Abstract

The paper attacks absolute phase estimation with a two-step approach: the first step applies an adaptive local denoising scheme to the modulo-2π noisy phase; the second step applies a robust phase unwrapping algorithm to the denoised modulo-2π phase obtained in the first step. The adaptive local modulo-2π phase denoising is a new algorithm based on local polynomial approximations. The zero-order and the first-order approximations of the phase are calculated in sliding windows of varying size. The zero-order approximation is used for pointwise adaptive window size selection, whereas the first-order approximation is used to filter the phase in the obtained windows. For phase unwrapping, we apply the recently introduced robust (in the sense of discontinuity preserving) PUMA unwrapping algorithm [IEEE Trans. Image Process. 16, 698 (2007)] to the denoised wrapped phase. Simulations give evidence that the proposed algorithm yields state-of-the-art performance, enabling strong noise attenuation while preserving image details.

© 2008 Optical Society of America

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References

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  1. L. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763-768 (1974).
    [CrossRef]
  2. H. Zebker and R. Goldstein, “Topographic mapping from interferometric synthetic aperture radar,” J. Geophys. Res. 91, 4993-4999 (1986).
    [CrossRef]
  3. P. Lauterbur, “Image formation by induced local interactions: examples employing nuclear magnetic resonance,” Nature 242, 190-191 (1973).
    [CrossRef]
  4. M. Hedley and D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177-181 (1992).
    [CrossRef] [PubMed]
  5. T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).
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    [CrossRef]
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    [CrossRef] [PubMed]
  8. D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (John Wiley & Sons, 1998).
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    [CrossRef]
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    [CrossRef] [PubMed]
  11. J. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393-2400(1995).
    [CrossRef]
  12. T. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692-2701(1997).
    [CrossRef]
  13. M. Costantini, “A novel phase unwrapping method based on network programing,” IEEE Trans. Geosci. Remote Sens. 36, 813-821 (1998).
    [CrossRef]
  14. M. Rivera, J. Marroquin, and R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381-8390 (1997).
    [CrossRef]
  15. J. Bioucas-Dias and J. Leitao, “The Z?M algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
    [CrossRef]
  16. J. Bioucas-Dias and G. Valadão, “Phase unwrapping via graph cuts,” IEEE Trans. Image Process. 16, 698-709 (2007).
    [CrossRef]
  17. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
    [CrossRef]
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    [CrossRef]
  19. M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917-1923 (1998).
    [CrossRef]
  20. Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imag. 15, 893-897 (1996).
    [CrossRef]
  21. V. Pascazio and G. Schirinzi, “Multifrequency InSAR height reconstruction through maximum likelihood estimation of local planes parameters,” IEEE Trans. Image Process. 11, 1478-1489 (2002).
    [CrossRef]
  22. M. Servin, F. J. Cuevas, D. Malacara, J. L. Marroguin, and R. Rodriguez-Vera, “Phase unwrapping through demodulation by use of the regularized phase-tracking technique,” Appl. Opt. 38, 1934-1941 (1999).
    [CrossRef]
  23. M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, 373-426.
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    [CrossRef]
  25. Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
    [CrossRef] [PubMed]
  26. J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
    [CrossRef]
  27. V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
    [CrossRef]
  28. L. L. Scharf, Statistical Signal Processing, Detection Estimation and Time Series Analysis (Addison-Wesley, 1991).

2008

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
[CrossRef]

2007

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

A. Patil and P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253-257 (2007).
[CrossRef]

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

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

2006

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

2005

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

2003

2002

J. Bioucas-Dias and J. Leitao, “The Z?M algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

V. Pascazio and G. Schirinzi, “Multifrequency InSAR height reconstruction through maximum likelihood estimation of local planes parameters,” IEEE Trans. Image Process. 11, 1478-1489 (2002).
[CrossRef]

2001

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, 373-426.

1999

1998

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

M. Servin, J. L. Marroquin, D. Malacara, and F. J. Cuevas, “Phase unwrapping with a regularized phase-tracking system,” Appl. Opt. 37, 1917-1923 (1998).
[CrossRef]

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (John Wiley & Sons, 1998).

