (PCA) and the least squares method (LSM) is proposed to extract the phase from interferograms with random phase shifts. The method estimates the initial phase by PCA, and then determines the correct global phase sign and reduces the residual phase error by LSM. Some factors that may influence the performance of the proposed method are analyzed and discussed, such as the number of frames used, the number of fringes in interferogram and the amplitude of random phase shifts. Numerical simulations and optical experiments are implemented to verify the effectiveness of this method. The proposed method is suitable for randomly phase-shifted interferograms.

© 2011 OSA

## 1. Introduction

1. L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. **48**(20), 3948–3960 (2009). [CrossRef] [PubMed]

2. Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. **24**(18), 3049–3052 (1985). [CrossRef] [PubMed]

9. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A **14**(4), 918–930 (1997). [CrossRef]

*K*, where

*K*is a positive integer). Thus it imposes a strict requirement on the piezo transducers (PZT)’s accuracy and the environmental stability. However, it is often difficult to meet the requirement exactly in practice, especially for the measurement of large-aperture optics.

10. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. **84**(3-4), 118–124 (1991). [CrossRef]

16. H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A **24**(1), 25–33 (2007). [CrossRef] [PubMed]

*et al*proposed the statistical self-calibrating algorithms to extract the phase distribution on the assumption of a constant fringe contrast [17

17. A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. **8**(7), 738–745 (1997). [CrossRef]

18. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. **9**(5), 744–750 (1998). [CrossRef]

19. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. **9**(9), 1451–1455 (1998). [CrossRef]

*et al*[20

20. L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. **29**(2), 183–185 (2004). [CrossRef] [PubMed]

*et al*[21

21. 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]

22. X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. **27**(2), 024215 (2010). [CrossRef]

*et al*[23

23. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. **34**(8), 1210–1212 (2009). [CrossRef] [PubMed]

*et al*[24

24. 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]

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

*et al*[26

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. **39**(4), 585–591 (2000). [CrossRef] [PubMed]

27. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A **17**(11), 2061–2066 (2000). [CrossRef] [PubMed]

*et al*[28

28. 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]

26. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. **39**(4), 585–591 (2000). [CrossRef] [PubMed]

28. 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]

## 2. Principal component analysis

*N*frames of randomly phase-shifted interferograms are collected and each image is reshaped into one column with size of

*M*, where

*M*means the number of pixels in each image. The intensity of the

*m*th pixel in the

*n*th interferogram is expressed asHere,

*and*B m

*m*th pixel.

*n*th frame. Then these

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

*N*frames without background intensity and can be approximately expressed asThe size of

*N*images can be expressed asThe covariance matrix of

*D*is a diagonal matrix consisting of

29. 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]

*Y*consists of the principal components of

*. The*I − μ I ,Y = [ y 1 , y 2 , ⋅ ⋅ ⋅ , y n , ⋅ ⋅ ⋅ , y N ] T

*i*th component of

*Y*is

29. 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]

30. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. **36**(12), 2215–2217 (2011). [CrossRef] [PubMed]

## 3. Least squares method

13. 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]

*N*and it is wrapped in the range from

10. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. **84**(3-4), 118–124 (1991). [CrossRef]

13. 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]

## 4. Numerical simulation and discussion

*N*frames are generated according to Eq. (1) by setting the parameters as follows:

*, and*B = 120

*,*− 1 ≤ x ≤ 1

*D*is a parameter that denotes the number of fringes in the filed of interferogram.

*n*>1, where

*D*= 1.5 and

*E*= 1. One typical interferogram and the given measured phase are shown in Figs. 1(a) and 1(b). The three interferograms are analyzed by the PCA method and the “PCA+LSM” method, and the extracted phases of them are shown in Figs. 1(c) and 1(e) respectively. By comparing Figs. 1(c) and 1(e) with the given measured phase, the residual phase errors of the PCA method and the “PCA+LSM” method are obtained and shown in Figs. 1(d) and 1(f), respectively. The RMS values of Figs. 1(d) and 1(f) are 0.0382 rad and 0.0157 rad. It can be concluded from Figs. 1(b)–(d) that, the phase extracted by PCA method has an incorrect global sign while the “PCA+LSM” method can correct the global sign. In addition, Figs. 1(d) and 1(f) indicate that the “PCA+LSM” method can reduce the residual phase error and thus has a smaller phase error than the PCA method.

