Abstract

A phase unwrapping algorithm for interferometric fringes based on the unscented Kalman filter (UKF) technique is proposed. The algorithm can bring about accurate phase unwrapping and good noise suppression simultaneously by incorporating the true phase and its derivative in the state vector estimation through the UKF process. Simulations indicate that the proposed algorithm has better accuracy than some widely employed phase unwrapping approaches in the same noise condition. Also, the time consumption of the algorithm is reasonably acceptable. Applications of the algorithm in our different optical interferometer systems are provided to demonstrate its practicability with good performance. We hope this algorithm can be a practical approach that can help to reduce the systematic errors significantly induced by phase unwrapping process for interferometric measurements such as wavefront distortion testing, surface figure testing of optics, etc.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
  2. D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
    [Crossref] [PubMed]
  3. D. Malacara, Optical Shop Testing, (Wiley, 2007).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  8. H. Zhong, J. Tang, and S. Zhang, “Phase quality map based on local multi-unwrapped results for two-dimensional phase unwrapping,” Appl. Opt. 54(4), 739–745 (2015).
    [Crossref] [PubMed]
  9. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30(25), 3627–3632 (1991).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  13. C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  24. I. Gurov, E. Ermolaeva, and A. Zakharov, “Analysis of low-coherence interference fringes by the Kalman filtering method,” J. Opt. Soc. Am. A 21(2), 242–251 (2004).
    [Crossref] [PubMed]
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    [Crossref]
  26. Z. Cheng, D. Liu, J. Luo, Y. Yang, Y. Zhou, Y. Zhang, L. Duan, L. Su, L. Yang, Y. Shen, K. Wang, and J. Bai, “Field-widened Michelson interferometer for spectral discrimination in high-spectral-resolution lidar: theoretical framework,” Opt. Express 23(9), 12117–12134 (2015).
    [Crossref] [PubMed]
  27. X. Chen, Y. Yang, C. Wang, D. Liu, J. Bai, and Y. Shen, “Aberration calibration in high-NA spherical surfaces measurement on point diffraction interferometry,” Appl. Opt. 54(13), 3877–3885 (2015).
    [Crossref]
  28. D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces,” Appl. Opt. 50(16), 2342–2348 (2011).
    [Crossref] [PubMed]
  29. T. Ling, D. Liu, Y. Yang, L. Sun, C. Tian, and Y. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. 38(14), 2493–2495 (2013).
    [Crossref] [PubMed]

2015 (4)

2014 (3)

2013 (1)

2012 (2)

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[Crossref] [PubMed]

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).
[Crossref]

2011 (3)

2010 (1)

2009 (1)

2008 (1)

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

2007 (1)

2004 (3)

I. Gurov, E. Ermolaeva, and A. Zakharov, “Analysis of low-coherence interference fringes by the Kalman filtering method,” J. Opt. Soc. Am. A 21(2), 242–251 (2004).
[Crossref] [PubMed]

I. P. Gurov and A. S. Zakharov, “Analysis of characteristics of interference fringes by nonlinear Kalman filtering,” Opt. Spectrosc. 96(2), 175–181 (2004).
[Crossref]

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92(3), 401–422 (2004).
[Crossref]

1999 (1)

1994 (1)

1991 (1)

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

1982 (1)

1974 (1)

Bai, J.

Bone, D. J.

Brangaccio, D. J.

Bruning, J. H.

Chen, C.

Chen, X.

Cheng, Z.

Cuevas, F. J.

Duan, L.

Ermolaeva, E.

Estrada, J. C.

Gallagher, J. E.

Ghiglia, D. C.

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Gurov, I.

Gurov, I. P.

I. P. Gurov and A. S. Zakharov, “Analysis of characteristics of interference fringes by nonlinear Kalman filtering,” Opt. Spectrosc. 96(2), 175–181 (2004).
[Crossref]

Herriott, D. R.

Ina, H.

Julier, S. J.

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92(3), 401–422 (2004).
[Crossref]

Knedlik, S.

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

Kobayashi, S.

Kulkarni, R.

Li, Y.

Ling, T.

Liu, D.

Liu, Z.

Loffeld, O.

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

Luo, J.

Luo, Y.

Madyastha, V.

Malacara, D.

Marroquin, J. L.

Navarro, M. A.

Nies, H.

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

Patil, A.

Quiroga, J. A.

Rastogi, P.

Reindl, L. M.

Rodriguez-Vera, R.

Romero, L. A.

Rosenfeld, D. P.

Servin, M.

Shen, Y.

Shi, T.

Su, L.

Sun, L.

Takeda, M.

Tang, J.

Tao, L.

Tian, C.

Uhlmann, J. K.

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92(3), 401–422 (2004).
[Crossref]

Vargas, J.

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).
[Crossref]

M. A. Navarro, J. C. Estrada, M. Servin, J. A. Quiroga, and J. Vargas, “Fast two-dimensional simultaneous phase unwrapping and low-pass filtering,” Opt. Express 20(3), 2556–2561 (2012).
[Crossref] [PubMed]

Wang, C.

