## Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum |

Optics Express, Vol. 19, Issue 8, pp. 7807-7815 (2011)

http://dx.doi.org/10.1364/OE.19.007807

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### Abstract

A blind self-calibrating algorithm for phase-shifting interferometry is presented, with which the nonlinear interaction introduced by phase shift errors, between the reconstructed phases and the reconstructed amplitudes of the reference wave, is measured with cross-bispectrum. Minimizing an objective function based on this cross-bispectrum allows accurately estimating the true phase shifts from only three interferograms in the absence of any supplementary assumptions and knowledge about these interferograms.

© 2011 OSA

## 1. Introduction

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

19. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-10-3743. [CrossRef] [PubMed]

20. 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]

## 2. Principle

### 2.1 Reconstructions of phase and beam amplitudes

*k*th interferogram is often described with a function of the formwhere

*K*is the number of phase shifts, |

*r*| and |

*o*| denote the amplitudes of the reference and object beams, respectively,

*φ*is the phase to be measured, and

*δ*are phase shifts with

_{k}*δ*

_{0}= 0. All these parameters except

*δ*are functions of pixel coordinates, and

_{k}*δ*for each fringe pattern is commonly a constant (assuming no tilt errors exist).

_{k}*K*= 3, by defining

*c*

_{0}= |

*r*|

^{2}+ |

*o*|

^{2},

*c*

_{1}= 2|

*r*||

*o*|cos

*φ*, and

*c*

_{2}= −2|

*r*||

*o*|sin

*φ*, Eq. (1) is restated asWe presume for the moment that the phase shifts,

*δ*, are known. Then solving the system based on Eq. (2) results in andFurther we have and

_{k}*φ*, |

*r*|, and |

*o*|. See Fig. 1 for example, (a) simulates a phase map and (b) is one of three interferograms generated with phase shifts being

*δ*= {0, 1.7, 2.9} and the amplitude of the reference wave being Gaussian-shaped (with a 70% decrease in magnitude at the corners). The size of the interferogram is 256 × 256 pixels. Because the true phase shifts are unknown in measurement practice, we use their nominal values

_{k}*δ*= {0, π/2, π} to recover

_{k}*φ*and |

*r*|. The resulting errors are displayed in (c) and (d), respectively, where the maximum phase error is 0.1676 radians and the RMS (root-mean-square) phase error is 0.1098 radians. From (c) and (d) we can observe that the shapes of artifacts in

*φ*and |

*r*| are similar to each other and to that of the fringes in (b), but they may have different spatial frequencies. This phenomenon typically reveals the effect of a nonlinear interaction between

*φ*and |

*r*| in the presence of phase shift errors. In other words, the phase shift errors introduce special correlations between

*φ*and |

*r*|. Because |

*r*| is mainly determined by the light source and the optics in the reference arm, whereas

*φ*depends on the optical path difference, naturally there should be no correlations between their distributions. For this reason, the actual phase shifts can be estimated by minimizing these correlations, but a problem arisen is how these correlations can be measured.

### 2.2 Cross-bispectrum as a measure of correlations

20. 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]

*f*

_{1}(

*t*) = cos(

*ωt*) and its square

*f*

_{2}(

*t*) = cos

^{2}(

*ωt*). The frequency of the first one is

*ω*. The second signal can be restated as

*f*

_{2}(

*t*) = [1 + cos(2

*ωt*)]/2, so its frequencies are 0 and 2

*ω*. The spectra of

*f*

_{1}(

*t*) and

*f*

_{2}(

*t*) are not overlapped, thus their cross power spectrum equals 0. This fact means that the cross power spectrum cannot describe the nonlinear correlations between these two signals].

*φ*the third-order correlations between

*ψ*and |

*r*| are estimated by a cross-bispectrum [21

21. K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. **7**(3), 284–294 (1981). [CrossRef]

*E*{·} is the expectation operator and * denotes complex conjugate;

*ω*denotes the spatial frequency in rad/sample; and

*ψ*and |

*r*|, respectively, with

*ψ*and |

*r*| being calculated using Eqs. (3-9). The purpose of subtracting the averages

*ψ*and |

*r*| is to make them zero-mean normalized. For a pair of phase shifts (

*δ*

_{1},

*δ*

_{2}), we select

*N*cross-sections in the fringe patterns, calculate the corresponding

*ψ*and |

*r*| first and then

*Ψ*(

*ω*) and

*R*(

*ω*). Finally we take the following average for estimating the cross-bispectrum:which is a two-dimensional complex-valued function. Because the angular frequency for a digital signal has a range from –π to π, Eq. (11) is defined in the hexagonal field |

*ω*

_{1}|<π, |

*ω*

_{2}|<π and |

*ω*

_{1}+

*ω*

_{2}|<π.

