Abstract

This paper presents a new phase unwrapping algorithm for wrapped phase fringes through combining a cubature information particle filter with an efficient local phase gradient estimator and an efficient quality-guided strategy based on heap-sort. The cubature information particle filter that not only is independent from noise statistics but also is not constrained by the nonlinearity of the model constructed is applied to retrieve unambiguous phase from modulus 2π wrapped fringe patterns through constructing a recursive cubature information particle filtering phase unwrapping procedure to perform simultaneously phase unwrapping and noise filtering for the first time to our knowledge, which can be expected to obtain more robust solutions from wrapped phase fringes. Phase gradient estimate is one of the key steps in almost all phase unwrapping algorithms and is directly related to the precision and the efficiency of phase unwrapping procedure. Accordingly, an efficient local phase gradient estimator that is more efficient than ones published previously is deduced to obtain phase gradient information required by the proposed algorithm, which can drastically decrease time consumption of unwrapping procedure and drastically improve the efficiency of the algorithm. The efficient quality-guided strategy based on heap-sort guarantees that the proposed algorithm efficiently unwraps wrapped pixels along the path from the high-reliance regions to the low-reliance regions of wrapped phase images. In addition, the accelerated version of the proposed algorithm is further developed through combing with reversible modulo wavelet operators to solve phase unwrapping problem of wrapped phase images in wavelet transform domain, which can reduce the amount of wrapped pixels that need to be unwrapped, and can further decrease time consumption of unwrapping procedure performing on wrapped phase images. This algorithm and its accelerated version under the frame of wavelet transform are demonstrated with various types of wrapped phase images, showing acceptable solutions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Iterated unscented Kalman filter for phase unwrapping of interferometric fringes

Xianming Xie
Opt. Express 24(17) 18872-18897 (2016)

Unscented information filtering phase unwrapping algorithm for interferometric fringe patterns

Xianming Xie and Gaoxing Dai
Appl. Opt. 56(34) 9423-9434 (2017)

