Abstract

We develop a new algorithm for interferometric synthetic aperture radar (SAR) phase unwrapping based on the first Green’s identity with the Green’s function representing a series in the eigenfunctions of the two-dimensional Helmholtz homogeneous differential equation. This provides closed-form solutions with use of one- or two-dimensional fast Fourier transforms. The algorithm is elaborated by using adaptive regularization of the interferometric phase gradient estimation. To diminish underestimation of the unwrapped phase typical of the linear phase unwrapping algorithms, the bias in the measured interferometric SAR phase is calculated in terms of the probability density function of the error in the processed interferometric SAR phase.

© 1999 Optical Society of America

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References

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  1. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  2. Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
    [CrossRef] [PubMed]
  3. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  4. S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
    [CrossRef]
  5. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
    [CrossRef]
  6. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
    [CrossRef]
  7. H. Takajo, T. Takahashi, “Noniterative methods for obtaining the exact solution for the normal equation in least-squares phase estimation from phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  8. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
    [CrossRef]
  9. E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
    [CrossRef]
  10. R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
    [CrossRef]
  11. U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
    [CrossRef]
  12. M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
    [CrossRef]
  13. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C.1977).
  14. U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33, 579–589 (1995).
    [CrossRef]
  15. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  16. J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).
  17. A. P. Prudnikov, Y. A. Brychkov, I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).
  18. S. Paquerault, H. Maı̂tre, J.-M. Nicolas, “Radarclinometry for ERS-1 data mapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 503–505.
  19. H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
    [CrossRef]
  20. V. N. Strakhov, G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).
  21. T. L. Ainsworth, J. S. Lee, “A joint adaptive interferometric phase unwrapping and filtering algorithm,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 71–73.
  22. T. J. Flynn, “Phase unwrapping algorithm using discontinuity optimization,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 80–82.
  23. H. A. Zebker, Y. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
    [CrossRef]
  24. J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
    [CrossRef]
  25. M. S. Seymour, I. G. Cumming, “Maximum likelihood estimation for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. IV (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 2272–2295.
  26. O. Loffeld, C. Arndt, “Estimating the derivative of modulo-mapped phases,” in Proceedings of International Conference on Acoustics, Speech and Signal Processing, Vol. IV (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1997), pp. 2841–2844.
  27. V. I. Dmitriev, Mathematical Models in the Theory of Geophysical Fields, (Nedra, Moscow, 1990) (in Russian).

1998 (3)

E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
[CrossRef]

H. A. Zebker, Y. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
[CrossRef]

1997 (1)

1996 (3)

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
[CrossRef]

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

1995 (1)

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

1994 (3)

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[CrossRef]

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

1993 (2)

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
[CrossRef]

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

1989 (1)

1988 (2)

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

H. Takajo, T. Takahashi, “Noniterative methods for obtaining the exact solution for the normal equation in least-squares phase estimation from phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
[CrossRef]

1981 (1)

V. N. Strakhov, G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).

Adam, N.

R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
[CrossRef]

Ainsworth, T. L.

T. L. Ainsworth, J. S. Lee, “A joint adaptive interferometric phase unwrapping and filtering algorithm,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 71–73.

Arndt, C.

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

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C.1977).

Bamler, R.

R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
[CrossRef]

Brychkov, Y. A.

A. P. Prudnikov, Y. A. Brychkov, I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).

Cumming, I. G.

M. S. Seymour, I. G. Cumming, “Maximum likelihood estimation for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. IV (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 2272–2295.

Davidson, G. W.

R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
[CrossRef]

Dmitriev, V. I.

V. I. Dmitriev, Mathematical Models in the Theory of Geophysical Fields, (Nedra, Moscow, 1990) (in Russian).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Flynn, T. J.

T. J. Flynn, “Phase unwrapping algorithm using discontinuity optimization,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 80–82.

Fornaro, G.

Franceschetti, G.

Ghiglia, D. C.

Goldstein, R. M.

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

Hensley, S.

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[CrossRef]

Hoppel, K. W.

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Lanari, R.

Lee, J. S.

T. L. Ainsworth, J. S. Lee, “A joint adaptive interferometric phase unwrapping and filtering algorithm,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 71–73.

Lee, J.-S.

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Liang, Z.-P.

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

Lim, J. S.

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

Loffeld, O.

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

Lu, Y.

Madsen, S. N.

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
[CrossRef]

Mai^tre, H.

E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

S. Paquerault, H. Maı̂tre, J.-M. Nicolas, “Radarclinometry for ERS-1 data mapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 503–505.

Mango, S. A.

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Martin, J.

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
[CrossRef]

Miachev, I. O.

A. P. Prudnikov, Y. A. Brychkov, I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).

Miller, A. R.

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Nicolas, J.-M.

E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

S. Paquerault, H. Maı̂tre, J.-M. Nicolas, “Radarclinometry for ERS-1 data mapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 503–505.

Paquerault, S.

S. Paquerault, H. Maı̂tre, J.-M. Nicolas, “Radarclinometry for ERS-1 data mapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 503–505.

Pritt, M. D.

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Prudnikov, A. P.

A. P. Prudnikov, Y. A. Brychkov, I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).

Romero, L. A.

Rosen, P. A.

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[CrossRef]

Sansosti, E.

Seymour, M. S.

M. S. Seymour, I. G. Cumming, “Maximum likelihood estimation for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. IV (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 2272–2295.

Shipman, J. S.

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Spagnolini, U.

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

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

Strakhov, V. N.

V. N. Strakhov, G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).

Takahashi, T.

Takajo, H.

Tesauro, M.

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C.1977).

Trouvé, E.

E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

Valiashko, G. M.

V. N. Strakhov, G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).

Werner, C. L.

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[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.

H. A. Zebker, Y. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
[CrossRef]

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[CrossRef]

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
[CrossRef]

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

Dokl. Acad. Nauk SSSR (1)

V. N. Strakhov, G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).

Geophysics (1)

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (8)

M. D. Pritt, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

H. A. Zebker, C. L. Werner, P. A. Rosen, S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
[CrossRef]

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

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
[CrossRef]

E. Trouvé, J.-M. Nicolas, H. Maı̂tre, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
[CrossRef]

R. Bamler, N. Adam, G. W. Davidson, “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 (1998).
[CrossRef]

J.-S. Lee, K. W. Hoppel, S. A. Mango, A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
[CrossRef]

IEEE Trans. Med. Imaging (1)

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

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

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 (10)

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C.1977).

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

J. S. Lim, Two-Dimensional Signal and Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1990).

A. P. Prudnikov, Y. A. Brychkov, I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).

S. Paquerault, H. Maı̂tre, J.-M. Nicolas, “Radarclinometry for ERS-1 data mapping,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1996), pp. 503–505.

T. L. Ainsworth, J. S. Lee, “A joint adaptive interferometric phase unwrapping and filtering algorithm,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 71–73.

T. J. Flynn, “Phase unwrapping algorithm using discontinuity optimization,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. I, T. I. Stein, ed. (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1998), pp. 80–82.

M. S. Seymour, I. G. Cumming, “Maximum likelihood estimation for SAR interferometry,” in Proceedings of the International Geoscience and Remote Sensing Symposium, Vol. IV (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 2272–2295.

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

V. I. Dmitriev, Mathematical Models in the Theory of Geophysical Fields, (Nedra, Moscow, 1990) (in Russian).

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Figures (10)

Fig. 1
Fig. 1

InSAR phase region of support: (a) without a region of InSAR phase inconsistency, (b) with a region H of InSAR phase inconsistency.

Fig. 2
Fig. 2

Unwrapping error caused by a local InSAR phase gradient degradation.

Fig. 3
Fig. 3

Improved resolution of differentiation by use of adaptive regularization (solid curve) compared with conventional regularization (dotted curve). The dotted curve is the exact derivative. The original signal was degraded by Gaussian noise with variance 0.01 before computation of the derivatives.

Fig. 4
Fig. 4

(a) Mean value and (b) variance of the error in the measured InSAR phase versus absolute value of the correlation coefficient and the exact monolook InSAR phase.

