OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 18, Iss. 2 — Feb. 1, 2001
  • pp: 338–351

Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization

Curtis W. Chen and Howard A. Zebker  »View Author Affiliations

JOSA A, Vol. 18, Issue 2, pp. 338-351 (2001)

View Full Text Article

Acrobat PDF (623 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



Interferometric radar techniques often necessitate two-dimensional (2-D) phase unwrapping, defined here as the estimation of unambiguous phase data from a 2-D array known only modulo 2π rad. We develop a maximum <i>a posteriori</i> probability (MAP) estimation approach for this problem, and we derive an algorithm that approximately maximizes the conditional probability of its phase-unwrapped solution given observable quantities such as wrapped phase, image intensity, and interferogram coherence. Examining topographic and differential interferometry separately, we derive simple, working models for the joint statistics of the estimated and the observed signals. We use generalized, nonlinear cost functions to reflect these probability relationships, and we employ nonlinear network-flow techniques to approximate MAP solutions. We apply our algorithm both to a topographic interferogram exhibiting rough terrain and layover and to a differential interferogram measuring the deformation from a large earthquake. The MAP solutions are complete and are more accurate than those of other tested algorithms.

© 2001 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

Curtis W. Chen and Howard A. Zebker, "Two-dimensional phase unwrapping with use of statistical models for cost functions in nonlinear optimization," J. Opt. Soc. Am. A 18, 338-351 (2001)

Sort:  Author  |  Year  |  Journal  |  Reset


  1. H. Zebker and R. Goldstein, “Topographic mapping from interferometric SAR observations,” J. Geophys. Res. 91, 4993–4999 (1986).
  2. A. K. Gabriel, R. M. Goldstein, and H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
  3. R. M. Goldstein and H. A. Zebker, “Interferometric radar measurements of ocean surface currents,” Nature (London) 328, 707–709 (1987).
  4. D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
  5. D. C. Ghiglia and L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
  6. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
  7. G. Fornaro, G. Franceschetti, R. Lanari, and E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
  8. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
  9. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
  10. R. M. Goldstein, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
  11. C. W. Chen and H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
  12. J. R. Buckland, J. M. Huntley, and S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
  13. R. K. Ahuja, T. L. Magnanti, and J. B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  14. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  15. M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, Calif., 1979).
  16. G. Carballo, “Statistically-based multiresolution network flow phase unwrapping for SAR interferometry,” Ph.D. dissertation (Royal Institute of Technology, Stockholm, Sweden, 2000).
  17. R. Bamler, N. Adam, G. W. Davidson, and 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 (1998).
  18. 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).
  19. H. A. Zebker and J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
  20. 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).
  21. F. K. Li and R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
  22. E. Rodriguez and J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
  23. R. Touzi, A. Lopes, J. Bruniquel, and P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
  24. B. Guindon, “Development of a shape-from-shading technique for the extraction of topographic models from individual spaceborne SAR images,” IEEE Trans. Geosci. Remote Sens. 28, 654–661 (1990).
  25. D. J. Goering, H. Chen, L. D. Hinzman, and D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
  26. A. Lopes, E. Nezry, R. Touzi, and H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
  27. C. J. Oliver and S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).
  28. L. M. H. Ulander, “Radiometric slope correction of synthetic aperture radar images,” IEEE Trans. Geosci. Remote Sens. 34, 1115–1122 (1996).
  29. F. T. Ulaby, R. K. Moore, and A. K. Fung, Microwave Remote Sensing, Active and Passive (Addison-Wesley, London, 1981).
  30. H. A. Zebker, P. A. Rosen, and S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
  31. J. L. Kennington and R. V. Helgason, Algorithms for Network Programming (Wiley, New York, 1980).
  32. S. Pallottino, “Shortest-path methods: complexity, interrelations and new propositions,” Networks 14, 257–267 (1984).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited