Abstract

A fresh phase unwrapping algorithm based on iterated unscented Kalman filter is proposed to estimate unambiguous unwrapped phase of interferometric fringes. This method is the result of combining an iterated unscented Kalman filter with a robust phase gradient estimator based on amended matrix pencil model, and an efficient quality-guided strategy based on heap sort. The iterated unscented Kalman filter that is one of the most robust methods under the Bayesian theorem frame in non-linear signal processing so far, is applied to perform simultaneously noise suppression and phase unwrapping of interferometric fringes for the first time, which can simplify the complexity and the difficulty of pre-filtering procedure followed by phase unwrapping procedure, and even can remove the pre-filtering procedure. The robust phase gradient estimator is used to efficiently and accurately obtain phase gradient information from interferometric fringes, which is needed for the iterated unscented Kalman filtering phase unwrapping model. The efficient quality-guided strategy is able to ensure that the proposed method fast unwraps wrapped pixels along the path from the high-quality area to the low-quality area of wrapped phase images, which can greatly improve the efficiency of phase unwrapping. Results obtained from synthetic data and real data show that the proposed method can obtain better solutions with an acceptable time consumption, with respect to some of the most used algorithms.

© 2016 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Practical phase unwrapping of interferometric fringes based on unscented Kalman filter technique

Zhongtao Cheng, Dong Liu, Yongying Yang, Tong Ling, Xiaoyu Chen, Lei Zhang, Jian Bai, Yibing Shen, Liang Miao, and Wei Huang
Opt. Express 23(25) 32337-32349 (2015)

Phase unwrapping algorithm using polynomial phase approximation and linear Kalman filter

Rishikesh Kulkarni and Pramod Rastogi
Appl. Opt. 57(4) 702-708 (2018)

References

  • View by:
  • |
  • |
  • |

  1. S. Fang, L. Meng, L. Wang, P. Yang, and M. Komori, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Appl. Opt. 50(28), 5446–5452 (2011).
    [Crossref] [PubMed]
  2. 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]
  3. 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]
  4. J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27(9), 1293–1301 (2009).
    [Crossref] [PubMed]
  5. 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]
  6. 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]
  7. 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]
  8. U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33(3), 579–589 (1995).
    [Crossref]
  9. 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]
  10. Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
    [Crossref]
  11. J. M. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16(3), 698–709 (2007).
    [Crossref] [PubMed]
  12. D. L. Zheng and F. P. Da, “A novel algorithm for branch cut phase unwrapping,” Opt. Lasers Eng. 49(5), 609–617 (2011).
    [Crossref]
  13. N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
    [Crossref] [PubMed]
  14. 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]
  15. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).
  16. 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]
  17. 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]
  18. W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–133 (1999).
    [Crossref]
  19. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36(3), 813–821 (1998).
    [Crossref]
  20. 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]
  21. D. Gao and F. Yin, “Mask cut optimization in two-dimensional phase unwrapping,” IEEE Geosci. Remote Sens. Lett. 9(3), 338–342 (2012).
    [Crossref]
  22. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14(10), 2692 (1997).
    [Crossref]
  23. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterated methods,” J. Opt. Soc. Am. A 11(1), 107–117 (1994).
    [Crossref]
  24. 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]
  25. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34(3), 728–738 (1996).
    [Crossref]
  26. 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]
  27. H. Nies, O. Loffeld, and R. Wang, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in Proceedings of IEEE Conference on International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.
    [Crossref]
  28. 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]
  29. 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), 1197–1206 (2009).
    [Crossref]
  30. X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 39(3), 705–709 (2011).
  31. R. G. Waghmare, P. Ram 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]
  32. 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]
  33. 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]
  34. 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]
  35. 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]
  36. R. Kulkarni and P. Rastogi, “Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation,” Meas. Sci. Technol. 27(6), 065203 (2016).
    [Crossref]
  37. E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
    [Crossref]
  38. E. W. Daniel, “Improved SAR interferometric processing using local phase slope correction,” Proc. SPIE 5427, 103–107 (2007).
  39. 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]
  40. 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]
  41. 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]
  42. J. F. Weng and Y. L. Lo, “Novel rotation algorithm for phase unwrapping applications,” Opt. Express 20(15), 16838–16860 (2012).
    [Crossref]
  43. 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]
  44. 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]
  45. R. H. Zhan and J. W. Wan, “Iterated unscented Kalman filter for passive target tracking,” IEEE Trans. Aerosp. Electron. Syst. 43(3), 1155–1163 (2007).
    [Crossref]
  46. S. Y. Chen and L. Yu, “Algorithm realization and its application evaluation of the iterated unscented Kalman filter,” Sys. Eng. Electron. 33(11), 2546–2553 (2011).
  47. F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
    [Crossref]
  48. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (The MIT Press, 2001).
  49. W. M. Yan, D. M. Li, and W. M. Wu, Data Structure (People's Posts and Telecommunications Press, 2015).
  50. S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92(3), 401–422 (2004).
    [Crossref]
  51. S. J. Julier, “The scaled unscented transformation,” in Proceedings of American Control Conference (American Automatic Control Council, 2002), pp. 4555–4559.

2016 (2)

R. G. Waghmare, P. Ram 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]

R. Kulkarni and P. Rastogi, “Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation,” Meas. Sci. Technol. 27(6), 065203 (2016).
[Crossref]

2015 (2)

2014 (2)

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 (6)

2011 (10)

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]

S. Y. Chen and L. Yu, “Algorithm realization and its application evaluation of the iterated unscented Kalman filter,” Sys. Eng. Electron. 33(11), 2546–2553 (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]

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 39(3), 705–709 (2011).

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

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]

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]

S. Fang, L. Meng, L. Wang, P. Yang, and M. Komori, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Appl. Opt. 50(28), 5446–5452 (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]

2010 (1)

2009 (2)

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

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), 1197–1206 (2009).
[Crossref]

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 (3)

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

R. H. Zhan and J. W. Wan, “Iterated unscented Kalman filter for passive target tracking,” IEEE Trans. Aerosp. Electron. Syst. 43(3), 1155–1163 (2007).
[Crossref]

E. W. Daniel, “Improved SAR interferometric processing using local phase slope correction,” Proc. SPIE 5427, 103–107 (2007).

2005 (1)

Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
[Crossref]

2004 (1)

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

2000 (1)

1999 (1)

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–133 (1999).
[Crossref]

1998 (4)

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

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]

E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
[Crossref]

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]

1997 (1)

1996 (3)

1995 (1)

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33(3), 579–589 (1995).
[Crossref]

1994 (1)

1992 (1)

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[Crossref] [PubMed]

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]

Asundi, A.

Bai, J.

Banoth, E.

Barbastathis, G.

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]

Braun, M.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[Crossref] [PubMed]

Cabral-Cano, E.

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, S. Y.

S. Y. Chen and L. Yu, “Algorithm realization and its application evaluation of the iterated unscented Kalman filter,” Sys. Eng. Electron. 33(11), 2546–2553 (2011).

Chen, X.

Chen, Z.

Cheng, Z.

Ching, N. H.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[Crossref] [PubMed]

Costantini, M.

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

Cumming, I.

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–133 (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]

Daniel, E. W.

E. W. Daniel, “Improved SAR interferometric processing using local phase slope correction,” Proc. SPIE 5427, 103–107 (2007).

De Veuster, C.

Ding, C. B.

F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
[Crossref]

Dixon, T. H.

Estrada, J. C.

Fang, S.

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]

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]

Gorthi, S. S.

He, G. T.

Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
[Crossref]

Hu, D. H.

F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
[Crossref]

Huang, H. Y. H.

Huang, L.

Huang, W.

Julier, S. J.

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

Komori, M.

Kulkarni, R.

R. Kulkarni and P. Rastogi, “Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation,” Meas. Sci. Technol. 27(6), 065203 (2016).
[Crossref]

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, F. F.

F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
[Crossref]

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), 1197–1206 (2009).
[Crossref]

Lu, Y. G.

Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
[Crossref]

Maitre, H.

E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
[Crossref]

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), 1197–1206 (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), 1197–1206 (2009).
[Crossref]

Meng, L.

Miao, L.

Mishra, D.

Navarro, M. A.

Nicolas, J.-M.

E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
[Crossref]

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.

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 39(3), 705–709 (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.

Ram Sukumar, P.

Rastogi, P.

R. Kulkarni and P. Rastogi, “Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation,” Meas. Sci. Technol. 27(6), 065203 (2016).
[Crossref]

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]

Renotte, Y.

Romero, L. A.

Rosenfeld, D.

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[Crossref] [PubMed]

Ruiz, P. D.

Sai Subrahmanyam, G. R.

Servin, M.

Shen, Y.

Singh, R. K.

Slangen, P.

Spagnolini, U.

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33(3), 579–589 (1995).
[Crossref]

Su, X.

Subrahmanyam, G. R. K. S.

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]

Tian, L.

Trouve, E.

E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
[Crossref]

Uhlmann, J. K.

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

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.

Wan, J. W.

R. H. Zhan and J. W. Wan, “Iterated unscented Kalman filter for passive target tracking,” IEEE Trans. Aerosp. Electron. Syst. 43(3), 1155–1163 (2007).
[Crossref]

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]

Wang, L.

Wang, X. Z.

Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
[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 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 39(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), 124–133 (1999).
[Crossref]

Yang, P.

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, L.

S. Y. Chen and L. Yu, “Algorithm realization and its application evaluation of the iterated unscented Kalman filter,” Sys. Eng. Electron. 33(11), 2546–2553 (2011).

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]

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.

Zhan, R. H.

R. H. Zhan and J. W. Wan, “Iterated unscented Kalman filter for passive target tracking,” IEEE Trans. Aerosp. Electron. Syst. 43(3), 1155–1163 (2007).
[Crossref]

Zhan, Y.

F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
[Crossref]

Zhang, L.

Zhang, Q.

Zhang, S.

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, 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]

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]

Appl. Opt. (9)

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]

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]

S. Fang, L. Meng, L. Wang, P. Yang, and M. Komori, “Quality-guided phase unwrapping algorithm based on reliability evaluation,” Appl. Opt. 50(28), 5446–5452 (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]

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]

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. 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]

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]

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]

IEEE Geosci. Remote Sens. Lett. (2)

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

IEEE Trans. Aerosp. Electron. Syst. (1)

R. H. Zhan and J. W. Wan, “Iterated unscented Kalman filter for passive target tracking,” IEEE Trans. Aerosp. Electron. Syst. 43(3), 1155–1163 (2007).
[Crossref]

IEEE Trans. Geosci. Remote Sens. (7)

E. Trouve, J.-M. Nicolas, and H. Maitre, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36(6), 1963–1965 (1998).
[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]

W. Xu and I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37(1), 124–133 (1999).
[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]

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), 1197–1206 (2009).
[Crossref]

U. Spagnolini, “2-D Phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33(3), 579–589 (1995).
[Crossref]

IEEE Trans. Image Process. (2)

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

N. H. Ching, D. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1(3), 355–365 (1992).
[Crossref] [PubMed]

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. Radars (1)

F. F. Li, Y. Zhan, D. H. Hu, and C. B. Ding, “A fast method for InSAR phase unwrapping based on quality guide,” J. Radars 1(2), 196–202 (2012).
[Crossref]

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]

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]

Meas. Sci. Technol. (1)

R. Kulkarni and P. Rastogi, “Simultaneous estimation of phase derivative and phase using parallel Kalman filter implementation,” Meas. Sci. Technol. 27(6), 065203 (2016).
[Crossref]

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. Eng. (1)

Y. G. Lu, X. Z. Wang, and G. T. He, “Phase unwrapping based on branch cut placing and reliability ordering,” Opt. Eng. 44(5), 055601 (2005).
[Crossref]

Opt. Express (7)

Opt. Lasers Eng. (1)

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

Proc. IEEE (1)

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

Proc. SPIE (1)

E. W. Daniel, “Improved SAR interferometric processing using local phase slope correction,” Proc. SPIE 5427, 103–107 (2007).

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]

Sys. Eng. Electron. (1)

S. Y. Chen and L. Yu, “Algorithm realization and its application evaluation of the iterated unscented Kalman filter,” Sys. Eng. Electron. 33(11), 2546–2553 (2011).

Tien Tzu Hsueh Pao (1)

X. M. Xie and Y. M. Pi, “Phase unwrapping: an unscented particle filtering approach,” Tien Tzu Hsueh Pao 39(3), 705–709 (2011).

Other (5)

H. Nies, O. Loffeld, and R. Wang, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in Proceedings 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).

T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms (The MIT Press, 2001).

W. M. Yan, D. M. Li, and W. M. Wu, Data Structure (People's Posts and Telecommunications Press, 2015).

S. J. Julier, “The scaled unscented transformation,” in Proceedings of American Control Conference (American Automatic Control Council, 2002), pp. 4555–4559.

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 (34)

Fig. 1
Fig. 1 Flowchart of unwrapping stage in the proposed method.
Fig. 2
Fig. 2 Synthetic interferogram over pyramid, (a) true phase surface; (b) noisy wrapped phase, and (c) filtered wrapped phase.
Fig. 3
Fig. 3 The BUT solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 4
Fig. 4 The NF solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 5
Fig. 5 The WLS solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 6
Fig. 6 The QGPU solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 7
Fig. 7 The EKFPU solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 8
Fig. 8 The EUKFPU solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 9
Fig. 9 The Solution of the IUKFPU (Niter = 0) method; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 10
Fig. 10 The Solution of the IUKFPU (Niter = 1) method; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 11
Fig. 11 The Solution of the IUKFPU (Niter = 2) method; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 12
Fig. 12 The Solution of the IUKFPU (Niter = 3) method; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 13
Fig. 13 Synthetic interferogram over pyramid, (a) true phase surface, and (b) noisy wrapped phase.
Fig. 14
Fig. 14 The Solution of the BUT method; (a) phase residues; (b) brand-cut lines, and (c) unwrapped phase.
Fig. 15
Fig. 15 The QGPU solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 16
Fig. 16 The EKFPU solution; (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 17
Fig. 17 The solution of the IUKFPU (Niter = 0); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 18
Fig. 18 The solution of the IUKFPU (Niter = 1); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 19
Fig. 19 The solution of IUKFPU (Niter = 2); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 20
Fig. 20 The solution of IUKFPU (Niter = 3); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 21
Fig. 21 Synthetic interferogram over peaks; (a) true phase surface; (b) noisy wrapped phase, and (c) phase residues.
Fig. 22
Fig. 22 The solution of IUKFPU (Niter = 0); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 23
Fig. 23 The solution of IUKFPU (Niter = 1); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 24
Fig. 24 The solution of IUKFPU (Niter = 2); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 25
Fig. 25 The solution of IUKFPU (Niter = 3); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 26
Fig. 26 Synthetic interferogram over cone; (a) true phase surface; (b) noisy wrapped phase, and (c) phase residues.
Fig. 27
Fig. 27 The solution of IUKFPU (Niter = 0); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 28
Fig. 28 The solution of IUKFPU (Niter = 1); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 29
Fig. 29 The solution of IUKFPU (Niter = 2); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 30
Fig. 30 The solution of IUKFPU (Niter = 3); (a) unwrapped phase; (b) unwrapped phase error, and (c) histogram of phase error.
Fig. 31
Fig. 31 The experimental topographic wrapped phase over Etna; (a) the experimental wrapped phase, and (b) phase residues.
Fig. 32
Fig. 32 The unwrapped phase obtained with the BUT, NF, WLS, EKFPU and QGPU method; (a) unwrapped phase of the BUT method; (b) unwrapped phase of the NF method; (c) unwrapped phase of the WLS method; (d) unwrapped phase of the EKFPU method, and (e) unwrapped phase of the QGPU method.
Fig. 33
Fig. 33 The rewrapped phase obtained with the NF, WLS, EKFPU and QGPU method; (a) rewrapped phase of the NF method; (b) rewrapped phase of the WLS method; (c) rewrapped phase of the EKFPU method, and (d) unwrapped phase of the QGPU method.
Fig. 34
Fig. 34 The solutions obtained with the IUKFPU method; (a) unwrapped phase (Niter = 0) ; (b) unwrapped phase (Niter = 1); (c) unwrapped phase (Niter = 2) ; (d) unwrapped phase (Niter = 3) ; (e) rewrapped phase (Niter = 0); (f) rewrapped phase (Niter = 1); (g) rewrapped phase (Niter = 2); (h) rewrapped phase (Niter = 3) .

Tables (7)

Tables Icon

Table 1 Phase unwrapping accuracy of different methods.

Tables Icon

Table 2 Phase unwrapping accuracy of the proposed method.

Tables Icon

Table 3 Run-time of different algorithms.

Tables Icon

Table 4 Run-time of the proposed method.

Tables Icon

Table 5 Comparison of phase unwrapping accuracy without pre-filtering procedure

Tables Icon

Table 6 Phase unwrapping accuracy of the proposed method without pre-filtering procedure.