References

  • View by:
  • |
  • |
  • |

  1. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).
  2. S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18(2), 560–565 (2010).
    [Crossref] [PubMed]
  3. B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt. 50(19), 3205–3220 (2011).
    [Crossref] [PubMed]
  4. M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
    [Crossref] [PubMed]
  5. H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
    [Crossref]
  6. 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]
  7. J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
    [Crossref] [PubMed]
  8. D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
    [Crossref]
  9. A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37(23), 5416–5420 (1998).
    [Crossref] [PubMed]
  10. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
    [Crossref] [PubMed]
  11. C. De Veuster, P. Slangen, Y. Renotte, L. Berwart, and Y. Lion, “Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms,” Appl. Opt. 35(2), 240–247 (1996).
    [Crossref] [PubMed]
  12. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–134 (1999).
    [Crossref]
  13. D. Gao and F. Yin, “Mask cut optimization in two-dimensional phase unwrapping,” IEEE Geosci. Remote Sens. Lett. 9(3), 338–342 (2012).
    [Crossref]
  14. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692 (1997).
    [Crossref]
  15. G. F. Carballo and P. W. Fieguth, “Probabilistic Cost Functions for Network Flow Phase Unwrapping,” IEEE Trans. Geosci. Remote Sens. 38(5), 2192–2201 (2000).
    [Crossref]
  16. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
    [Crossref]
  17. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17(3), 401–414 (2000).
    [Crossref] [PubMed]
  18. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11(1), 107–117 (1994).
    [Crossref]
  19. G. H. Kaufmann, G. E. Galizzi, and P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of digital speckle-pattern interferometry phase maps,” Appl. Opt. 37(14), 3076–3084 (1998).
    [Crossref] [PubMed]
  20. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34(3), 728–738 (1996).
    [Crossref]
  21. D. C. Ghiglia and L. A. Romero, “Minimum LP-norm two dimensional phase unwrapping,” J. Opt. Soc. Am. A 13(10), 1999–2013 (1996).
    [Crossref]
  22. H. Nies, O. Loffeld, and R. Wang, “Phase unwrapping using 2D-Kalman filter potential and limitations,” inProceedings of IEEE Conference on International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.
    [Crossref]
  23. O. Loffeld, H. Nies, S. Knedlik, and W. Yu, “Phase unwrapping for SAR interferometry: A data fusion approach by Kalman filtering,” IEEE Trans. Geosci. Remote Sens. 46(1), 47–58 (2008).
    [Crossref]
  24. X. Xianming and P. Yiming, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar & Navigation 5(3), 296–304 (2011).
    [Crossref]
  25. X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22(3), 365–372 (2011).
    [Crossref]
  26. 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]
  27. X. M. Xie and Q. N. Zeng, “Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy,” Appl. Opt. 54(31), 9294–9307 (2015).
    [Crossref] [PubMed]
  28. R. G. Waghmare, D. Mishra, G. R. Sai Subrahmanyam, E. Banoth, and S. S. Gorthi, “Signal tracking approach for phase estimation in digital holographic interferometry,” Appl. Opt. 53(19), 4150–4157 (2014).
    [Crossref] [PubMed]
  29. Z. Cheng, D. Liu, Y. Yang, T. Ling, X. Chen, L. Zhang, J. Bai, Y. Shen, L. Miao, and W. Huang, “Practical phase unwrapping of interferometric fringes based on unscented Kalman filter technique,” Opt. Express 23(25), 32337–32349 (2015).
    [Crossref] [PubMed]
  30. X. Xie, “Iterated unscented Kalman filter for phase unwrapping of interferometric fringes,” Opt. Express 24(17), 18872–18897 (2016).
    [Crossref] [PubMed]
  31. J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
    [Crossref]
  32. X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 9(3), 705–709 (2011).
  33. R. G. Waghmare, P. R. Sukumar, G. R. K. S. Subrahmanyam, R. K. Singh, and D. Mishra, “Particle-filter-based phase estimation in digital holographic interferometry,” J. Opt. Soc. Am. A 33(3), 326–332 (2016).
    [Crossref] [PubMed]
  34. X. Xie and G. Dai, “Unscented information filtering phase unwrapping algorithm for interferometric fringe patterns,” Appl. Opt. 56(34), 9423–9434 (2017).
    [Crossref] [PubMed]
  35. D. Blinder, H. Ottevaere, A. Munteanu, and P. Schelkens, “Efficient multiscale phase unwrapping methodology with modulo wavelet transform,” Opt. Express 24(20), 23094–23108 (2016).
    [Crossref] [PubMed]
  36. H. Y. H. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G. Barbastathis, “Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising,” Opt. Express 20(13), 14075–14089 (2012).
    [Crossref] [PubMed]
  37. H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
    [Crossref]
  38. J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
    [Crossref] [PubMed]
  39. S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
    [Crossref] [PubMed]
  40. 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]
  41. 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]
  42. J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19(6), 5126–5133 (2011).
    [Crossref] [PubMed]
  43. 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]
  44. J. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express 20(10), 10896–10920 (2012).
    [Crossref] [PubMed]
  45. J. F. Weng and Y. L. Lo, “Novel rotation algorithm for phase unwrapping applications,” Opt. Express 20(15), 16838–16860 (2012).
    [Crossref]
  46. H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
    [Crossref]
  47. K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).
  48. M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
    [Crossref]
  49. X. M. Xie, “New particle filtering phase unwrapping algorithm for wrapped fringe pattern,” Opt. Lasers Eng. 116, 55–67 (2019).
    [Crossref]
  50. J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
    [Crossref]
  51. R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
    [Crossref]

2019 (1)

X. M. Xie, “New particle filtering phase unwrapping algorithm for wrapped fringe pattern,” Opt. Lasers Eng. 116, 55–67 (2019).
[Crossref]

2017 (1)

2016 (3)

2015 (2)

2014 (3)

2013 (1)

H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
[Crossref]

2012 (5)

2011 (9)

B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt. 50(19), 3205–3220 (2011).
[Crossref] [PubMed]

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
[Crossref]

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 9(3), 705–709 (2011).

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

X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22(3), 365–372 (2011).
[Crossref]

J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19(6), 5126–5133 (2011).
[Crossref] [PubMed]

K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).