*N*), the number of fringes in interferogram (

*D*), and the amplitude of random phase shifts (

*E*). These factors are analyzed and discussed as follows. Here we define the phase-shift extraction error as the average of the absolute values of the difference between the extracted phase shifts and the given phase shifts over

*N*frames.

*D*= 1.5 and

*E*= 1, and let

*N*vary from 3 to 20. Then we use the “PCA+LSM” method to analyze these interferograms. The phase-shift extraction error and the phase error for different

*N*are obtained and shown in Figs. 2(a) and 2(b), respectively. Meanwhile, the PCA method is also used for comparison, and the phase error is given in Fig. 2(b). Figure 2(a) shows that the phase-shift extraction error oscillates between 0.026 rad and 0.031 rad as

*N*increases from 3 to 20. Figure 2(b) shows that the phase errors of the PCA method and “PCA+LSM” method decreases slowly with the increasing of

*N,*but the latter is less than one third of the former for a given

*N*. It can be also concluded from Figs. 2(a) and 2(b) that, for

*D*= 1.5,

*E*= 1 and

*N*≥3, the phase-shift extraction error and the RMS value of phase error of the proposed method are less than 0.031 rad and 0.012 rad, respectively.

*N*= 3 and

*E*= 1 and let

*D*vary from 1 to 20. The PCA method and the “PCA+LSM” method are applied and the results are shown in Fig. 3 . Since the approximations of Eqs. (5)–(6) are more close to the truth for a larger

*D*, the phase-shift extraction error and the phase error decrease with the increasing of

*D*, just as shown in Figs. 3 (a) and 3(b). Figure 3(b) indicates that the “PCA+LSM” method is more accurate than the PCA method, but the results of them will converge slowly as

*D*increases. Figures 3(a) and 3(b) also show that the phase-shift extraction error and the RMS value of phase error of the “PCA+LSM” method are less than 0.035 rad and 0.03 rad for

*D*>1.

*N*= 3 and

*D*= 2.5, and let

*E*vary from 0.1 to 3. The PCA method and the “PCA+LSM” method are used and the phase errors of them are shown in Fig. 4(a) . Figure 4(a) shows that the phase errors are kept under a small value for

*E*<1. For

*N*= 3 and

*E*<1, the relative phase shifts meet the condition that

**and**0 .57 < θ 2 − θ 1 < 2.57

*N*= 3 and

*E*>1.5, the relative phase shifts

*kπ,*where

*k*= 0, 1, 2….In this case, the three interferograms become more interdependent and finally it results in a big phase error, just as shown in Fig. 4 (a). This problem can be solved by increasing the number of frames used. Figure 4(b) shows the phase errors of the two methods at different

*E*for

*N*= 5 and

*D*= 2.5. For

*N*≥5, it is easy to ensure that there are two or more frames whose relative phase shifts are not close to 2

*kπ.*Thus there is no big phase error in Fig. 4(b) even if

*E*>1.5. It can be concluded from Figs. 4(a) and 4(b) that we should collect more frames to improve the performance of the proposed method if the phase shifts are completely random.

24. 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]

*N*= 3 and

*E*= 1 and let

*D*vary from 0.25 to 5. The phase-shift extraction errors of our method and Gao’s method are obtained and shown in Fig. 5 . It shows that the phase-shift extraction error of our method decreases with the increasing of

*D*and the error is less than 0.12 for

*D*and the error is less than 0.12 for

## 5. Experiment

13. 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]

## 6. Conclusion

## Acknowledgments

## References and links

1. | L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt. |

2. | Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt. |

3. | P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt. |

4. | P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt. |

5. | K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A |

6. | Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. |

7. | Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett. |

8. | 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 |

9. | K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A |

10. | K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. |

11. | G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt. |

12. | B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. |

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

14. | 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. |

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

16. | H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A |

17. | A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol. |

18. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol. |

19. | A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol. |

20. | L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett. |

21. | 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. |

22. | X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett. |

23. | X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett. |

24. | 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. |

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

26. | X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. |

27. | X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A |

28. | 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 |

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

30. | J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. |

31. | J. Xu, L. Sun, Y. Li, and Y. Li, “Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts,” Opt. Express |

32. | |

33. | |

34. | http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem. |

**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: July 19, 2011

Revised Manuscript: September 7, 2011

Manuscript Accepted: September 20, 2011

Published: October 3, 2011

**Citation**

Jiancheng Xu, Weimin Jin, Liqun Chai, and Qiao Xu, "Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method," Opt. Express **19**, 20483-20492 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20483

