OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 8 — Apr. 11, 2011
  • pp: 7807–7815
« Show journal navigation

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

Hongwei Guo  »View Author Affiliations


Optics Express, Vol. 19, Issue 8, pp. 7807-7815 (2011)
http://dx.doi.org/10.1364/OE.19.007807


View Full Text Article

Acrobat PDF (1170 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

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

In phase-shifting interferometry, miscalibration of phase shifter is a major factor decreasing measurement accuracy, and a number of algorithms have been developed for solving this problem. For example, the least-squares methods allow minimizing the measurement error induced by random phase shift errors [1

1. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]

,2

2. J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

], and averaging techniques and some optimized techniques can also compress the influence of phase shift errors [3

3. 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]

5

5. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]

]. A kind of more effective methods is to retrieve phases and phase shifts simultaneously, so that the errors caused by miscalibrations of phase shifters can be eliminated [6

6. 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]

11

11. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007). [CrossRef]

]. With these algorithms, however, capturing at least five interferogarms has been demonstrated to be the sufficient and necessary condition under which the phases and phase shifts are uniquely determined [11

11. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007). [CrossRef]

]. In recent years, some attempts have been made in order to break through this limitation, and several algorithms using three or four interferogarms have been proposed. These algorithms are useful in the situation that only fewer than five fringe patterns are available, for example in spatial phase-shifting interferometry [12

12. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). [CrossRef] [PubMed]

] or when measuring a moveable profile so that the image capturing time is very tight. Among these algorithms, the algorithm of Cai et al. [13

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. 28(19), 1808–1810 (2003). [CrossRef] [PubMed]

] exploits the statistics of fringes and is effective when the amplitudes of beams have been known prior to measurements. Also proposed by Cai et al. [14

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. 233(1-3), 21–26 (2004). [CrossRef]

], the technique assumes the amplitude of the reference wave to be a constant over fringe patterns. The methods proposed by Xu et al. [15

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. 31(13), 1966–1968 (2006). [CrossRef] [PubMed]

] and by Gao et al. [16

16. 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]

] are also based on the same assumption that the amplitudes of the reference waves are uniform. The least-squares iterative algorithm of Wang and Han [17

17. 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]

] can give satisfactory results when background intensities and modulations are evenly distributed. All these algorithms allow estimating phase shifts from only three interferograms, but the supplementary constraints in them, such as those concerning uniformities of the beam amplitudes or background intensities, imply that the number of unknowns decreases. These algorithms share a drawback that the supplementary restraints are usually very strong and not always satisfied in measurement practice, typically when the beam is not perfectly shaped or when directly a Gaussian beam is used.

Without any supplementary constraints, it is not easy to determine the unknowns such as phases and phase shifts when only three fringe patterns are captured, because the aforementioned sufficient and necessary condition is valid. In other words, the number of independent equations in this case is smaller than that of unknowns, thus making the equation system under-determined. A practically effective solution for this problem was proposed by Goldberg and Bokor [18

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

]. It uses Fourier transforms to determine the phase shift between two interferograms with a carrier frequency. In the technique of Soloviev and Vdovin [19

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]

], properly tilting the reference mirror and using Hough transforms of the image differences can estimate the phase shifts between consecutive frames. The algorithm in 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]

]. permits blindly estimating phase shifts by minimizing the cross power spectrum between the background and fringes. However, it is not stable enough when the problem is poorly conditioned. The reason is that the crosstalk introduced by phase shift errors between the calculated background and fringes is not linear but complicatedly nonlinear; and it is impossible to fully describe such nonlinear interactions by simply using a second order correlation (e.g. cross power spectrum). This reason makes the reliability of this algorithm significantly decreased.

From the aforementioned facts, we know that deriving a self-calibrating algorithm depending on only three interferograms in the absence of any preconditions or knowledge about interferograms still remains substantially challenging. To solve this problem, this paper presents, to the best of our knowledge, a novel blind self-calibrating algorithm, that minimizes the high order correlations between phases and amplitudes of the reference wave.

2. Principle

2.1 Reconstructions of phase and beam amplitudes

In phase-shifting interferometry, the intensities of the kth interferogram is often described with a function of the form
Ik=|r|2+|o|2+2|r||o|cos(ϕ+δk),k=0,1,,K1
(1)
where 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 δk are phase shifts with δ 0 = 0. All these parameters except δk are functions of pixel coordinates, and δk for each fringe pattern is commonly a constant (assuming no tilt errors exist).

When K = 3, by defining c 0 = |r|2 + |o|2, c 1 = 2|r||o|cosφ, and c 2 = −2|r||o|sinφ, Eq. (1) is restated as
Ik=c0+c1cosδk+c2sinδk,k=0, 1, 2.
(2)
We presume for the moment that the phase shifts, δk, are known. Then solving the system based on Eq. (2) results in
c0=I0sin(δ2δ1)I1sinδ2+I2sinδ1sin(δ2δ1)sinδ2+sinδ1,
(3)
c1=I0(sinδ1sinδ2)+I1sinδ2I2sinδ1sin(δ2δ1)sinδ2+sinδ1,
(4)
and
c2=I0(cosδ2cosδ1)+I1(1cosδ2)I2(1cosδ1)sin(δ2δ1)sinδ2+sinδ1.
(5)
Further we have
ϕ=arctan(c2c1),
(6)
|r|=c0+c02c12c222,
(7)
and

|o|=c0c02c12c222.
(8)

Through Eqs. (3-8), the errors in phase shifts will introduce distortions in the reconstructed φ, |r|, and |o|. See Fig. 1
Fig. 1 Numerical simulation results. (a) Phase map. (b) One of three interferograms. (c) The errors in the reconstructed amplitudes of the reference wave and (d) the errors in the reconstructed phase map by using nominal phase shifts.
for example, (a) simulates a phase map and (b) is one of three interferograms generated with phase shifts being δk = {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 δk = {0, π/2, π} to recover 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

As mentioned in Section 1, the algorithm 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]

] employs a cross power spectrum to estimate the correlations between the background intensities and fringes. This algorithm is not stable enough, because the cross power spectrum can only measure the second-order correlations between two signals and is blind to higher order ones introduced by nonlinear interactions [For example, there are two signals, i.e. f 1(t) = cos(ωt) and its square f 2(t) = cos2(ωt). The frequency of the first one is ω. The second signal can be restated as f 2(t) = [1 + cos(2ωt)]/2, so its frequencies are 0 and 2ω. 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].

We overcome the above problem by using high-order correlations. Considering the cis function of φ
ψ=exp(iϕ)=cosϕ+isinϕ,
(9)
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]

]
B(ω1,ω2)=E{Ψ(ω1)Ψ(ω2)R(ω1+ω2)},
(10)
where E{·} is the expectation operator and * denotes complex conjugate; ω denotes the spatial frequency in rad/sample; and Ψ(ω)=FT(ψψ¯) and R(ω)=FT(|r||r|¯) are one-dimensional Fourier transforms of ψ and |r|, respectively, with ψ and |r| being calculated using Eqs. (3-9). The purpose of subtracting the averages ψ¯ and |r|¯ from 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:
B^(δ1,δ2)(ω1,ω2)=n=1NΨn(ω1)Ψ(ω2)nRn(ω1+ω2),
(11)
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.

When (δ 1, δ 2) = (1.7, 2.9), the magnitudes of B^ are illustrated in Fig. 2 (b). Although B^ is a measure for the third-order correlations, it is not convenient to compare. To solve this problem, we reasonably define a single-valued objective function of the form
β(δ1,δ2)=u=ππv=ππ|B^(δ1,δ2)(ω1,ω2)|.
(12)
Our task is to find the phase shifts (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

3. Experiments

The second row of Fig. 3 illustrates the reconstructed results by using the nominal phase shifts, with the left panel being the amplitude distribution of the reference wave and the right one being the phase map (the tilt has been removed). From both the reference wave amplitudes and the phases, we see the ripple-like artifacts typically induced by phase shift errors. In the reconstructed phase map, the RMS and PV (peak-to-valley) phase values are 0.1435 and 2.0150 radians, respectively. The third row is similar to the second row but is obtained by using the phase shifts estimated with the proposed algorithm. The artifacts appearing in the second row have been significantly compressed in the third row with the proposed algorithm. Correspondingly, the RMS and PV values of the phases are reduced to 0.1315 and 1.9003 radians, respectively.

Similarly to Fig. 3, the second and third rows of Fig. 4 illustrate the reconstructed results by using the nominal phase shifts and the estimated ones, respectively. In them, the left panels are the amplitude distributions of the reference wave and the right ones are the recovered phase maps (the tilt has been removed). The artifacts in the second row induced by the phase shift errors have also been significantly compressed in the third row.

4. Discussions

There are issues, for example the stability, worth discussion regarding this algorithm. When true phase shifts being close to multiples of 2π, very small phase shift errors will introduce large errors in the calculated phases, and the problem becomes poorly conditioned. The performance of this algorithm in this case have been evaluated with a numerical simulation, in which true phase shifts (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]

] fails in estimating phase shifts because of the reason aforementioned. In fact, the proposed algorithm is effective even if only a fraction of a fringe is present when measuring a flat with a small curvature, but the accuracy may decrease. Tilting the reference mirror can introduce more fringes into the interferograms thus guaranteeing the accuracy.

The bispectrum methods are generally known to be insensitive to independent random noise. For the interferograms simulated in Section 2, the true phase shifts are (δ 1, δ 2) = (1.7, 2.9), and the estimated phase shifts in the presence of noise are listed in Table 1

Table 1. Estimated phase shifts in the presence of noise

table-icon
View This Table
, 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

In conclusion, we have proposed a blind self-calibrating algorithm, which estimates phase shifts by minimizing the high-order correlations, between the reconstructed phases and the reconstructed amplitudes of the reference wave, introduced by phase shift errors. This algorithm has several advantages over others. First, it allows estimating the random phase shifts from only three fringe patterns. Second, supplementary assumptions (like uniform amplitude of beam) and knowledge about interferograms are not required. Third, it offers a high accuracy and a high stability, as well as a satisfactory computational efficiency.

Acknowledgement

This work was supported by the Mechatronics Engineering Innovation Group Project from Shanghai Education Commission, and the author also acknowledges the support of the China Scholarship Council.

References and links

1.

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]

2.

J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

3.

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]

4.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl., “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]

5.

Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]

6.

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]

7.

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]

8.

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]

9.

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

10.

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]

11.

H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007). [CrossRef]

12.

O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). [CrossRef] [PubMed]

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. 28(19), 1808–1810 (2003). [CrossRef] [PubMed]

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. 233(1-3), 21–26 (2004). [CrossRef]

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. 31(13), 1966–1968 (2006). [CrossRef] [PubMed]

16.

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]

17.

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]

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]

21.

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

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


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982). [CrossRef] [PubMed]
  2. J. E. Greivenkamp, “Generlized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).
  3. 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]
  4. 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]
  5. Y. Surrel, “Phase stepping: a new self-calibrating algorithm,” Appl. Opt. 32(19), 3598–3600 (1993). [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. 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]
  9. X. Chen, M. Gramaglia, and J. A. Yeazell, “Phase-shifting interferometry with uncalibrated phase shifts,” Appl. Opt. 39(4), 585–591 (2000). [CrossRef]
  10. 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]
  11. H. Guo, Z. Zhao, and M. Chen, “Efficient iterative algorithm for phase-shifting interferometry,” Opt. Lasers Eng. 45(2), 281–292 (2007). [CrossRef]
  12. O. Y. Kwon, “Multichannel phase-shifted interferometer,” Opt. Lett. 9(2), 59–61 (1984). [CrossRef] [PubMed]
  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. 28(19), 1808–1810 (2003). [CrossRef] [PubMed]
  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. 233(1-3), 21–26 (2004). [CrossRef]
  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. 31(13), 1966–1968 (2006). [CrossRef] [PubMed]
  16. 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]
  17. 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]
  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]
  21. K. S. Lii and K. N. Helland, “Cross-bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7(3), 284–294 (1981). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 
Fig. 4
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited