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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 17234–17248
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Phase extraction from interferograms with unknown tilt phase shifts based on a regularized optical flow method

Fa Zeng, Qiaofeng Tan, Huarong Gu, and Guofan Jin  »View Author Affiliations


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

The temporal phase-shifting interferometry (PSI) has been generally accepted as the most accurate technique for wave-front reconstruction based on automated interferogram analysis [1

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]

]. PSI electronically records a series of interferograms while the reference phase of the interferometer is changed, so as to extract the phase information encoded in the variations in the intensity pattern of the recorded interferograms [2

2. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 14, pp. 547–666.

]. However, the accuracy of PSI is limited by the phase-shifting uncertainties resulting from the imperfect conditions in the real testing environments, such as the miscalibration, non-linear responses, aging of the phase shifter [3

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]

], and the mechanical vibration [1

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]

].

It should be pointed out that the extracted phase by the proposed method would have indetermination in the global sign if no prior knowledge about the phase shifts is available. However, such indetermination is not a particular problem of our method, because it is common to any asynchronous approach, if no information is given about the phase-shifts [8

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

].

The rest of the paper is organized as follows: the principle of the proposed method is provided in detail in section 2, simulation and experimental results are reported in section 3 and section 4, respectively. Finally, conclusions are drawn in Section 5.

2. Principle

In practice, if possible, it is always preferable to take more than three interferograms for the phase extraction. The reasons for it can be easily understood. On one hand, for pixels where one or both of the phase shifts are equal or close to integer numbers of 2π, Eq. (2) will be invalid and the corresponding phase extraction result will be untruthful, as the equivalent number of interferograms is reduced for those pixels; On the other hand, as a rule of thumb, the PSI methods’ resistance ability to noise can be improved as more interferograms are involved.

Figure 1
Fig. 1 The flow chart of the proposed method.
shows the flow chart of the proposed method in this paper.

Obviously, the accurate estimation of all the phase-shifting plane parameters {kxn,kyn,dn;n=2,3,...N}[see Eq. (1)] would be crucial to the performance of the phase extraction procedure. As we will see below, the proposed method would provide an accurate and fast solution for it.

Next, we will first briefly review the regularized optical flow method for the two-step interferometry, and then further introduce the procedures of the proposed method step by step. To be simple and intuitive, the descriptions in sections 2.1-2.4 are based on the case of two interferograms.

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

Supposing there are two interferogramsI1,I2, and their high-pass filtered background-suppressed versions are denoted as I˜1 and I˜2. Then we have:I˜1B(x,y)cos[φ(x,y)], I˜2=B(x,y)cos[φ(x,y)+δ(x,y)], where δ(x,y)represents the phase shift distribution between the I1 and I2, which is located at the phase-shifting plane: δ(x,y)=kxx+kyy+d.

The paper [10

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]

] has put forward a regularized optical flow method to extract the wrapped demodulation phase from two interferograms with an arbitrary unknown but constant phase shift inside the range(0,2π), with a single exception of δ=π, i.e. kx=ky=0 and dπ are assumed. First, the fringe direction is obtained with a regularized optical flow method in an iterative way
uk+1=u¯kIx[Ixu¯k+Iyv¯k+It]/(ρ2+Ix2+Iy2)vk+1=v¯kIy[Ixu¯k+Iyv¯k+It]/(ρ2+Ix2+Iy2)
(3)
η=arctan(ν/u)
(4)
where Ixand Iy represent the derivatives of I˜1with respect to x,y, while Itrepresents the difference between the two background-suppressed interferograms, i.e. It=I˜2I˜1; uk+1 and vk+1are the velocity components obtained in the iteration k+1, whileu¯kandv¯k correspond to the mean value of uand v in a defined neighborhood; ρis the regularizing parameter that weighs the smoothness of uand v;ηrepresents the fringe direction map. Then, the modulation phase can be extracted using the spiral phase transform
SPT{}=FT1{(ωx+iωyωx2+ωy2)FT{}}
(5)
φest=arctan(iexp(iη)SPT{I˜1}I˜1)
(6)
where SPT{}denotes the spiral phase transform; ωxand ωyare the coordinates in the spectral domain; φest is the extracted phase by the regularized optical flow method.

2.2 Computation of the wrapped phase shift distribution between two interferograms

Actually, the phase extraction method given in the paper [10

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]

] can provide us with more information than the authors revealed.

On the other hand, the phase shift distribution can also be analyzed as described above even if it is non-uniform across the whole field, since the regularized optical flow method is implemented in a local way. However, if the phase shift is equal to kπ(kZ)somewhere in the field, the computed wrapped phase shift distribution δest(x,y)would include some flip lines. In this case, it should be correctly unwrapped inside the approximate range(0,2π), which can be implemented as follows.

2.3 Unwrap the phase shift distribution

δ˜est(x,y)={δ¯est(x,y),BM2(x,y)=12πδ¯est(x,y),BM2(x,y)=0
(14)

2.4 Estimation of the tilt phase shift parameters

From the expression δ(x,y)=kxx+kyy+drelated to the phase-shifting plane, we can deduce that the data in any row (column) of the complex signal exp[jδ(x,y)]is single-tone, with frequency kxrad/pixel(kyrad/pixel). Thus, we can estimate the phase-shifting plane parameterskx,kyfrom exp[jδ˜est(x,y)] using an efficient 2-D frequency estimation method given in the paper [16

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]

]. If the estimated results are denoted as kx,estandky,est, then the estimated parameter d can be further obtained as
dest=angle{x,yexp[jδ˜est(x,y)]exp[j(kx,estx+ky,esty)]}
(15)
where angle{}denotes the operation of extracting the phase angle from a complex number. Then the estimated phase-shifting plane between the interferograms I1,I2can be obtained as

δ˜2(x,y)=kx,estx+ky,esty+dest
(16)

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

Supposing there are totallyNphase-shifted interferograms, then all the phase-shifting planesδi(x,y),2iN can be estimated in order with the procedures introduced in section 2.1-2.4. However, it can be noted that as to the specific phase-shifting plane, the selection of Eq. (12) or Eq. (13), will give rise to two different estimation results with the same phase-shifting distribution but with an opposite global sign. As a result, the estimated phase-shifting planes will have at most2N1possible arrangements {±δ˜2(x,y),±δ˜3(x,y),,±δ˜N(x,y)}, and the majority of them will give erroneous phase extraction results, i.e., only the two arrangement which are closest to the true phase-shifting arrangement {δ2(x,y),δ3(x,y),,δN(x,y)} or its opposite version {δ2(x,y),δ3(x,y),,δN(x,y)}, will achieve the accurate estimation of the modulation phase φ(x,y) or its opposite version φ(x,y). Here we put forward a simple way to find the interested arrangement. Considering the case of three phase-shifted interferograms Ii,Ij,Ik(1i,j,kN), where the estimated coarse phase distributions [see Eq. (6)], the intermediate binary maps [see Eqs. (12) or (13)], and the estimated phase-shifting planes from the two-step phase-shifted interferograms {Ii,Ij}, {Ii,Ik}are denoted as{φest,ij(x,y),φest,ik(x,y)}, {BM2ij,BM2ik}, and {δ˜ij(x,y),δ˜ik(x,y)}, respectively. Then we can define
φ˜est,ij(x,y)={φest,ij(x,y),BM2ij(x,y)=12πφest,ij(x,y),BM2ij(x,y)=0
(17)
The φ˜est,ik(x,y)is defined withφest,ik(x,y)and the mapBM2ikin the similar way. Subsequently, we further define
val1=|x,y{exp[jφ˜est,ij(x,y)]exp[jφ˜est,ik(x,y)]}|val2=|x,y{exp[jφ˜est,ij(x,y)]exp[jφ˜est,ik(x,y)]}|
(18)
if val1>val2, the arrangements {δ˜ij(x,y),δ˜ik(x,y)} or {δ˜ij(x,y),δ˜ik(x,y)}will be adopted, otherwise the arrangements {δ˜ij(x,y),δ˜ik(x,y)}or {δ˜ij(x,y),δ˜ik(x,y)}will be adopted. When there are more than three phase-shifted interferograms, the phase-shifting arrangement can be searched in the similar way.

2.6 Phase extraction with the least-squares method

3. Simulation results

As the accurate estimation of the phase-shifting plane parameters is crucial to the performance of the phase extraction procedure, a series of computation simulations have been carried out to verify the effectiveness of phase-shifting plane estimation by the proposed method. Some results will be given in the following paragraphs.