Fig. 5
Fig. 5

Initial interferometric data: (a) interferogram of the Bern region as registered by ERS-1 ©CNES; the dashed lines are used to count the number of fringes per length unit; (b) coherence corresponding to the same region ©CNES. The size of both images is 256×256.

Fig. 6
Fig. 6

Unwrapping the InSAR phase by using the GF5: (a) enlargement of the unwrapped phase corresponding to the lower left quarter of the InSAR phase image in Fig. 5(a), (b) rewrapped phase, (c) residual phase; there are many residual fringes.

Fig. 7
Fig. 7

InSAR phase in Fig. 5(a) unwrapped by use of the developed method: (a) enlargement of the unwrapped phase corresponding to the lower left quarter of the InSAR phase image in Fig. 5(a), (b) rewrapped phase, (c) residual phase; the number of residual fringes is substantially diminished.

Fig. 8
Fig. 8

Initial interferometric data: (a) interferogram of the Bern region as registered by ERS-1 ©CNES, (b) coherence corresponding to the same region ©CNES. The size of both images is 512×512. InSAR phase and coherence in Fig. 5 correspond to the upper right corners of these images.

Fig. 9
Fig. 9

InSAR phase in Fig. 8(a) unwrapped by use of the developed method: (a) unwrapped phase, (b) rewrapped phase.

Fig. 10
Fig. 10

Gradient-continuation-algorithm performance: (a) initial phase, (b) phase unwrapped by use of the GF, (c) result of the gradient-continuation algorithm. Initial and reconstructed profiles along the thick solid curves through point y=25 in the x direction and through point x=49 in the y direction are shown in the background of the coordinate boxes.

Tables (1)

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Table 1 Computational Costs of Phase Unwrapping Methods

Equations (92)

