Abstract

The effect of noise-induced phase inconsistency on interferometric phase information content is studied. Phase inconsistencies, or residues, hinder a correct unwrapping of the phase signal. The probability of noise-induced phase inconsistencies is obtained as a function of the signal-to-noise ratio. Moreover, a two-dimensional noise-residue filter, intended to be applied as a preprocessing step before phase unwrapping, is proposed. The method is based on the observation that the noise creates adjacent phase inconsistencies mainly in interferometric phase images. A local analysis leads to a set of rules to be applied to reduce noise-induced phase inconsistencies. The filter performances are tested on noisy synthetic and real phase data.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Creath, “Phase-measurement in interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, pp. 351–393.
  2. C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).
  3. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  4. K. Itoh, “Analysis of the phase unwrapping algorithm,” Appl. Opt. 21, 2470 (1982).
    [CrossRef] [PubMed]
  5. Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic aperture radar interferometry,” Appl. Opt. 33, 201–208 (1994).
    [CrossRef] [PubMed]
  6. P. G. Charette, I. W. Hunter, “Robust phase unwrapping method for phase images,” Appl. Opt. 35, 3506–3513 (1996).
    [CrossRef] [PubMed]
  7. J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
    [CrossRef]
  8. R. M. Goldstein, C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
    [CrossRef]
  9. D. Just, R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
    [CrossRef] [PubMed]
  10. R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. London, Ser. A 449, 567–589 (1995).
    [CrossRef]
  11. C. Rothjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997–2008 (1995).
    [CrossRef]
  12. K. Po Ho, J. M. Kahn, “Exact probability-density function for phase-measurement interferometry,” J. Opt. Soc. Am. A 12, 1984–1989 (1995).
    [CrossRef]
  13. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  14. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  15. O. Loffeld, C. Arndt, “Estimating the derivative of the modulo-mapped phases,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Vol. 4 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2841–2844.
  16. R. Cusack, J. M. Huntley, A. T. Goldrein, “Improved noise-immune phase unwrapping algorithm,” Appl. Opt. 34, 781–789 (1995).
    [CrossRef] [PubMed]
  17. B. Friedlander, J. M. Francos, “Model-based phase method for unwrapping,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
    [CrossRef]
  18. L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia, “A new regularization scheme for phase unwrapping,” Appl. Opt. 37, 3053–3058 (1998).
    [CrossRef]
  19. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
    [CrossRef]
  20. R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
    [CrossRef]
  21. D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.
  22. D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

1998 (3)

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

R. M. Goldstein, C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia, “A new regularization scheme for phase unwrapping,” Appl. Opt. 37, 3053–3058 (1998).
[CrossRef]

1996 (4)

P. G. Charette, I. W. Hunter, “Robust phase unwrapping method for phase images,” Appl. Opt. 35, 3506–3513 (1996).
[CrossRef] [PubMed]

B. Friedlander, J. M. Francos, “Model-based phase method for unwrapping,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

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

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

1995 (4)

1994 (2)

1991 (1)

1988 (1)

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

1982 (1)

Adam, N.

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

Ainsworth, T. L.

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Arndt, C.

O. Loffeld, C. Arndt, “Estimating the derivative of the modulo-mapped phases,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Vol. 4 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2841–2844.

Bamler, R.

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

D. Just, R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
[CrossRef] [PubMed]

D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

Blacknell, D.

R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. London, Ser. A 449, 567–589 (1995).
[CrossRef]

Bone, D. J.

Charette, P. G.

Chiarantini, L.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Coppo, P.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Creath, K.

K. Creath, “Phase-measurement in interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, pp. 351–393.

Cusack, R.

Davidson, G. W.

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

Eichel, P. H.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

Francos, J. M.

B. Friedlander, J. M. Francos, “Model-based phase method for unwrapping,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Friedlander, B.

B. Friedlander, J. M. Francos, “Model-based phase method for unwrapping,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

Ghiglia, D. C.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

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

Goldrein, A. T.

Goldstein, R. M.

R. M. Goldstein, C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

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

Grunes, M. R.

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Guerriero, L.

Hunter, I. W.

Huntley, J. M.

Itoh, K.

Jakowatz, C. V.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

Just, D.

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

D. Just, R. Bamler, “Phase statistics of interferograms with applications to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
[CrossRef] [PubMed]

D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

Kahn, J. M.

Lee, J. J.

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Lin, Q.

Loffeld, O.

O. Loffeld, C. Arndt, “Estimating the derivative of the modulo-mapped phases,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Vol. 4 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2841–2844.

Luni, G.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Nico, G.

Papathanassiou, K. P.