Wang, D.

Wang, K.

Wang, L.

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

White, A. D.

Wu, H.

Xianming, X.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[Crossref]

Xie, X.

Yang, L.

Yang, Y.

X. Chen, Y. Yang, C. Wang, D. Liu, J. Bai, and Y. Shen, “Aberration calibration in high-NA spherical surfaces measurement on point diffraction interferometry,” Appl. Opt. 54(13), 3877–3885 (2015).
[Crossref]

Z. Cheng, D. Liu, J. Luo, Y. Yang, Y. Zhou, Y. Zhang, L. Duan, L. Su, L. Yang, Y. Shen, K. Wang, and J. Bai, “Field-widened Michelson interferometer for spectral discrimination in high-spectral-resolution lidar: theoretical framework,” Opt. Express 23(9), 12117–12134 (2015).
[Crossref] [PubMed]

L. Zhang, C. Tian, D. Liu, T. Shi, Y. Yang, H. Wu, and Y. Shen, “Non-null annular subaperture stitching interferometry for steep aspheric measurement,” Appl. Opt. 53(25), 5755–5762 (2014).
[Crossref] [PubMed]

T. Ling, D. Liu, Y. Yang, L. Sun, C. Tian, and Y. Shen, “Off-axis cyclic radial shearing interferometer for measurement of centrally blocked transient wavefront,” Opt. Lett. 38(14), 2493–2495 (2013).
[Crossref] [PubMed]

D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces,” Appl. Opt. 50(16), 2342–2348 (2011).
[Crossref] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
[Crossref] [PubMed]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
[Crossref] [PubMed]

Yiming, P.

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[Crossref]

Yu, W.

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

Zakharov, A.

Zakharov, A. S.

I. P. Gurov and A. S. Zakharov, “Analysis of characteristics of interference fringes by nonlinear Kalman filtering,” Opt. Spectrosc. 96(2), 175–181 (2004).
[Crossref]

Zander, T. E.

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Zhang, L.

Zhang, S.

Zhang, W.

Zhang, Y.

Zhong, H.

Zhou, Y.

Zhuo, Y.

Appl. Opt. (10)

L. Zhang, C. Tian, D. Liu, T. Shi, Y. Yang, H. Wu, and Y. Shen, “Non-null annular subaperture stitching interferometry for steep aspheric measurement,” Appl. Opt. 53(25), 5755–5762 (2014).
[Crossref] [PubMed]

D. Liu, Y. Yang, L. Wang, and Y. Zhuo, “Real time diagnosis of transient pulse laser with high repetition by radial shearing interferometer,” Appl. Opt. 46(34), 8305–8314 (2007).
[Crossref] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital Wavefront Measuring Interferometer for Testing Optical Surfaces and Lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[Crossref] [PubMed]

H. Zhong, J. Tang, and S. Zhang, “Phase quality map based on local multi-unwrapped results for two-dimensional phase unwrapping,” Appl. Opt. 54(4), 739–745 (2015).
[Crossref] [PubMed]

D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30(25), 3627–3632 (1991).
[Crossref] [PubMed]

M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroquin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38(10), 1934–1941 (1999).
[Crossref] [PubMed]

C. Tian, Y. Yang, D. Liu, Y. Luo, and Y. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
[Crossref] [PubMed]

X. Xie and Y. Li, “Enhanced phase unwrapping algorithm based on unscented Kalman filter, enhanced phase gradient estimator, and path-following strategy,” Appl. Opt. 53(18), 4049–4060 (2014).
[Crossref] [PubMed]

X. Chen, Y. Yang, C. Wang, D. Liu, J. Bai, and Y. Shen, “Aberration calibration in high-NA spherical surfaces measurement on point diffraction interferometry,” Appl. Opt. 54(13), 3877–3885 (2015).
[Crossref]

D. Wang, Y. Yang, C. Chen, and Y. Zhuo, “Point diffraction interferometer with adjustable fringe contrast for testing spherical surfaces,” Appl. Opt. 50(16), 2342–2348 (2011).
[Crossref] [PubMed]

IEEE Trans. Geosci. Rem. Sens. (1)

O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry - A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Rem. Sens. 46(1), 47–58 (2008).
[Crossref]

IET Radar Sonar Navig. (1)

X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5(3), 296–304 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lasers Eng. (1)

J. C. Estrada, M. Servin, and J. Vargas, “2D simultaneous phase unwrapping and filtering: A review and comparison,” Opt. Lasers Eng. 50(8), 1026–1029 (2012).
[Crossref]

Opt. Lett. (4)

Opt. Spectrosc. (1)

I. P. Gurov and A. S. Zakharov, “Analysis of characteristics of interference fringes by nonlinear Kalman filtering,” Opt. Spectrosc. 96(2), 175–181 (2004).
[Crossref]

Proc. IEEE (1)

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92(3), 401–422 (2004).
[Crossref]

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: Two-dimensional phase unwrapping,” Radio Sci. 23(4), 713–720 (1988).
[Crossref]

