## Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity

JOSA A, Vol. 16, Issue 2, pp. 378-395 (1999)

http://dx.doi.org/10.1364/JOSAA.16.000378

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### 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

**OCIS Codes**

(120.3180) Instrumentation, measurement, and metrology : Interferometry

(280.6730) Remote sensing and sensors : Synthetic aperture radar

(350.5030) Other areas of optics : Phase

**Citation**

Igor Lyuboshenko and Henri Maître, "Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity," J. Opt. Soc. Am. A **16**, 378-395 (1999)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-2-378

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### References

- D. C. Ghiglia and L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
- Z.-P. Liang, “A model-based method for phase unwrapping,” IEEE Trans. Med. Imaging 15, 893–897 (1996).
- R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
- S. N. Madsen, H. A. Zebker, and J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sens. 31, 246–256 (1993).
- G. Fornaro, G. Franceschetti, and R. Lanari, “Interferometric phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 720–727 (1996).
- G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, “Robust phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
- H. Takajo and 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).
- G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, and M. Tesauro, “Global and local phase-unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
- E. Trouvé, J.-M. Nicolas, and H. Maître, “Improving phase unwrapping by the use of local frequency estimates,” IEEE Trans. Geosci. Remote Sens. 36, 1963–1972 (1998).
- R. Bamler, N. Adam, and 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).
- U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
- M. D. Pritt and J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
- A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (Winston, Washington, D.C. 1977).
- U. Spagnolini, “2-D phase unwrapping and instantaneous frequency estimation,” IEEE Trans. Geosci. Remote Sens. 33, 579–589 (1995).
- P. M. Morse and 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, and I. O. Miachev, Integrals and Series (Gordon & Breach, New York, 1986).
- S. Paquerault, H. Maître, and 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.
- H. A. Zebker, C. L. Werner, P. A. Rosen, and S. Hensley, “Accuracy of topographic maps derived from ERS-1 interferometric radar,” IEEE Trans. Geosci. Remote Sens. 32, 823–836 (1994).
- V. N. Strakhov and G. M. Valiashko, “Adaptive regularization algorithms for linear ill-posed problems,” Dokl. Acad. Nauk SSSR 259, 546–548 (1981) (in Russian).
- T. L. Ainsworth and 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.
- H. A. Zebker and Y. Lu, “Phase unwrapping algorithms for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
- J.-S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32, 1017–1028 (1994).
- M. S. Seymour and 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 and 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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