2010 (2)

2009 (2)

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
[Crossref]

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[Crossref] [PubMed]

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. Remote Sens. 46(1), 47–58 (2008).
[Crossref]

2007 (2)

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
[Crossref] [PubMed]

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
[Crossref] [PubMed]

2002 (2)

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

2000 (2)

G. F. Carballo and P. W. Fieguth, “Probabilistic Cost Functions for Network Flow Phase Unwrapping,” IEEE Trans. Geosci. Remote Sens. 38(5), 2192–2201 (2000).
[Crossref]

C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17(3), 401–414 (2000).
[Crossref] [PubMed]

1999 (2)

1998 (3)

1997 (1)

1996 (3)

1995 (1)

R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
[Crossref]

1994 (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]

1978 (1)

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Arulampalam, M. S.

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

Asundi, A.

Bai, J.

Banoth, E.

Barbastathis, G.

Bellaire, R. L.

R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
[Crossref]

Berwart, L.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
[Crossref] [PubMed]

Blinder, D.

Burton, D. R.

Cabral-Cano, E.

Carballo, G. F.

G. F. Carballo and P. W. Fieguth, “Probabilistic Cost Functions for Network Flow Phase Unwrapping,” IEEE Trans. Geosci. Remote Sens. 38(5), 2192–2201 (2000).
[Crossref]

Chandra, K. P. B.

K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).

Chen, C. W.

Chen, M.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

Chen, X.

Chen, Z.

Cheng, Z.

Clapp, T.

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
[Crossref]

Cuevas, F. J.

Cumming, I.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–134 (1999).
[Crossref]

Da, F. P.

D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
[Crossref]

Dai, G.

De Veuster, C.

Dixon, T. H.

Estrada, J. C.

Fieguth, P. W.

G. F. Carballo and P. W. Fieguth, “Probabilistic Cost Functions for Network Flow Phase Unwrapping,” IEEE Trans. Geosci. Remote Sens. 38(5), 2192–2201 (2000).
[Crossref]

Flynn, T. J.

Galizzi, G. E.

Gao, D.

D. Gao and F. Yin, “Mask cut optimization in two-dimensional phase unwrapping,” IEEE Geosci. Remote Sens. Lett. 9(3), 338–342 (2012).
[Crossref]

Gdeisat, M. A.

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]

Gordon, N.

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

Gorthi, S. S.

Gu, D. W.

K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).

Herráez, M. A.

Huang, H. Y. H.

Huang, L.

Huang, W.

Kamen, E. W.

R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
[Crossref]

Kaufmann, G. H.

Kemao, Q.

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. Remote Sens. 46(1), 47–58 (2008).
[Crossref]

Lalor, M. J.

Langley, J.

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[Crossref] [PubMed]

Li, Y.

Ling, T.

Lion, Y.

Liu, D.

Liu, F. F.

H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
[Crossref]

Liu, Y.

Lo, Y. L.

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. Remote Sens. 46(1), 47–58 (2008).
[Crossref]

Lopez-Sanchez, J. M.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
[Crossref]

Luo, Y.

Malacara, D.

Marroquin, J. L.

Martinez-Espla, J. J.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
[Crossref]

Martinez-Marin, T.

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
[Crossref]

Maskell, S.

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

Miao, L.

Mishra, D.

Moré, J. J.

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Munteanu, A.

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. Remote Sens. 46(1), 47–58 (2008).
[Crossref]

Osmanoglu, B.

Ottevaere, H.

Pi, Y. M.

X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22(3), 365–372 (2011).
[Crossref]

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 9(3), 705–709 (2011).

Postlethwaite, I.

K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).

Pritt, M. D.

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34(3), 728–738 (1996).
[Crossref]

Quiroga, J. A.

Rajshekhar, G.

Rastogi, P.

Renotte, Y.

Rodriguez-Vera, R.

Romero, L. A.

Ruiz, P. D.

Sai Subrahmanyam, G. R.

Schelkens, P.

Servin, M.

Shen, Y.

Singh, R. K.

Slangen, P.

Su, X.

Subrahmanyam, G. R. K. S.

Sukumar, P. R.