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ρc=E[SASB*]{E[|SA|2]E[|SB|2]}1/2=γ exp(jθ),
ψ=(θp+eθ)2π,
ψ=argl=1MSA(l)SB*(l),
SdS(ϕ2g+ϕ·g)=Cdcϕ gnS,
g(r-r)=-12πln|r-r|,
2g(r-r)=-δ(r-r),r, rS,
ϕ(r)=SdSψ(r)·g(r-r)-Cdcϕ(r) g(r-r)nS.
12ϕ(rc)=SdSψ(r)·g(r-rc)-Cdcϕ(r) g(r-rc)nS,
NGF=3F2D(2N)+2NF1D(2N)+4(N+1)NitF1D(2N),
Γ(r, r)=m,k=0vmk20 Fmk(r)Fmk(r)vmk2,
2Fmk(r)+vmk2Fmk(r)=0,m, k(0, ).
S={(x, y) : x, y[0, a[}.
Fmk(x, y)=2a-1 cos(ξmx)cos(ηky),
vmk2=ξm2+ηk2,m, k(0, ),
Fmk(r)nSrC=0.
g˜(r, r)=4π-2m,k=0m2+k20×cos(ξmx)cos(ηky)cos(ξmx)cos(ηky)m2+k2,
g˜(r, r)nSrC=0,rS.
ϕ(r)=SdSψ(r)·g˜(r, r).
g˜(r, r)=4π-2m,k=0m2+k20N-1×cos(ξmx)cos(ηky)cos(ξmx)cos(ηky)m2+k2,
Φˆ(m, k)=T{mX(m, k)+kY(m, k)}4πN(m2+k2),
X(m, k)=Xˆ(ξm, ηk)+X˜(ξm,-ηk),
Y(m, k)=Yˆ(ξm, ηk)-Yˆ(ξm,-ηk),
ϕ(l, n)=4N2R{Φ(l, n)+Φ(l,-n)},
l, n=[0, N-1],
e(r)=8πβLNm=0N-1k=0N-1Fmk(r)×ηk sin(ηku)cos(ξmν)ξm2+ηk2sinc(ξmL),
g˜(r, r)=4π2k=0m2+k20 cos(ηky)cos(ηky)×m=0 cos(ξmx)cos(ξmx)m2+k2.
m=1 cos(mx)m2+k2=π2kcosh[k(π-x)]sinh(kπ)-12k2,
0x2π,
2m=1 cos(ξmx)cos(ξmx)m2+k2+1k2=πkfk(x, x),
fk(x, x)=1sinh(kπ)×cosh(kπ-ηkx)cosh(ηkx),x>xcosh(kπ-ηkx)cosh(ηkx),x<x.
yg˜(r, r)=-2N-1k=1N-1 cos(ηky)fk(x, x)sin(ηky),
xg˜(r, r)=-2N-1m=1N-1 cos(ξmx)fm(y, y)sin(ξmx).
ϕ(x, y)=2N-1k=1N-1 cos(ηky)0afk(x, x)rx(ηk)dx+2N-1m=1N-1 cos(ξmx)0afm(y, y)ry(ξm)dy,
rx(ηk)=ηkR{Ux(ηk)},
ry(ξm)=ξmR{Uy(ξm)},
m, k=1,, N-1.
cosh(kπ-ηkx)/sinh(kπ)exp(-ηkx),
cosh(ηkx)/sinh(kπ)exp(-kπ+ηkx).
ϕ(r)=U(r)S0dSψ(r)·g˜(r, r)+Bdcg˜(r, r) ϕ(r)nH,
U(r)=1/2,rH2/3,rB1,rS0,
F[ϕ(r)]=SdSF[ϕ(r)]·g˜(r, r)
ϕ(r)=F-1SdSF[ϕ(r)]·g˜(r, r).
Φx(η)=iηUx(η)Rx(η, α),
Φy(ξ)=iξUy(ξ)Ry(ξ, α),
Rx(η, α)=[1+αη2|Ux(η)|¯-2]-1,
Rˆx(η, α)=[1+αη2]-1.
rx(ηk)=ηkR{Ux(ηk)}Rx(ηk, α),
ry(ξm)=ξmR{Uy(ξm)}Ry(ξm, α),
m, k=1,, N-1,
p(γ, M, ϵ)=Γ(M+0.5)(1-γ2)Mβ2π1/2Γ(M)(1-β2)M+0.5+(1-γ2)M2πF(M, 1; 0.5; β2),
E{Δ|θp}=-2π sign(θp)-π-π+|θp|p(γ, M, ϵ)dϵ,
D{Δ|θp}=D{γ, M}-E{Δ|θp}2-2πE{Δ||θp|}+4π-π-π+|θp|ϵp(γ, M, ϵ)dϵ,
D{γ, M}CR=1-γ22Mγ2.
4.γ=0, |θp(0, π):p(0, M, ϵ)=1/2π, E{Δ|θp}=-θp, D{Δ|θp}=π2/3;
γ=1, |θp|(0, π):p(1, M, ϵ)=δ(ϵ), E{Δ|θp}=0, D{Δ|θp}=0.
ψˆ=ψ-E{Δ|θp},
g˜x(r, r)=-2π-1m=0N-1k=0N-1Fmk(r)cos(ηky)m2+k2×m sin(ξmx),