1997

1996

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imag. 15, 893-897 (1996).
[CrossRef]

1995

J. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393-2400(1995).
[CrossRef]

1992

M. Hedley and D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177-181 (1992).
[CrossRef] [PubMed]

1991

L. L. Scharf, Statistical Signal Processing, Detection Estimation and Time Series Analysis (Addison-Wesley, 1991).

1988

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

1986

H. Zebker and R. Goldstein, “Topographic mapping from interferometric synthetic aperture radar,” J. Geophys. Res. 91, 4993-4999 (1986).
[CrossRef]

1982

1974

L. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763-768 (1974).
[CrossRef]

1973

P. Lauterbur, “Image formation by induced local interactions: examples employing nuclear magnetic resonance,” Nature 242, 190-191 (1973).
[CrossRef]

Astola, J.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
[CrossRef]

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

Bioucas-Dias, J.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

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

J. Bioucas-Dias and J. Leitao, “The Z?M algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

Chang, S. W.

Costantini, M.

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

Cuevas, F. J.

Egiazarian, K.

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
[CrossRef]

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

Feng, L.

Flynn, T.

Gauthier, P.

Ghiglia, D.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (John Wiley & Sons, 1998).

Goldstein, R.

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

H. Zebker and R. Goldstein, “Topographic mapping from interferometric synthetic aperture radar,” J. Geophys. Res. 91, 4993-4999 (1986).
[CrossRef]

Graham, L.

L. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763-768 (1974).
[CrossRef]

Hedley, M.

M. Hedley and D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177-181 (1992).
[CrossRef] [PubMed]

Hong, C. K.

Itoh, K.

Katkovnik, V.

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
[CrossRef]

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

Kemao, Q.

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Q. Kemao, L. T. H. Nam, L. Feng, and S. H. Soon, “Comparative analysis on some filters for wrapped phase maps,” Appl. Opt. 46, 7412-7418 (2007).
[CrossRef] [PubMed]

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

Kujawinska, M.

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, 373-426.

Lauterbur, P.

P. Lauterbur, “Image formation by induced local interactions: examples employing nuclear magnetic resonance,” Nature 242, 190-191 (1973).
[CrossRef]

Leitao, J.

J. Bioucas-Dias and J. Leitao, “The Z?M algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

Liang, Z.-P.

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imag. 15, 893-897 (1996).
[CrossRef]

Malacara, D.

Marroguin, J. L.

Marroquin, J.

M. Rivera, J. Marroquin, and R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381-8390 (1997).
[CrossRef]

J. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393-2400(1995).
[CrossRef]

Marroquin, J. L.

Nam, L. T. H.

Pascazio, V.

V. Pascazio and G. Schirinzi, “Multifrequency InSAR height reconstruction through maximum likelihood estimation of local planes parameters,” IEEE Trans. Image Process. 11, 1478-1489 (2002).
[CrossRef]

Patil, A.

A. Patil and P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253-257 (2007).
[CrossRef]

Pritt, M.

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (John Wiley & Sons, 1998).

Rastogi, P.

A. Patil and P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253-257 (2007).
[CrossRef]

Rivera, M.

M. Rivera, J. Marroquin, and R. Rodriguez-Vera, “Fast algorithm for integrating inconsistent gradient fields,” Appl. Opt. 36, 8381-8390 (1997).
[CrossRef]

J. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393-2400(1995).
[CrossRef]

Rodriguez-Vera, R.

Rosenfeld, D.

M. Hedley and D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177-181 (1992).
[CrossRef] [PubMed]

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing, Detection Estimation and Time Series Analysis (Addison-Wesley, 1991).

Schirinzi, G.

V. Pascazio and G. Schirinzi, “Multifrequency InSAR height reconstruction through maximum likelihood estimation of local planes parameters,” IEEE Trans. Image Process. 11, 1478-1489 (2002).
[CrossRef]

Servin, M.

Soon, S. H.

Stetson, K.

Valadão, G.

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

Wahid, J.

Werner, C.

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

Yun, H. Y.