Tang, J.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

Tang, J. S.

H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
[Crossref]

Tian, C.

Tian, L.

Valadão, G.

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
[Crossref] [PubMed]

Vargas, J.

Waghmare, R. G.

Wang, H. Y.

H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
[Crossref]

Wdowinski, S.

Weng, J. F.

Wensen, Z.

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]

Xianming, X.

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

Xie, X.

Xie, X. M.

X. M. Xie, “New particle filtering phase unwrapping algorithm for wrapped fringe pattern,” Opt. Lasers Eng. 116, 55–67 (2019).
[Crossref]

X. M. Xie and Q. N. Zeng, “Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy,” Appl. Opt. 54(31), 9294–9307 (2015).
[Crossref] [PubMed]

X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22(3), 365–372 (2011).
[Crossref]

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 9(3), 705–709 (2011).

Xu, W.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–134 (1999).
[Crossref]

Yang, Y.

Yiming, P.

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

Yin, F.

D. Gao and F. Yin, “Mask cut optimization in two-dimensional phase unwrapping,” IEEE Geosci. Remote Sens. Lett. 9(3), 338–342 (2012).
[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. Remote Sens. 46(1), 47–58 (2008).
[Crossref]

Yuqing, S.

Zabin, S. M.

R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
[Crossref]

Zebker, H. A.

C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17(3), 401–414 (2000).
[Crossref] [PubMed]

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]

Zeng, Q. N.

Zhang, L.

Zhang, Q.

Zhang, S.

H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
[Crossref]

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

Zhang, X. B.

H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
[Crossref]

Zhang, Z.

Zhao, M.

Zhao, Q.

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[Crossref] [PubMed]

Zheng, D. L.

D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
[Crossref]

Zhong, H.

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

Zhong, H. P.

H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
[Crossref]

Zhu, Q. F.

H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
[Crossref]

Zhuo, Y.

Am. Control Conf. (1)

K. P. B. Chandra, D. W. Gu, and I. Postlethwaite, “Cubature Information Filter and its Applications,” Am. Control Conf. 47(3), 3609–3614(2011).

Appl. Opt. (12)

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]

B. Osmanoglu, T. H. Dixon, S. Wdowinski, and E. Cabral-Cano, “On the importance of path for phase unwrapping in synthetic aperture radar interferometry,” Appl. Opt. 50(19), 3205–3220 (2011).
[Crossref] [PubMed]

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

A. Asundi and Z. Wensen, “Fast phase-unwrapping algorithm based on a gray-scale mask and flood fill,” Appl. Opt. 37(23), 5416–5420 (1998).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

C. De Veuster, P. Slangen, Y. Renotte, L. Berwart, and Y. Lion, “Disk-growing algorithm for phase-map unwrapping: application to speckle interferograms,” Appl. Opt. 35(2), 240–247 (1996).
[Crossref] [PubMed]

G. H. Kaufmann, G. E. Galizzi, and P. D. Ruiz, “Evaluation of a preconditioned conjugate-gradient algorithm for weighted least-squares unwrapping of digital speckle-pattern interferometry phase maps,” Appl. Opt. 37(14), 3076–3084 (1998).
[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. M. Xie and Q. N. Zeng, “Efficient and robust phase unwrapping algorithm based on unscented Kalman filter, the strategy of quantizing paths-guided map, and pixel classification strategy,” Appl. Opt. 54(31), 9294–9307 (2015).
[Crossref] [PubMed]

R. G. Waghmare, D. Mishra, G. R. Sai Subrahmanyam, E. Banoth, and S. S. Gorthi, “Signal tracking approach for phase estimation in digital holographic interferometry,” Appl. Opt. 53(19), 4150–4157 (2014).
[Crossref] [PubMed]

X. Xie and G. Dai, “Unscented information filtering phase unwrapping algorithm for interferometric fringe patterns,” Appl. Opt. 56(34), 9423–9434 (2017).
[Crossref] [PubMed]

IEEE Geosci. Remote Sens. Lett. (3)

D. Gao and F. Yin, “Mask cut optimization in two-dimensional phase unwrapping,” IEEE Geosci. Remote Sens. Lett. 9(3), 338–342 (2012).
[Crossref]