We also have evaluated the performance of our method by testing some other phase-shifting planes with different tilt-shift amounts and different orientations. The simulation parameters are all the same as in Fig. 2, except for the simulated phase-shifting plane. Specifically, four different phase-shifting planes are considered, whereδ3(x,y)=δ2(x,y)/2,δ4(x,y)=2δ2(x,y), δ5(x,y)=0.0011x+0.0243y+1.583, andδ6(x,y)=0.0164x+0.0179y+1.583, i.e., the corresponding tilt-shift amounts across the field are 0.70λ,2.80λ,1.40λand1.40λ, respectively. Actually, δ5(x,y) and δ6(x,y)are obtained by rotating δ2(x,y)anticlockwise in the x-y coordinate system with 0.25πand0.5πrad, respectively. Herein, the definition ofδ2(x,y)is the same as shown in Fig. 2(c). The residual errors of the estimation results in PV and RMS measures are shown in Table 1

Table 1. Residual Errors of the Estimated Phase-shifting Planes

table-icon
View This Table
. As can be seen from Table 1, all the residual PV errors and RMS errors are within0.016λand0.004λ, respectively.

The above simulations demonstrate that the proposed method can well estimated the tilt phase-shifting planes, and the estimation performance is robust to noise, the orientation and the amplitude of the phase-shifting plane. As explained before, the phase-shifting plane estimation result will face a global sign ambiguity problem. To reasonably evaluate the estimation performance of the proposed method, the simulation results given in Fig. 1 and Table 1 are all assigned with the correct signs.

4. Experimental results

For further verification of the feasibility of the proposed method, we have applied it to the experimental interferograms. Two set of experimental results will be provided below.

In the first experiment, the data of interferograms are obtained from the paper [10

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]

], which include totally nine real phase-shifted interferograms with unknown but constant phase shifts (the image size of them is481×641). In this case, we only need to estimate the parametersdi. In Fig. 3
Fig. 3 (a) The first interferogram used in this experiment; (b) The wrapped reference phase map by the AIA method using all the interferograms; (c) The wrapped phase extraction result by the AIA method with eight interferograms (i.e., one of the interferograms is excluded); (d) The wrapped phase extraction result by the proposed method using all the interferograms; (e) The wrapped phase difference between (b) and (c), after removing the bias; (f) The wrapped phase difference between (b) and (d), after removing the bias. The data shown in (b)-(f) are in radians.
we show the reference phase obtained by the AIA method [4

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]

] using all the nine interferograms (b), as well as the phase extraction results by the AIA method but with only eight interferograms (c) and the proposed method (d). As the extracted phase by the AIA method would include a trivial constant bias, the residual errors of the latter two results after removing the bias would be0.05λin PV measure, 0.006λin RMS measure [see Fig. 3(e)], and0.024λin PV measure, 0.007λin 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

We have proposed a novel method for extracting phase distribution from interferograms with unknown tilt phase shifts. The proposed method can estimate the unknown tilt phase shift between two temporal phase-shifted interferograms, by extending the regularized optical flow method provided in the paper [10

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]

] with the spatial image processing and frequency estimation technology. With all the estimated tilt phase shifts, we can further obtain the phase extraction result with the least-squares method. Both simulation and experimental results have proved the feasibility of the proposed method. The proposed method is expected to be used in a testing environment with low frequency and high amplitude vibration, where costly and accurate phase-shifting devices are not longer required for steady-state measures.

Acknowledgments

We are grateful to Professor Javier Vargas at Centro Nacional de Biotecnología-CSIC, Spain, for his kind help about the phase-shifting methods and providing us with partial experimental data of interferograms used in this paper. The research was partially supported by the National Basic Research Program of China under grant No. 2011CB706701. Thanks also go to the anonymous reviewers for their valuable comments and suggestions.

References and links

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]

2.

D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 14, pp. 547–666.

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]

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]

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. 33(8), 776–778 (2008). [CrossRef] [PubMed]

6.

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]

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. 36(6), 996–998 (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]

9.

H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express 19(8), 7807–7815 (2011). [CrossRef] [PubMed]

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]

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]

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]

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]

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]

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, Digital Image Processing Using MATLAB (Prentice Hall, 2004).

18.

L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, Upper Saddle River, New Jersey, USA, 2001).

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

  1. 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]
  2. D. Malacara, Optical Shop Testing, 3rd ed. (Wiley, 2007), Chap. 14, pp. 547–666.
  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]
  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]
  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.33(8), 776–778 (2008). [CrossRef] [PubMed]
  6. 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]
  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.36(6), 996–998 (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]
  9. H. Guo, “Blind self-calibrating algorithm for phase-shifting interferometry by use of cross-bispectrum,” Opt. Express19(8), 7807–7815 (2011). [CrossRef] [PubMed]
  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]
  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]
  13. 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]
  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]
  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]
  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, Digital Image Processing Using MATLAB (Prentice Hall, 2004).
  18. L. G. Shapiro and G. C. Stockman, Computer Vision (Prentice Hall, Upper Saddle River, New Jersey, USA, 2001).

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