*δ*

_{1},

*δ*

_{2}) = (1.7, 2.9), the magnitudes of

*δ*

_{1},

*δ*

_{2}) that minimize

*β*(

*δ*

_{1},

*δ*

_{2}). Figure 2(c) plots

*β*as a function of (

*δ*

_{1},

*δ*

_{2}). On the diagonal,

*δ*

_{1}=

*δ*

_{2}, hence the phase map cannot be recovered. The field of (

*δ*

_{1},

*δ*

_{2}) is split, by this diagonal, into two triangular segments, which are symmetrical to each other regarding the point (π, π). In the lower triangle, this function has a minimum when the phase shifts take the estimated phase shifts (

*δ*

_{1},

*δ*

_{2}) = (1.6992, 2.8981), with which the maximum and RMS errors in the reconstructed phase map become 0.0016 and 0.0012 radians, repectively. Using another minimum within the upper triangle can also recover the phases correctly but the signs are opposite.

### 2.3 Search procedure

*c*

_{0}

^{2}≥Σ(

*c*

_{1}

^{2}+

*c*

_{2}

^{2}) [i.e. Σ(|

*r*|

^{2}+ |

*o*|

^{2})≥Σ2|

*r*||

*o*|] can be used as a criterion to exclude the phase shifts with very large errors before calculating the corresponding

*ω*

_{1}=

*ω*

_{2}as shown in Fig. 2(b). Thus we need only to compute the half-plane of Fig. 2(b) below

*ω*

_{1}=

*ω*

_{2}. Third, noting that

*β*(

*δ*

_{1},

*δ*

_{2}) is symmetrical regarding the point (π, π), searching only the lower triangle when

*δ*

_{1}<

*δ*

_{2}(or the upper one when

*δ*

_{1}>

*δ*

_{2}) can decrease the computational time to a half. Finally, we employ a cascade search strategy, with which several search cycles with increasing resolutions are performed in the cascade way. For example, in the first cycle we seek the minimum over the whole search region by using a search step of 0.02π, and in the second cycle we search for the minimum over a very small region centered at the previous searching result with a resolution 0.01π (halving the previous search step). In this way, the resolution can achieve 10

^{−4}level after 7 cycles, and the computational time for one simulation is less than a minute.

## 3. Experiments

*δ*= {0, 2π/3, 4π/3} and the size of each interferogram being 256 × 256 pixels. Figure 3(b) illustrates the calculated

_{k}*β*(

*δ*

_{1},

*δ*

_{2}), from which we see that the search regions have been contracted. In the low triangle, the minimum is located at the position (

*δ*

_{1},

*δ*

_{2}) = (2.0044, 3.9617), which deviate from the nominal phase shifts.

*δ*= {0, 2π/3, 4π/3}, but the phase shifts estimated with the proposed algorithm are (

_{k}*δ*

_{1},

*δ*

_{2}) = (2.0593, 4.4251) which minimize the objective function

*β*as shown in Fig. 4(b). Reviewing Section 2.3, the search regions are contracted by excluding the phase shift pairs that may make the recalculated intensities negative. In this example, however, the fringe modulation is relatively small because the measured grating has a much lower reflectance than the reference plate, and the contracted search regions in Fig. 4(b), as a result, are much larger than that in Fig. 3(b).

## 4. Discussions

*δ*

_{1},

*δ*

_{2}) = (0.01π, 1.99π), the estimated results are (0.0314, 6.2518) and the errors are very small; whereas the method from Ref [20

20. 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]

*δ*

_{1},

*δ*

_{2}) = (1.7, 2.9), and the estimated phase shifts in the presence of noise are listed in Table 1 , where the noise SD 0.001 corresponds to 0.255 gray levels in a 8-bit image and so forth. From this table we see that large noise (e.g. noise SD≥0.01) inversely affects the accuracy of the proposed algorithm, because the noise added to the fringe patterns, after a sequence of nonlinear computational operations with Eqs. (3-9), will not be independent and unbiased. For severely noisy interferograms, we can filter them before implementing the proposed algorithm, since the bispectrum methods are unaffected by a linear transform like low-pass filtering.

## 5. Conclusion

## Acknowledgement

## References and links

1. | C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. |

2. | J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. |

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

4. | J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. |

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

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

7. | C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. |

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

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

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

11. | H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. |

12. | O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. |

13. | L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. |

14. | L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. |

15. | 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. |

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

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

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

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

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

21. | K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. |

**OCIS Codes**

(100.2650) Image processing : 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: February 7, 2011

Revised Manuscript: March 30, 2011

Manuscript Accepted: March 30, 2011

Published: April 7, 2011

**Citation**

Hongwei Guo, "Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum," Opt. Express **19**, 7807-7815 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7807

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### References

- C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]
- J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
- 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]
- J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, N. Streibl, and ., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]
- Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]
- K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shifting interferometry,” Opt. Commun. 84(3-4), 118–124 (1991). [CrossRef]
- C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992). [CrossRef]
- I.-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]
- X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]
- 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]
- H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007). [CrossRef]
- O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). [CrossRef] [PubMed]
- L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28(19), 1808–1810 (2003). [CrossRef] [PubMed]
- L. Z. Cai, Q. Liu, and X. L. Yang, “Simultaneous digital correction of amplitude and phase errors of retrieved wave-front in phase-shifting interferometry with arbitrary phase shift errors,” Opt. Commun. 233(1-3), 21–26 (2004). [CrossRef]
- 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]
- 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]
- 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]
- K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2894 (2001). [CrossRef]
- 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-10-3743 . [CrossRef] [PubMed]
- 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]
- K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981). [CrossRef]

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