H. Zhong, J. Tang, S. Zhang, and M. Chen, “An improved quality-guided phase-unwrapping algorithm based on priority queue,” IEEE Geosci. Remote Sens. Lett. 8(2), 364–368 (2011).
[Crossref]

H. P. Zhong, J. S. Tang, S. Zhang, and X. B. Zhang, “A Quality-Guided and Local Minimum Discontinuity Based Phase Unwrapping Algorithm for InSAR/InSAS Interferograms,” IEEE Geosci. Remote Sens. Lett. 11(1), 215–219 (2014).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (6)

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–134 (1999).
[Crossref]

G. F. Carballo and P. W. Fieguth, “Probabilistic Cost Functions for Network Flow Phase Unwrapping,” IEEE Trans. Geosci. Remote Sens. 38(5), 2192–2201 (2000).
[Crossref]

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
[Crossref]

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

M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34(3), 728–738 (1996).
[Crossref]

J. J. Martinez-Espla, T. Martinez-Marin, and J. M. Lopez-Sanchez, “A particle filter approach for InSAR phase filtering and unwrapping,” IEEE Trans. Geosci. Remote Sens. 47(4), 1197–1211 (2009).
[Crossref]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
[Crossref] [PubMed]

IEEE Trans. Signal Process. (1)

M. S. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50(2), 174–188 (2002).
[Crossref]

IET Radar Sonar & Navigation (1)

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

J. Opt. Soc. Am. A (5)

J. Syst. Eng. Electron. (1)

X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22(3), 365–372 (2011).
[Crossref]

Lect. Notes Math. (1)

J. J. Moré, “The Levenberg-Marquardt algorithm: Implementation and theory,” Lect. Notes Math. 630, 105–116 (1978).
[Crossref]

Magn. Reson. Imaging (1)

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
[Crossref] [PubMed]

Opt. Commun. (1)

H. Y. Wang, F. F. Liu, and Q. F. Zhu, “Improvement of phase unwrapping algorithm based on image segmentation and merging,” Opt. Commun. 308(11), 218–223 (2013).
[Crossref]

Opt. Express (10)

S. Yuqing, “Robust phase unwrapping by spinning iteration,” Opt. Express 15(13), 8059–8064 (2007).
[Crossref] [PubMed]

J. C. Estrada, M. Servin, and J. A. Quiroga, “Noise robust linear dynamic system for phase unwrapping and smoothing,” Opt. Express 19(6), 5126–5133 (2011).
[Crossref] [PubMed]

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. F. Weng and Y. L. Lo, “Integration of robust filters and phase unwrapping algorithms for image reconstruction of objects containing height discontinuities,” Opt. Express 20(10), 10896–10920 (2012).
[Crossref] [PubMed]

J. F. Weng and Y. L. Lo, “Novel rotation algorithm for phase unwrapping applications,” Opt. Express 20(15), 16838–16860 (2012).
[Crossref]

D. Blinder, H. Ottevaere, A. Munteanu, and P. Schelkens, “Efficient multiscale phase unwrapping methodology with modulo wavelet transform,” Opt. Express 24(20), 23094–23108 (2016).
[Crossref] [PubMed]

H. Y. H. Huang, L. Tian, Z. Zhang, Y. Liu, Z. Chen, and G. Barbastathis, “Path-independent phase unwrapping using phase gradient and total-variation (TV) denoising,” Opt. Express 20(13), 14075–14089 (2012).
[Crossref] [PubMed]

Z. Cheng, D. Liu, Y. Yang, T. Ling, X. Chen, L. Zhang, J. Bai, Y. Shen, L. Miao, and W. Huang, “Practical phase unwrapping of interferometric fringes based on unscented Kalman filter technique,” Opt. Express 23(25), 32337–32349 (2015).
[Crossref] [PubMed]

X. Xie, “Iterated unscented Kalman filter for phase unwrapping of interferometric fringes,” Opt. Express 24(17), 18872–18897 (2016).
[Crossref] [PubMed]