Zebker, H.

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

H. Zebker and R. Goldstein, “Topographic mapping from interferometric synthetic aperture radar,” J. Geophys. Res. 91, 4993-4999 (1986).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

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

IEEE Trans. Image Process.

J. Bioucas-Dias and J. Leitao, “The Z?M algorithm: a method for interferometric image reconstruction in SAR/SAS,” IEEE Trans. Image Process. 11, 408-422 (2002).
[CrossRef]

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

V. Pascazio and G. Schirinzi, “Multifrequency InSAR height reconstruction through maximum likelihood estimation of local planes parameters,” IEEE Trans. Image Process. 11, 1478-1489 (2002).
[CrossRef]

V. Katkovnik, J. Astola, and K. Egiazarian, “Phase local approximation (PhaseLa) technique for phase unwrap from noisy data,” IEEE Trans. Image Process. 17, 833-846 (2008).
[CrossRef]

IEEE Trans. Med. Imag.

Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imag. 15, 893-897 (1996).
[CrossRef]

J. Geophys. Res.

H. Zebker and R. Goldstein, “Topographic mapping from interferometric synthetic aperture radar,” J. Geophys. Res. 91, 4993-4999 (1986).
[CrossRef]

J. Opt. Soc. Am.

J. Marroquin and M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. 12, 2393-2400(1995).
[CrossRef]

J. Opt. Soc. Am. A

Magn. Reson. Med.

M. Hedley and D. Rosenfeld, “A new two-dimensional phase unwrapping algorithm for MRI images,” Magn. Reson. Med. 24, 177-181 (1992).
[CrossRef] [PubMed]

Nature

P. Lauterbur, “Image formation by induced local interactions: examples employing nuclear magnetic resonance,” Nature 242, 190-191 (1973).
[CrossRef]

Opt. Lasers Eng.

A. Patil and P. Rastogi, “Moving ahead with phase,” Opt. Lasers Eng. 45, 253-257 (2007).
[CrossRef]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

Proc. IEEE

L. Graham, “Synthetic interferometer radar for topographic mapping,” Proc. IEEE 62, 763-768 (1974).
[CrossRef]

Radio Sci.

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

Other

V. Katkovnik, K. Egiazarian, and J. Astola, Local Approximation Techniques in Signal and Image Processing (SPIE, 2006).
[CrossRef]

D. Ghiglia and M. Pritt, Two-Dimensional Phase Unwrapping. Theory, Algorithms, and Software (John Wiley & Sons, 1998).

T. Kreis, Handbook of Holographic Interferometry: Optical and Digital Methods (Wiley-VCH, 2005).

J. Bioucas-Dias, V. Katkovnik, J. Astola, and K. Egiazarian, “Adaptive local phase approximations and global unwrapping,” in 3DTV Conference: The True Vision -- Capture, Transmission and Display of 3D Video (IEEE, 2008), pp. 253-258.
[CrossRef]

M. Servin and M. Kujawinska, “Modern fringe pattern analysis in interferometry,” Handbook of Optical Engineering, D. Malacara and B. J. Thompson, eds. (Dekker, 2001), Chap. 12, 373-426.

L. L. Scharf, Statistical Signal Processing, Detection Estimation and Time Series Analysis (Addison-Wesley, 1991).

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Figures (9)

Fig. 1
Fig. 1

Illustration of the observed phase model (1): φ is the true phase, ϕ is the observed phase, and ϕ n is the phase component of ϕ due to noise vector n.

Fig. 2
Fig. 2

PEARLS estimation results for a Gaussian shaped surface with σ = 0.5 corresponding to SNR = 3 dB . The window size parameter h shown in part (b) illustrates the ICI ability to locally adapt the amount of smoothness: the larger windows are selected in areas were a first-order polynomial is a good approximation to the data and vice versa. The denoised wrapped phase, shown in part (c) is clearly cleaner; the SNR improvement is of 10.8 dB . The error of the reconstructed image in part (d) is RMSE = 0.15 rad .