Tables Icon

Table 7 Phase unwrapping accuracy of the proposed method without pre-filtering procedure.

Equations (17)

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

ϕ ˜ (m,n)= [ϕ(m,n)] 2π ±k2π(π,π],
x(m,n)=x(a,s)+ g (m,n)|(a,s) + ε (m,n)|(a,s) =f[x(a,s)]+ ε (m,n)|(a,s) 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),
g (m,n)|(a,s) = ϕ ¯ y(a,s) (ma)+ ϕ ¯ x(a,s) (ns),
x (m,n)= (a,s)Ψ d (a,s)x[(m,n)|(a,s)],
x[(m,n)|(a,s)]= x (a,s)+ g (m,n)|(a,s) d(a,s)= [ P xx (a,s) 1 SNR(a,s) ] 1 (a,s)Ψ [ P xx (a,s) 1 SNR(a,s) ] 1 ,
SNR(a,s)== γ (a,s) 1 γ (a,s) ,
P xx (m,n)= (a,s)Ψ d(a,s) P xx (a,s).
χ 0 (m,n)= x (m,n) χ 1 (m,n)= x (m,n)+ (1+λ) P xx (m,n) χ 2 (m,n)= x (m,n) (1+λ) P xx (m,n) ,
λ= α 2 (1+κ)1,
x ˜ (m,n)= j=0 2 w j m χ j (m,n) P ˜ xx (m,n)= j=0 2 w j c [ χ j (m,n) x ˜ (m,n)] , [ χ j (m,n) x ˜ (m,n)] T + (a,s)Ψ d(a,s) Q (m,n)|(a,s)
w 0 m =λ/(1+λ), w 0 c =λ/(1+λ)+(1 α 2 +β), w j m = w j c =1/[2(1+λ)]
Q (m,n)|(a,s) =G( ϕ ¯ y(a,s) ) (ma) 2 +G( ϕ ¯ x(a,s) ) (ns) 2 ,
G( ϕ ¯ y(a,s) )=4 π 2 1 γ (a,s) 2 γ (a,s) 2 Κ n ( Κ m 2 1) , G( ϕ ¯ x(a,s) )=4 π 2 1 γ (a,s) 2 γ (a,s) 2 Κ m ( Κ n 2 1)
ξ j (m,n)=h[ χ j (m,n)] y (m,n)= j=0 2 w j m ξ j (m,n) P yy (m,n)= j=0 2 w j c [ ξ j (m,n) y (m,n)] [ ξ j (m,n) y (m,n)] T +R(m,n) , P xy (m,n)= j=0 2 w j c [ χ j (m,n) x ˜ (m,n)] [ ξ j (m,n) y (m,n)] T ρ(m,n)= P xy (m,n)/ P yy (m,n) x 0 (m,n)= x ˜ (m,n)+ρ(m,n)[y(m,n) y (m,n)] P xx 0 (m,n)= P ˜ xx (m,n)ρ(m,n) P yy (m,n)ρ(m,n)
R(m,n)= 1 SNR(m,n) = 1 γ (m,n) γ (m,n) ,
χ 0 i+1 (m,n)= x i (m,n) χ 1 i+1 (m,n)= x i (m,n)+ (1+λ) P xx i (m,n) χ 2 i+1 (m,n)= x i (m,n) (1+λ) P xx i (m,n) , x i+1 (m,n)= j=0 2 w j m χ j i+1 (m,n)
ξ j i+1 (m,n)=h[ χ j i+1 (m,n)] y i+1 (m,n)= j=0 2 w j m ξ j i+1 (m,n) P yy i+1 (m,n)= j=0 2 w j c [ ξ j i+1 (m,n) y i+1 (m,n)] , [ ξ j i+1 (m,n) y i+1 (m,n)] T +R(m,n) P xy i+1 (m,n)= j=0 2 w j c [ χ j i+1 (m,n) x i+1 (m,n)] [ ξ j i+1 (m,n) y i+1 (m,n)] T ρ i+1 (m,n)= P xy i+1 (m,n)/ P yy i+1 (m,n) x i+1 (m,n)= x ˜ (m,n)+ ρ i+1 (m,n)[y(m,n) y i+1 (m,n)] P xx i+1 (m,n)= P xx i (m,n) ρ i+1 (m,n) P yy i+1 (m,n) ρ i+1 (m,n)

Metrics