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Pasquariello, G.

Po Ho, K.

Pritt, M. D.

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

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

Quegan, S.

R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. London, Ser. A 449, 567–589 (1995).
[CrossRef]

Reigber, A.

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

Rothjen, C.

Rudolf, H.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Schwäbisch, M

D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

Sieber, A. J.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Stramaglia, S.

Tarchi, D.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

Thompson, P. A.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

Tough, R. J. A.

R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. London, Ser. A 449, 567–589 (1995).
[CrossRef]

Vesecky, J. F.

Wahl, D. E.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

Werner, C. L.

R. M. Goldstein, C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

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

Zebker, H. A.

Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic aperture radar interferometry,” Appl. Opt. 33, 201–208 (1994).
[CrossRef] [PubMed]

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

Appl. Opt. (7)

Geophys. Res. Lett. (1)

R. M. Goldstein, C. L. Werner, “Radar interferogram filtering for geophysical applications,” Geophys. Res. Lett. 25, 4035–4038 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (3)

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

R. Bamler, N. Adam, G. W. Davidson, D. Just, “Noise-induced slope distortion in 2-D phase unwrapping by linear estimators with application to SAR interferometry,” IEEE Trans. Geosci. Remote Sens. 36, 913–921 (1996).
[CrossRef]

J. J. Lee, K. P. Papathanassiou, T. L. Ainsworth, M. R. Grunes, A. Reigber, “A new technique for noise filtering of SAR interferometric phase images,” IEEE Trans. Geosci. Remote Sens. 36, 1456–1465 (1998).
[CrossRef]

IEEE Trans. Signal Process. (1)

B. Friedlander, J. M. Francos, “Model-based phase method for unwrapping,” IEEE Trans. Signal Process. 44, 2999–3007 (1996).
[CrossRef]

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

Proc. R. Soc. London, Ser. A (1)

R. J. A. Tough, D. Blacknell, S. Quegan, “A statistical description of polarimetric and interferometric synthetic aperture radar data,” Proc. R. Soc. London, Ser. A 449, 567–589 (1995).
[CrossRef]

Radio Sci. (1)

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

Other (6)

O. Loffeld, C. Arndt, “Estimating the derivative of the modulo-mapped phases,” in Proceedings of International Conference on Acoustics, Speech, and Signal Processing, Vol. 4 (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2841–2844.

D. Tarchi, H. Rudolf, G. Luni, L. Chiarantini, P. Coppo, A. J. Sieber, “SAR interferometry for structural changes detection: a demonstration test on a dam,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1999), pp. 592–594.

D. Just, N. Adam, M Schwäbisch, R. Bamler, “Comparison of phase unwrapping algorithms for SAR interferograms,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. 1, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1995), pp. 767–769.

K. Creath, “Phase-measurement in interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, New York, 1988), Vol. 26, pp. 351–393.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Boston, 1996).

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 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 (8)

Fig. 1
Fig. 1

Probability density functions of (a) the interferometric phase for the case f=0 and (b) the interferometric phase derivative for the case f1=f2=0, corresponding to different coherence values. From the upper curve to the lower one, γ=0.9, 0.7, 0.5, 0.3, and 0.0. The phase unit is in radians.

Fig. 2
Fig. 2

The circulations [Eq. (9)], calculated on a noisy phase image with γ=0.6, are depicted as elements of the three-dimensional space [-2, 2]3. The three independent phase gradients are expressed in π units. The volume occupied by all the phase gradient triads is depicted in (a), whereas only those that do not cause residues are depicted in (b). Of all phase gradient triads giving rise to residues, those having three or one g in module greater than π are shown in (c) and (d), respectively.

Fig. 3
Fig. 3

Residue probability as a function of the coherence γ: calculated curves and Monte Carlo simulations. See Subsection 2.B for details.

Fig. 4
Fig. 4

Phase configuration characterized by (a) two adjacent opposite-sign residues and (b) two diagonal opposite-sign residues.

Fig. 5
Fig. 5

Unwrapping of the original and filtered phase signal for different coherence values. LMS solution: (a) γ=0.8, (c) γ=0.7, (e) γ=0.6, and (g) γ=0.5. Goldstein’s solution: (b) γ=0.8, (d) γ=0.7, (f) γ=0.6, and (h) γ=0.5. The diagonal lines fi,i from the unwrapping of the signal f after the filtering (continuous curves) are compared with those unwrapping before the filtering (dashed curves). The original absolute phase signal is depicted in all figures.

Fig. 6
Fig. 6

(a) Real radar interferogram with a large patch of decorrelation; residue map (b) before the filtering and (c) after the filtering.