g˜y(r, r)=-2π-1m=0N-1k=0N-1Fmk(r)cos(ξmx)m2+k2×k sin(ηky),
ϕ(x, y)=-2π-1m=0N-1k=0N-1Fmk(r)m2+k2×mS xψ(r)sin(ξmx)×cos(ηky)dS+kS yψ(r)×cos(ξmx)sin(ηky)dS.
R(m, k)=12Sm xψ(r)+k yψ(r)×sin(ξmx+ηky)dS+12Sm xψ(r)-k yψ(r)×sin(ξmx-ηky)dS.
Φˆ(m, k)=-R(m, k)2πN(m2+k2)=F{mX(m, k)+kY(m, k)}4πN(m2+k2)
X(m, k)=Xˆ(ξm, ηk)+Xˆ(ξm,-ηk),
Y(m, k)=Yˆ(ξm, ηk)-Yˆ(ξm,-ηk),
ϕ(l, n)=4N2R[Φ(l, n)+Φ(l,-n)],
l, n=[0, N-1],
xψ(x, y)=xϕ(x, y),
yψ(x, y)=yϕ(x, y)-2πβδ(y-u)×rectx-ν2L.
e(r)=4βm=0N-1k=0N-1 kFmk(r)m2+k2Sδ(y-u)×rectx-ν2Lcos(ξmx)sin(ηky)dS=4βm=0N-1k=0N-1 kFmk(r)m2+k2sin(ηku)×ν-Lν+L cos(ξmx)dx=8πβLNm=0N-1k=0N-1Fmk(r)×ηk sin(ηku)cos(ξmν)ξm2+ηk2sinc(ξmL),
NGF=14N2 log2(2N)+Nit(4N2+4N)×log2(2N)+2(2N)2N2({14+4Nit}log2 N+22+4Nit),
NHF1=4N2 log2(2N)2+4(2N)2+2N2 log2(2N)2=N2(12 log2 N+28).
ϕ(1)(x, y)=2N-1k=1N-1fk(1)(x)cos(ηky)×0xfk(2)(x)rx(ηk)dx+2N-1k=1N-1fk(2)(x)cos(ηky)×xafk(1)(x)rx(ηk)dx,
fk(1)(x)=[sinh(kπ)]-1/2 cosh(kπ-ηkx),
fk(2)(x)=[sinh(kπ)]-1/2 cosh(ηkx).
NHF2=2N2(log2 N+1)+8N2+2N2(log2 N+1)+2N2=N2(4 log2 N+14).
ϕ(r)=S0dSψ(r)·g˜(r, r)-Bdcϕ(r) g˜(r, r)nS,
g˜(r, r)nS=-g˜(r, r)nH,rB,rS0,
ϕ(r)=S0dSψ(r)·g˜(r, r)+Bdcϕ(r) g˜(r, r)nH.
HdSϕ(r)2g˜(r, r)=-ϕ(r)forrH-0.5ϕ(r)forrB0forrS0,
Bdcϕ(r) g˜(r, r)nH
=G(r)-ϕ(r)forrHG(r)-0.5ϕ(r)forrBG(r)forrS0,
G(r)=Bdcg˜(r, r) ϕ(r)nH-HdSg˜(r, r)2ϕ(r).
ψ=δ+2πforδ(-2π,-π]δforδ(-π, π]δ-2πforδ(π, 2π].
pψ(ϵ)=-2π2πpψ|δ(ϵ|u)pδ(u)du=[pδ(ϵ-2π)+pδ(ϵ)+pδ(ϵ+2π)]rectϵ2π.
pψ(ϵ)=[p(γ, M, ϵ-θp-2π)+p(γ, M, ϵ-θp)+p(γ, M, ϵ-θp+2π)]rectϵ2π.
pΔ|θp(ϵ)=[p(γ, M, ϵ-2π)+p(γ, M, ϵ)+p(γ, M, ϵ+2π)]rectϵ+θp2π.
E{Δ|θp}=-θp-π-θp+πϵpΔ|θp(ϵ)dϵ.
D{Δ|θp}=-θp-π-θp+π(ϵ-E)2pΔ|θp(ϵ)dϵ.
D{Δ|θp}=D(1){Δ|θp}+D(2){Δ|θp}+D(3){Δ|θp},
D(1){Δ|θp}=-θp-π-θp+π(ϵ-E)2p(γ, M, ϵ-2π)dϵ=0,
D(2){Δ|θp}=-θp-π-θp+π(ϵ-E)2p(γ, M, ϵ)dϵ=-π-θp+π(ϵ-E)2p(γ, M, ϵ)dϵ,
D(3){Δ|θp}=-θp-π-θp+π(ϵ-E)2p(γ, M, ϵ+2π)dϵ=-θp+ππ(ϵ-2π-E)2p(γ, M, ϵ)dϵ=-θp+ππ(ϵ-E)2p(γ, M, ϵ)dϵ-2(π+E)E+4π-π-π+θpϵp(γ, M, ϵ)dϵ.
D{Δ|θp}=-ππ(ϵ-E)2p(γ, M, ϵ)dϵ-2(π+E)E+4π-π-π+θpϵp(γ, M, ϵ)dϵ=-ππϵ2p(γ, M, ϵ)dϵ-2E-ππϵp(γ, M, ϵ)dϵ+E2-ππp(γ, M, ϵ)dϵ-2(π+E)E+4π-π-π+θpϵp(γ, M, ϵ)dϵ.

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