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### References

- L. L. Deck, “Suppressing phase errors from vibration in phase-shifting interferometry,” Appl. Opt.48(20), 3948–3960 (2009). [CrossRef] [PubMed]
- Y.-Y. Cheng and J. C. Wyant, “Phase shifter calibration in phase-shifting interferometry,” Appl. Opt.24(18), 3049–3052 (1985). [CrossRef] [PubMed]
- P. Hariharan, B. F. Oreb, and T. Eiju, “Digital phase-shifting interferometry: a simple error-compensating phase calculation algorithm,” Appl. Opt.26(13), 2504–2506 (1987). [CrossRef] [PubMed]
- P. L. Wizinowich, “Phase shifting interferometry in the presence of vibration: a new algorithm and system,” Appl. Opt.29(22), 3271–3279 (1990). [CrossRef] [PubMed]
- K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A9(10), 1740–1748 (1992). [CrossRef]
- Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt.32(19), 3598–3600 (1993). [CrossRef] [PubMed]
- Y. Ishii and R. Onodera, “Phase-extraction algorithm in laser-diode phase-shifting interferometry,” Opt. Lett.20(18), 1883–1885 (1995). [CrossRef] [PubMed]
- 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. A12(4), 761–768 (1995). [CrossRef]
- K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A14(4), 918–930 (1997). [CrossRef]
- K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun.84(3-4), 118–124 (1991). [CrossRef]
- G.-S. Han and S.-W. Kim, “Numerical correction of reference phases in phase-shifting interferometry by iterative least-squares fitting,” Appl. Opt.33(31), 7321–7325 (1994). [CrossRef] [PubMed]
- B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng.34(1), 183–188 (1995). [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, 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]
- 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]
- H. Guo, Y. Yu, and M. Chen, “Blind phase shift estimation in phase-shifting interferometry,” J. Opt. Soc. Am. A24(1), 25–33 (2007). [CrossRef] [PubMed]
- A. Dobroiu, P. C. Logofatu, D. Apostol, and V. Damian, “Statistical self-calibrating algorithm for three-sample phase-shift interferometry,” Meas. Sci. Technol.8(7), 738–745 (1997). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Self-calibrating algorithm for three-sample phase-shift interferometry by contrast leveling,” Meas. Sci. Technol.9(5), 744–750 (1998). [CrossRef]
- A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Statistical self-calibrating algorithm for phase-shift interferometry based on a smoothness assessment of the intensity offset map,” Meas. Sci. Technol.9(9), 1451–1455 (1998). [CrossRef]
- L. Z. Cai, Q. Liu, and X. L. Yang, “Generalized phase-shifting interferometry with arbitrary unknown phase steps for diffraction objects,” Opt. Lett.29(2), 183–185 (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]
- X. Xian-Feng, C. Lu-Zhong, W. Yu-Rong, and L. Dai-Lin, “Accurate phase shift extraction for generalized phase-shifting interferometry,” Chin. Phys. Lett.27(2), 024215 (2010). [CrossRef]
- X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, J. P. Guo, and Y. R. Wang, “Wavefront reconstruction and three-dimensional shape measurement by two-step dc-term-suppressed phase-shifted intensities,” Opt. Lett.34(8), 1210–1212 (2009). [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]
- K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt.40(17), 2886–2894 (2001). [CrossRef] [PubMed]
- X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt.39(4), 585–591 (2000). [CrossRef] [PubMed]
- X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shift calibration algorithm for phase-shifting interferometry,” J. Opt. Soc. Am. A17(11), 2061–2066 (2000). [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. Express18(23), 24368–24378 (2010). [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]
- J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett.36(12), 2215–2217 (2011). [CrossRef] [PubMed]
- J. Xu, L. Sun, Y. Li, and Y. Li, “Principal component analysis of multiple-beam Fizeau interferograms with random phase shifts,” Opt. Express19(15), 14464–14472 (2011). [CrossRef] [PubMed]
- http://en.wikipedia.org/wiki/Principal_component_analysis .
- http://en.wikipedia.org/wiki/Singular_value_decomposition .
- http://en.wikipedia.org/wiki/Karhunen%E2%80%93Lo%C3%A8ve_theorem .

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