Abstract

This paper presents an enhanced phase unwrapping algorithm by combining an unscented Kalman filter, an enhanced local phase gradient estimator based on an amended matrix pencil model, and a path-following strategy. This technology is able to accurately unwrap seriously noisy wrapped phase images by applying the unscented Kalman filter to simultaneously perform noise suppression and phase unwrapping along the path from the high-quality region to the low-quality region of the wrapped phase images. Results obtained with synthetic data and real data validate the effectiveness of the proposed method and show improved performance of this new algorithm with respect to some of the most used algorithms.

© 2014 Optical Society of America

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    [CrossRef]
  21. X. M. Xie and Y. M. Pi, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5, 296–304 (2011).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  28. X. M. Xie and Y. M. Pi, “Phase noise filtering and phase unwrapping method based on unscented Kalman filter,” J. Syst. Eng. Electron. 22, 365–372 (2011).
  29. I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
    [CrossRef]
  30. T. Emmanuel and N. Jean-Marie, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
    [CrossRef]
  31. E. W. Daniel, “Improved SAR interferometric processing using local phase slope correction,” Proc. SPIE 5427, 103–107 (2007).
  32. U. Spagnolini, “2D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33, 579–589 (1994).
    [CrossRef]
  33. Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40, 2267–2280 (1992).
    [CrossRef]
  34. B. Tang, “Novel method for two-dimensional sinusoidal frequency efficient estimation,” J. Southwest Petroleum Inst. 20, 78–80 (1998).
  35. E. A. Wan and R. V. Merwe, “The unscented Kalman Filter for nonlinear estimation,” in The IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (IEEE, 2000), pp. 153–158.
  36. S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
    [CrossRef]

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, 218–223 (2013).
[CrossRef]

2012 (4)

2011 (6)

2010 (1)

2009 (2)

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27, 1293–1301 (2009).
[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, 1197–1211 (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, 47–58 (2008).
[CrossRef]

2007 (2)

Y. G. Lua, X. Z. Wang, and X. P. Zhang, “Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map,” Optik 118, 62–66 (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, 055601 (2005).
[CrossRef]

2004 (1)

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

2003 (1)

1999 (1)

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

1998 (4)

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 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, 3076–3084 (1998).
[CrossRef]

B. Tang, “Novel method for two-dimensional sinusoidal frequency efficient estimation,” J. Southwest Petroleum Inst. 20, 78–80 (1998).

T. Emmanuel and N. Jean-Marie, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

1997 (1)

1996 (2)

G. Fornaro, G. Franceshetti, and R. Lanari, “Interferometric SAR phase unwrapping using Greens formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

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, 240–247 (1996).
[CrossRef]

1994 (2)

1992 (2)

Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40, 2267–2280 (1992).
[CrossRef]

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

1988 (1)

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

Arines, J.

Asundi, A.

Berwart, L.

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, 355–365 (1992).
[CrossRef]

Cabral-Cano, E.

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, 355–365 (1992).
[CrossRef]

Costantini, M.

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 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, 124–134 (1999).
[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.

Dixon, T. H.

Emmanuel, T.

T. Emmanuel and N. Jean-Marie, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

Estrada, J. C.

Fang, S.

Flynn, T. J.

Fornaro, G.

G. Fornaro, G. Franceshetti, and R. Lanari, “Interferometric SAR phase unwrapping using Greens formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Franceshetti, G.

G. Fornaro, G. Franceshetti, and R. Lanari, “Interferometric SAR phase unwrapping using Greens formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Galizzi, G. E.

Ghiglia, D. C.

Goldstein, R. M.

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

Gorthi, S. S.

Gurov, I.

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

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, 055601 (2005).
[CrossRef]

Hua, Y.

Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40, 2267–2280 (1992).
[CrossRef]

Huang, H. Y. H.

Huang, L.

Jean-Marie, N.

T. Emmanuel and N. Jean-Marie, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

Julier, S. J.

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 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, 47–58 (2008).
[CrossRef]

Komori, M.

Lanari, R.

G. Fornaro, G. Franceshetti, and R. Lanari, “Interferometric SAR phase unwrapping using Greens formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

Langley, J.

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

Lion, Y.

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, 218–223 (2013).
[CrossRef]

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, 47–58 (2008).
[CrossRef]

H. Nies, O. Loffeld, and W. Robert, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.

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, 1197–1211 (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, 055601 (2005).
[CrossRef]

Lua, Y. G.

Y. G. Lua, X. Z. Wang, and X. P. Zhang, “Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map,” Optik 118, 62–66 (2007).
[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, 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, 1197–1211 (2009).
[CrossRef]

Meng, L.

Merwe, R. V.

E. A. Wan and R. V. Merwe, “The unscented Kalman Filter for nonlinear estimation,” in The IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (IEEE, 2000), pp. 153–158.

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, 47–58 (2008).
[CrossRef]

H. Nies, O. Loffeld, and W. Robert, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.

Osmanoglu, B.

Pi, Y. M.

X. M. Xie and Y. M. Pi, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5, 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, 365–372 (2011).

Pritt, M. D.

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

Quiroga, J. A.

Rajshekhar, G.

Rastogi, P.

Renotte, Y.

Robert, W.

H. Nies, O. Loffeld, and W. Robert, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.

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, 355–365 (1992).
[CrossRef]

Ruiz, P. D.

Servin, M.

Slangen, P.

Spagnolini, U.

U. Spagnolini, “2D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33, 579–589 (1994).
[CrossRef]

Su, X. Y.

Tang, B.

B. Tang, “Novel method for two-dimensional sinusoidal frequency efficient estimation,” J. Southwest Petroleum Inst. 20, 78–80 (1998).

Tian, L.

Uhlmann, J. K.

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

Vargas, J.

Volynsky, M.

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

Wan, E. A.

E. A. Wan and R. V. Merwe, “The unscented Kalman Filter for nonlinear estimation,” in The IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (IEEE, 2000), pp. 153–158.

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, 218–223 (2013).
[CrossRef]

Wang, L.

Wang, X. Z.

Y. G. Lua, X. Z. Wang, and X. P. Zhang, “Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map,” Optik 118, 62–66 (2007).
[CrossRef]

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

Wdowinski, S.

Weng, J. F.

Werner, C. L.

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

Xie, X. M.

X. M. Xie and Y. M. Pi, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5, 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, 365–372 (2011).

Xu, W.

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

Yang, P.

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, 47–58 (2008).
[CrossRef]

Zerber, H. A.

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

Zhang, Q. C.

Zhang, X. P.

Y. G. Lua, X. Z. Wang, and X. P. Zhang, “Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map,” Optik 118, 62–66 (2007).
[CrossRef]

Zhang, Z.

Zhao, M.

Zhao, Q.

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27, 1293–1301 (2009).
[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, 218–223 (2013).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Geosci. Remote Sens. (7)

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, 1197–1211 (2009).
[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, 47–58 (2008).
[CrossRef]

T. Emmanuel and N. Jean-Marie, “Improving phase unwrapping techniques by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

U. Spagnolini, “2D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33, 579–589 (1994).
[CrossRef]

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

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

G. Fornaro, G. Franceshetti, and R. Lanari, “Interferometric SAR phase unwrapping using Greens formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

IEEE Trans. Image Process. (1)

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

IEEE Trans. Signal Process. (1)

Y. Hua, “Estimating two-dimensional frequencies by matrix enhancement and matrix pencil,” IEEE Trans. Signal Process. 40, 2267–2280 (1992).
[CrossRef]

IET Radar Sonar Navig. (1)

X. M. Xie and Y. M. Pi, “Multi-baseline phase unwrapping algorithm based on the unscented Kalman filter,” IET Radar Sonar Navig. 5, 296–304 (2011).
[CrossRef]

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

J. Southwest Petroleum Inst. (1)

B. Tang, “Novel method for two-dimensional sinusoidal frequency efficient estimation,” J. Southwest Petroleum Inst. 20, 78–80 (1998).

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, 365–372 (2011).

Magn. Reson. Imaging (1)

J. Langley and Q. Zhao, “Unwrapping magnetic resonance phase maps with Chebyshev polynomials,” Magn. Reson. Imaging 27, 1293–1301 (2009).
[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, 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, 055601 (2005).
[CrossRef]

Opt. Express (5)

Opt. Lasers Eng. (1)

I. Gurov and M. Volynsky, “Interference fringe analysis based on recurrence computational algorithms,” Opt. Lasers Eng. 50, 514–521 (2012).
[CrossRef]

Optik (1)

Y. G. Lua, X. Z. Wang, and X. P. Zhang, “Weighted least-squares phase unwrapping algorithm based on derivative variance correlation map,” Optik 118, 62–66 (2007).
[CrossRef]

Proc. IEEE (1)

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 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. Zerber, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

Other (3)

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

E. A. Wan and R. V. Merwe, “The unscented Kalman Filter for nonlinear estimation,” in The IEEE 2000 Adaptive Systems for Signal Processing, Communications, and Control Symposium (IEEE, 2000), pp. 153–158.

H. Nies, O. Loffeld, and W. Robert, “Phase unwrapping using 2D-Kalman filter potential and limitations,” in IEEE International Geoscience and Remote Sensing Symposium (IEEE, 2008), paper IV1213.

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Figures (16)

Fig. 1.
Fig. 1.

Simulated wrapped phase image. (a) Noisy wrapped phase. (b) True unambiguous unwrapped phase.

Fig. 2.
Fig. 2.

BUT solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 3.
Fig. 3.

WLS solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 4.
Fig. 4.

EKFPU solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 5.
Fig. 5.

UKFPU solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 6.
Fig. 6.

TUKFPU solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 7.
Fig. 7.

Noisy wrapped phase image. (a) Noisy wrapped phase. (b) True unambiguous unwrapped phase.

Fig. 8.
Fig. 8.

TUKFPU solution. (a) Unwrapped phase. (b) Unwrapped phase error. (c) Histogram of the unwrapped phase error.

Fig. 9.
Fig. 9.

Experimental topographic wrapped phase over Etna. (a) Experimental topographic wrapped phase. (b) Phase residues map.

Fig. 10.
Fig. 10.

BUT solution for the experimental topographic wrapped phase over Etna. (a) Unwrapped phase. (b) Zoom of partial region of unwrapping failure.

Fig. 11.
Fig. 11.

WLS solution for the experimental topographic wrapped phase over Etna. (a) Unwrapped phase. (b) Rewrapped phase.

Fig. 12.
Fig. 12.

EKFPU solution for the experimental topographic wrapped phase over Etna. (a) Unwrapped phase. (b) Rewrapped phase.

Fig. 13.
Fig. 13.

TUKFPU solution for the experimental topographic wrapped phase over Etna. (a) Unwrapped phase. (b) Rewrapped phase.

Fig. 14.
Fig. 14.

UKFPU solution for the experimental topographic wrapped phase over Etna. (a) Unwrapped phase. (b) Rewrapped phase.

Fig. 15.
Fig. 15.

TUKFPU solution for whole wrapped phase over Etna. (a) Whole wrapped phase over Etna. (b) Unwrapped phase. (c) Rewrapped phase.

Fig. 16.
Fig. 16.

TUKFPU solution for the experimental topographic wrapped phase over Three Gorges area. (a) Experimental topographic wrapped phase over Three Gorges area. (b) Unwrapped phase. (c) Rewrapped phase.