Fig. 3
Fig. 3

As in Fig. 2, for the random surface shown in part (a), the denoised wrapped phase, shown in part (c), is clearly cleaner; the SNR improvement is ISNR = 4.8 dB . The error of the reconstructed image in part (d) is RMSE = 0.33 rad .

Fig. 4
Fig. 4

As in Fig. 2, for the Gaussian shaped surface with a quarter set to zero shown in part (a), the noise variance is set to σ = 0.5 corresponding to SNR = 3 dB .

Fig. 5
Fig. 5

Impact of denoising on PUMA unwrapping.

Fig. 6
Fig. 6

Contour map (rad) of the terrain used to generate the InSAR data. The surface, a digital terrain elevation model of mountainous terrain around Longs Peak, Colorodo has been resampled in the SAR slant plane; it can be therefore directly compared with the estimated surfaces. (Data distributed with [8]).

Fig. 7
Fig. 7

Results obtained on a simulated SAR data based on a real surface with quality maps. Parts (a), (b), and (c) are as in Fig. 2. The improvement in the SNR of estimate shown in part (c) is ISNR = 4 dB . Part (d) shows the histogram of the absolute phase error in the set X X 0 . The correspondent estimation error is RMSE = 0.2 rad .

Fig. 8
Fig. 8

Quality map relative to the interferogram shown in Fig. 7, part (a), computed using the phase derivative and thresholding procedures presented in [8], Chap. 3. Black color signals pixels where the interferogram is of low quality [8].

Fig. 9
Fig. 9

True and estimated phases. The values of the estimates phase in the set X 0 were extrapolated from the neighbors in X X 0 .

Tables (1)

Tables Icon

Table 1 RMSE (in rad.) for PEARLS, PhaseLa, and Z π M Algorithms

Equations (19)

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

z = A exp ( j φ ) + n , A > 0 ,
ϕ = angle ( z ) = W ( φ + ϕ n ) , ϕ [ π , π ) ,
φ ( u , v | c ) = p T ( u , v ) c ,
φ ( x + x s , y + x s ) φ ( x s , y s | c ) .
c ^ = arg min c L h ( c ) ,
L h ( c ) = 1 2 s w h , s | z ϕ ( x + x s , y + y s ) exp ( j φ ( x s , y s | c ) | 2 = s w h , s { 1 cos [ ϕ ( x + x s , y + y s ) φ ( x s , y s | c ) ] } ,
( c ^ 2 , c ^ 3 ) arg max c 2 , c 3 | F h ( c 2 , c 3 ) | ,
c ^ 1 = angle F h ( c ^ 2 , c ^ 3 ) ,
F h ( c 2 , c 3 ) = s w h , s z ϕ ( x + x s , y + y s ) e j ( c 2 x s + c 3 y s ) .
φ ^ h ( x , y ) = c ^ 1 ( x , y ) ,
φ ^ h ( x , y ) = c ^ 1 ( x , y ) = angle [ F h ( 0 , 0 ) ] .
F h ( c 2 , c 3 ) = s w h , s z ϕ ( x + x s , y + y s ) e j ( c 2 x s + c 3 y s ) = e j c 1 s w h , s e j [ ( c 2 c 2 ) x s + ( c 3 c 3 ) y s ] = e j c 1 W h [ ( c 2 c 2 ) x s + ( c 3 c 3 ) y s ) ] ,
φ ^ h ( x , y ) = angle [ F h ( 0 , 0 ) ] = c 1 + angle [ W h ( c 2 , c 3 ) ] ;
φ ^ h ( x , y ) = angle [ F h ( c ^ 2 , c ^ 3 ) ] = c 1 .
σ h 2 = σ 2 s w h , s 2 ( s w h , s ) 2 .
Q h = { φ ^ h Γ · σ h , φ ^ h + Γ · σ h } ,
φ ^ h ( x , y ) = angle [ s w h , s z ϕ ( x + x s , y + y s ) ] ;
ISNR = 10 log 10 e j ϕ e j φ 2 e j φ ^ h + e j φ 2 ,
min φ i , i X 0 p q ( φ p φ q ) 2 subject to :     φ i = φ ^ i , i X X 0 ,

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