Other (3)

T. J. Flynn, “Consistent 2-D phase unwrapping guided by a quality map,” in Proceedings of the 1996 International Geoscience and Remote Sensing Symposium. Part 3 (of 4), May 28,1996- May 31, 1996(IEEE, Lincoln, NE, USA, 1996), pp. 2057–2059.
[Crossref]

D. C. Ghiglia. and M. D. Pritt, Two-Dimensional Phase Unwrapping:Theory, Algorithm, and Software,(Wiley, 1990).

D. Malacara, Optical Shop Testing, (Wiley, 2007).

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Figures (8)

Fig. 1
Fig. 1 Two recommended phase unwrapping strategies for the proposed algorithm: (a) column-by-column unwrapping and (b) the region growing unwrapping.
Fig. 2
Fig. 2 Simulation validation of the proposed phase unwrapping algorithm: (a) the wrapped phase with the SNR of 15dB, (b) the unwrapped phase by the proposed algorithm, and (c) the phase unwrapping error. The unit of the phase is radian here.
Fig. 3
Fig. 3 Simulation validation of the proposed phase unwrapping algorithm: (a) the wrapped phase with the SNR of 10dB, (b) the unwrapped phase by the proposed algorithm, and (c) the phase unwrapping error. The unit of the phase is radian here.
Fig. 4
Fig. 4 Performance comparisons between several typical phase unwrapping algorithms. (a) is the simulated original phase map, (b) is the noise-corrupted wrapped phase map with SNR of 5dB, (c)-(g) are the histograms of residual unwrapping errors by the DCT-LS, the QG-PF, the RPU, the RPT, and the proposed algorithm, respectively. The unit of the phase is radian here.
Fig. 5
Fig. 5 The application of the proposed algorithm in the field-widened Michelson interferometer system. (a) one of the original interferogram for the 4-step phase-shifting demodulation, (b) the wrapped phase map after the 4-step phase-shifting demodulation, and (c) the unwrapped phase map through the proposed method. The unit of the phase is radian here.
Fig. 6
Fig. 6 The application of the proposed algorithm in the phase-shift point diffraction interferometer system. (a) one of the original interferogram for the phase-shifting demodulation, (b) the wrapped phase map after demodulation, and (c) the unwrapped phase map through the proposed method. The unit of the phase is radian here.
Fig. 7
Fig. 7 The application of the proposed algorithm in the off-axis cyclic radial shearing interferometer system. (a) the original spatial carrier interferogram, (b) the wrapped phase map after the Fourier transform demodulation, and (c) the unwrapped phase map through the proposed method. The unit of the phase is radian here.
Fig. 8
Fig. 8 The application of the proposed algorithm for unwrapping a complicated phase map generated in the non-null annular sub-aperture stitching interferometry. (a) one of the original interferogram for the phase-shifting demodulation, (b) the wrapped phase map after demodulation, and (c) the unwrapped phase map through the proposed method. The unit of the phase is radian here.

Tables (1)

Tables Icon

Table 1 Details of the phase unwrapping accuracy and time consumption through different algorithms.

Equations (16)

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{ C( x,y )=cos( ϕ 2π ) S( x,y )=sin( ϕ 2π ) ,
ϕ( k+1 )=ϕ( k )+ ϕ ( k )Δk.
x k+1 =f( x k , v k )=F x k + v k ,
F=[ 1 1 0 1 ],
y k =h( x k , n k )=H( x k )+ n k .
H( x )=[ C( k ) S( k ) ]=[ cos( x 1 ) sin( x 1 ) ],
x k a = [ x k T v k T n k T ] T .
P k a =[ P x 0 0 0 P v 0 0 0 P n ].
X i,k1 a ={ x k1 a ,i=0, x k1 a + ( ( N+λ ) P k1 a ) i ,i=1,...,N, x k1 a ( ( N+λ ) P k1 a ) i ,i=N+1,...,2N.
X i,k1 a = [ X i,k1 x X i,k1 v X i,k1 n ] T ,
W 0 (m) =λ/ ( N+λ ) , W 0 (c) =λ/ ( N+λ ) +( 1 α 2 +β ), W i (c) = W i (m) =1/ {2( N+λ )},i=1,...2N.
X i,k|k1 x =f[ X i,k1 x , X i,k1 v ],i=0,1,...,2N. x k = i=0 2N W i (m) X i,k|k1 x , P x k = i=0 2N W i (c) [ X i,k|k1 x x k ] [ X i,k|k1 x x k ] T ,
y i,k|k1 =h[ X i,k|k1 x , X i,k1 n ],i=0,1,...,2N, y k = i=0 2N W i (m) y i,k|k1 , P y k = i=0 2N W i (c) [ y i,k|k1 y k ] [ y i,k|k1 y k ] T .
P x k y k = i=0 2N W i (c) [ X i,k|k1 x x k ] [ y i,k|k1 y k ] T , K k = P x k y k P y k 1 .
x k = x k + K k ( y k y k ).
P x k = P x k K k P y k K k T .

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