S. S. Gorthi, G. Rajshekhar, and P. Rastogi, “Strain estimation in digital holographic interferometry using piecewise polynomial phase approximation based method,” Opt. Express 18(2), 560–565 (2010).
[Crossref] [PubMed]

Opt. Lasers Eng. (2)

D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
[Crossref]

X. M. Xie, “New particle filtering phase unwrapping algorithm for wrapped fringe pattern,” Opt. Lasers Eng. 116, 55–67 (2019).
[Crossref]

Proc. SPIE (1)

R. L. Bellaire, E. W. Kamen, and S. M. Zabin, “New nonlinear iterated filter with applications to target tracking,” Proc. SPIE 2561, 240–251 (1995).
[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]

Tien Tzu Hsueh Pao (1)

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 9(3), 705–709 (2011).

Other (2)

H. Nies, O. Loffeld, and R. Wang, “Phase unwrapping using 2D-Kalman filter potential and limitations,” inProceedings of IEEE Conference on International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.
[Crossref]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Flowchart of unwrapping stage of the accelerated CIPFPU algorithm
Fig. 2
Fig. 2 Synthetic wrapped phase images; (a) the true unwrapped phase map with the size of 512 × 512; (b) the true unwrapped phase map with the size of 768 × 768; (c) the true unwrapped phase map with the size of 1024 × 1024; (d) the wrapped phase image at 5.39dB, corresponding to Fig. 2(a); (e) the wrapped phase image at 3.36dB, corresponding to Fig. 2(b); (f) the wrapped phase image at 1.72dB, corresponding to Fig. 2(c).
Fig. 3
Fig. 3 Solutions with different PU methods executing on Fig. 2(d); Each row represents the results for different algorithms: “the QGPU method”, “the IUKFPU method”, “the AF2DPU method”, “the CIPFPU method”, “the accelerated CIPFPU method”; Columns (left), (middle) and (right) refer to the unwrapped phase, the PU errors, and the histogram of the PU errors, respectively.
Fig. 4
Fig. 4 Solutions with different PU methods executing on Fig. 2(e); Each row represents the results for different algorithms: “the QGPU method”, “the IUKFPU method”, “the AF2DPU method”, “the CIPFPU method”, “the accelerated CIPFPU method”; Columns (left), (middle) and (right) refer to the unwrapped phase, the PU errors, and the histogram of the PU errors, respectively.
Fig. 5
Fig. 5 Solutions with different PU methods executing on Fig. 2(f); Each row represents the results for different algorithms: “the QGPU method”, “the IUKFPU method”, “the AF2DPU method”, “the CIPFPU method”, “the accelerated CIPFPU method”; Columns (left), (middle) and (right) refer to the unwrapped phase, the PU errors, and the histogram of the PU errors, respectively.
Fig. 6
Fig. 6 Noisy MEMS with the size of 1024 × 1024.
Fig. 7
Fig. 7 Solutions with different PU methods executing on Fig. 6; Each row represents the results for different algorithms: “the QGPU method”, “the IUKFPU method”, “the AF2DPU method”, “the CIPFPU method”, “the accelerated CIPFPU method”; Columns (left) and (right) refer to the unwrapped phase, the rewrapped phase of the unwrapped phase, respectively.

Tables (3)

Tables Icon

Table 1 Mean Square Root Errors (in radian) for Different Algorithms over Synthetic Images