Fig. 7
Fig. 7

LMS solution (shown rewrapped) of the interferogram in Fig. 6: (a) before and (b) after the filtering.

Fig. 8
Fig. 8

Interferogram and the corresponding residue map: (a) and (b) original noisy data; (c) and (d) complex average filter; (e) and (f) sigma filter; and (g) and (h) proposed filter.

Tables (1)

Tables Icon

Table 1 Twenty Rules Needed to Correct the Phase Gradient Field Locally to Eliminate Adjacent Noise-Induced Residues

Equations (48)

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

W:fRg(-π, π].
g=W[f+n],
SNR=k22σn2,
γ=E(z1z2*)[E(|z1|2)E(|z2|2)]1/2,
ˆf(i, j)=f(i, j)+n(i, j),
ˆf(i, j)=W{ig(i, j)}W{jg(i, j)}.
׈f(i, j)=×n(i, j)=^jf(i+1, j)-^jf(i, j)-^if(i, j+1)+^if(i, j)0.
pg|f(g)=12π1-γ21-γ2cos(g-f )×1+γ cos(g-f )arccos[-γ cos(g-f )][1-γ2cos(g-f )]1/2
r(i, j)=׈f(i, j)=12πW{ig(i, j)}+W{jg(i+1, j)}-W{ig(i, j+1)}-W{jg(i, j)}.
pf1,f2(Δ)=-+pg1|f1(g1)pg2|f3(Δ+g1)dg1=pg1|f1(-g1)*pg2|f2(g2),
pf2,f3(Δ2|Δ1)=pf2(-g2|Δ1) * pg3|f3(g3),
pf2(-g2|Δ1)=pg2|f2(-g2),max(-π, -π+Δ1)g2min(π, π+Δ1)0,otherwise.
pf1,f2,f3(Δ1, Δ2)=pf2,f3(Δ2|Δ1)pf1, f2(Δ1).
pf1,f2,f3,f4(Δ1, Δ2, Δ3)
=pf3,f4(Δ3|Δ1, Δ2)pf1,f2,f3(Δ1, δ2),
pf3, f4(Δ3|Δ1, Δ2)=pf3(-g3|Δ1, Δ2)*pg4|f4(g4),
pf3(-g3|Δ1, Δ2)
=pg3|f3(-g3),for gIg3gIIifΔ10gIIIg3gIVifΔ1>00,otherwise
gI=max(-π+Δ2,-π),
gII=min(π+Δ1+Δ2, π),
gIII=max(-π+Δ1+Δ2, -π),
gIV=min(π+Δ2, π).
pf1,f2(Δ)=-+dmexp{imΔ/2}.
dm1NΦ(m)0mN/2Φ(N-m)-N/2mN-1,
pf1,f2(n)=pf1,f2(Δ=nξ),
pf1,f2(n)=4πN pf1(-n)*pf2(n).
pf1,f2(n)=4πNl=0N-1pf1(l)pf2(l+n).
F{pf1,f2}=4πNF{pf1}F{pf2}.
Δ1a=g(2)-g(1),Δ1b=g(3)-g(2),
Δ2a=g(5)-g(2),Δ2b=g(6)-g(3),
Δ3a=g(4)-g(5),Δ3b=g(5)-g(6),
Δ4a=g(1)-g(5),Δ4b=g(2)-g(5).
Ra:W[Δ1a]+W[Δ2a]+W[δ3a]+W[Δ4a]=2π,
Rb:W[Δ1b]+W[Δ2b]+W[Δ3b]+W[Δ4b]=-2π.
W[Δ2a]W[Δ2a]+2π,W[Δ4b]W[Δ4b]-2π.
Δ1=g(2)-g(1),Δ5=g(6)-g(3),
Δ2=g(3)-g(2),Δ6=g(4)-g(5),
Δ3=g(1)-g(4),Δ7=g(5)-g(6),
Δ4=g(5)-g(2),Δ8=g(4)-g(7),
Δ9=g(8)-g(5),
Δ10=g(9)-g(6),
Δ11=g(7)-g(8),
Δ12=g(8)-g(9).
R0 : W[Δ1]+W[Δ4]+W[Δ6]+W[Δ3]=0,
R : W[Δ2]+W[Δ5]+W[Δ7]-W[Δ4]=-2π,
R : -W[Δ6]+W[Δ9]+W[Δ11]+W[Δ8]=2π,
R0 : -W[Δ7]+W[Δ10]+W[Δ12]-W[Δ9]=0,
W[Δ4]W[Δ4]+2π,W[Δ6]W[Δ6]-2π.

Metrics