Tables (3)

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Table 1. Comparison of Phase Unwrapping Accuracy

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Table 2. Run Time of Phase Unwrapping Algorithms

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Table 3. Phase Unwrapping Accuracy and Run Time with Different Sizes of the EPGE Window

Equations (20)

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x(k+1)=x(k)+gk+w(k)=f[x(k)]+w(k)y(k+1)={sin[x(k+1)]cos[x(k+1)]}+{v1(k+1)v2(k+1)}=h[x(k+1)]+V(k+1),
E[w(k)]=0;Q(k)=E[w(k)w(k)T],
V(k)={v1(k)v2(k)},E[v1(k)]=E[v2(k)]=0R(k)=E[V(k)V(k)T]=[σ(k),00,σ(k)],σ(k)=1SNRk.
y(m,n)=exp[jϕ˜(m,n)]exp{j2π[fy(m,n)m+fx(m,n)n]}+V(m,n)=exp[jϕy(m,n)m+ϕx(m,n)n)]+V(m,n),
Y(m,n)=[yΓ(Lm,Ln),yΓ(Lm,Ln+1),···,yΓ(Lm,Ln)yΓ(Lm+1,Ln),yΓ(Lm+1,Ln+1),···,yΓ(Lm+1,Ln)yΓ(Lm,Ln),yΓ(Lm,Ln+1),···,yΓ(Lm,Ln)],
Y(m,n)=UYΣYVYH,
ΣY=[σ1,0,,00,σ2,,00,0,,σLn[0](LmLn)×Ln],
Y⃗(m,n)=UYΣ⃗YVYH,Σ⃗Y=[σ⃗,0,,0[0](Lm1)×Ln],σ⃗=σ11Ln1j=2Lnσj.
X0=X(1:Lm1,1:Ln1)X1=X(2:Lm,1:Ln1)X2=X(1:Lm1,2:Ln),
X0=U0sΣ0sV0sH+U0nΣ0nV0nH,
X¯0=U0sTX0V0sX¯1=U0sTX1V0sX¯2=U0sTX2V0s.
ϕ¯y(m,n)=Arg[conj(X¯1+X¯0)]ϕ¯x(m,n)=Arg[conj(X¯2+X¯0)],
g=ϕ(a,s)ϕ(m,n)=ϕ¯y(m,n)(am)+ϕ¯x(m,n)(sn),
χi(k+1)=f[χi(k)],x⃗(k+1)=i=02wimχi(k+1)Pxx(k+1)=i=02wic[χi(k+1)x⃗(k+1)]·[χi(k+1)x⃗(k+1)]T+Q(k),
w0m=λ/(1+λ),w0c=λ/(1+λ)+(1α2+β)wic=wim=1/[2(1+λ)],λ=α2(N+κ)N.
χ0(k)=x⃗(k)χ1(k)=x⃗(k)+(1+λ)Pxx(k)χ2(k)=x⃗(k)(1+λ)Pxx(k).
ξi(k+1)=h[χi(k+1)]y⃗(k+1)=i=02Nbimξi(k+1)Pyy(k+1)=i=02Nbic[ξi(k+1)y⃗(k+1)]·[ξi(k+1)y⃗(k+1)]T+R(k+1)Pxy(k+1)=i=02Nbic[χi(k+1)x⃗(k+1)]·[ξi(k+1)y⃗(k+1)]TΠ(k+1)=Pxy(k+1)/Pyy(k+1)x⃗(k+1)=x⃗(k+1)+Π(k+1)[y(k+1)y⃗(k+1)]Pxx(k+1)=Pxx(k+1)Π(k+1)Pxy(k+1)Π(k+1)T,
χi(m,n)=(a,s)(BG)d(a,s)[χi(m,n)|(a,s)]Pxx(m,n)=i=02Nwic[χi(m,n)x⃗(m,n)][χi(m,n)x⃗(m,n)]T+(a,s)(BG)d(a,s)Q(a,s)x⃗(m,n)=i=02Nwimχi(m,n),
[χi(m,n)|(a,s)]=χi(a,s)+ϕ¯y(a,s)(ma)+ϕ¯x(a,s)(ns).
d(a,s)=[Pxx(a,s)×1SNR(a,s)]1(a,s)(BG)[Pxx(a,s)×1SNR(a,s)]1,

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