Tables Icon

Table 2 Run-time (in second) for Different Algorithms over Synthetic Images

Tables Icon

Table 3 Run-time (in second) for Different Algorithms over Experimental Images

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

Δ ¯ (k,l) r = 1 (2 κ 1 + 1 1 )(2 κ 2 + 1 1 ) u=k κ 1 k+ κ 1 v=l κ 2 l+ κ 2 Δ (u,v) r Δ ¯ (k,l) c = 1 (2 κ 1 + 1 1 )(2 κ 2 + 1 1 ) u=k κ 1 k+ κ 1 v=l κ 2 l+ κ 2 Δ (u,v) c , η 1 =2 κ 1 +1, η 2 =2 κ 2 +1
Δ (u,v) r =Wrap[φ(u+1,v)φ(u,v)] Δ (u,v) c =Wrap[φ(u,v+1)φ(u,v)],
Wrap[x]={ x2π x>π x |x|<π x+2π x<π ,
δ( Δ ¯ (k,l) r )=4 π 2 1 γ (k,l) 2 γ (k,l) 2 η 2 ( η 1 2 1) , δ( Δ ¯ (k,l) c )=4 π 2 1 γ (k,l) 2 γ (k,l) 2 η 1 ( η 2 2 1)
x(m,n)= 1 Κ (k,l)Ω [x(k,l)+(mk) Δ ¯ (k,l) r +(nl) Δ ¯ (k,l) c ] +ε(m,n) = F Ω x(k,l)+ε(m,n) y(m,n)={ sin[x(m,n)] cos[x(m,n)] }+{ v 1 (m,n) v 2 (m,n) }=h[x(m,n)]+v(m,n)
Q (m,n) = 1 Κ 2 (k,l)Ω [δ( Δ ¯ (k,l) r ) (mk) 2 +δ( Δ ¯ (k,l) c ) (nl) 2 ] , R (m,n) =[ σ R,(m,n) 2 , 0 0, σ R,(m,n) 2 ], σ R,(m,n) 2 = 1 SN R (m,n) = 1 γ (m,n) 2 2 γ (m,n) 2
x (m,n)= 1 Κ (k,l)Ω [ x (k,l)+(mk) Δ ¯ (k,l) r +(nl) Δ ¯ (k,l) c ] , P xx (m,n)= 1 Κ 2 P xx (k,l)= S ˜ (m,n)× [ S ˜ (m,n)] T
χ ˜ j (m,n)= S ˜ (m,n) ξ j + x (m,n) x + (m,n)= j=1 m x w j χ ˜ j (m,n) P xx + (m,n)= j=1 m x w j χ ˜ j (m,n) [ χ ˜ j (m,n)] T x + (m,n) [ x + (m,n)] T + Q (m,n) ,
ξ j = m x 2 [ 1 ] j ,(j=1,2,... m x ) w j = 1 m x ,(j=1,2,... m x )
[ 1 ]=[ 1 0 ... 0 0 1 ... 0 ... 0 0 ... 1 1 0 ... 0 0 1 0 0 ... 0 0 ... 1 ],
P ˜ xx (m,n)={ I P xx + (m,n)× [ P xx + (m,n)+ u 1 I ] 1 }× P xx + (m,n)= S (m,n) [ S (m,n)] T ,
Z( m,n )= [ P ˜ xx (m,n)] 1 z(m,n)=Z( m,n ) x + (m,n),
χ j (m,n)= S (m,n) ξ j + x + (m,n) η j (m,n)=h[ χ j (m,n)] y (m,n)= j=1 m x w j η j (m,n) P xy (m,n)= j=1 m x w j χ j (m,n) [ η j (m,n)] T x + (m,n) [ y (m,n)] T ,
ε(m,n)=y(m,n) y (m,n) i(m,n)=Z( m,n ) P xy (m,n) [ R (m,n) ] 1 {ε(m,n)+ [ P xy (m,n)] T [Z( m,n )] T x + (m,n)}, I(m,n)=Z( m,n ) P xy (m,n) [ R (m,n) ] 1 [ P xy (m,n)] T [Z( m,n )] T
{ z ( m,n )=z( m,n )+i(m,n) Z ( m,n )=Z( m,n )+I(m,n) P xx (m,n)= [ Z ( m,n )] T x ( m,n )= z ( m,n )/ Z ( m,n ) ,
w (m,n) i = 1 | γ (m,n) | 2 2π 1 1 | γ (m,n) | 2 cos 2 ( φ (m,n) x (m,n) i ) {1+ | γ (m,n) |cos( φ (m,n) x (m,n) i )arccos[| γ (m,n) |cos( φ (m,n) x (m,n) i )] [1 | γ (m,n) | 2 cos 2 ( φ (m,n) x (m,n) i )] 1 2 }, w ¯ (m,n) i = w (m,n) i i=1 N s w (m,n) i
x (m,n) = i=1 N s w ¯ (m,n) i x (m,n) i ,

Metrics