Abstract

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 a posteriori 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

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  1. H. Zebker, R. Goldstein, “Topographic mapping from interferometric SAR observations,” J. Geophys. Res. 91, 4993–4999 (1986).
    [CrossRef]
  2. A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
    [CrossRef]
  3. R. M. Goldstein, H. A. Zebker, “Interferometric radar measurements of ocean surface currents,” Nature (London) 328, 707–709 (1987).
    [CrossRef]
  4. D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
    [CrossRef]
  5. D. C. Ghiglia, 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).
    [CrossRef]
  6. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sens. 34, 728–738 (1996).
    [CrossRef]
  7. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
    [CrossRef]
  8. T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
    [CrossRef]
  9. M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
    [CrossRef]
  10. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  11. C. W. Chen, H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
    [CrossRef]
  12. J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
    [CrossRef] [PubMed]
  13. R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  14. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).
  15. M. R. Garey, 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, 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).
    [CrossRef]
  18. 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]
  19. H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
    [CrossRef]
  20. 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]
  21. F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
    [CrossRef]
  22. E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
    [CrossRef]
  23. R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
    [CrossRef]
  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).
    [CrossRef]
  25. D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
    [CrossRef]
  26. A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
    [CrossRef]
  27. C. J. Oliver, 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).
    [CrossRef]
  29. F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing, Active and Passive (Addison-Wesley, London, 1981).
  30. H. A. Zebker, P. A. Rosen, S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
    [CrossRef]
  31. J. L. Kennington, 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).
    [CrossRef]

2000 (1)

1999 (1)

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

1998 (3)

M. Costantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
[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 (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 (2)

T. J. Flynn, “Two-dimensional phase unwrapping with minimum weighted discontinuity,” J. Opt. Soc. Am. A 14, 2692–2701 (1997).
[CrossRef]

H. A. Zebker, P. A. Rosen, S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
[CrossRef]

1996 (4)

L. M. H. Ulander, “Radiometric slope correction of synthetic aperture radar images,” IEEE Trans. Geosci. Remote Sens. 34, 1115–1122 (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]

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

D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
[CrossRef]

1995 (2)

J. R. Buckland, J. M. Huntley, S. R. E. Turner, “Unwrapping noisy phase maps by use of a minimum-cost-matching algorithm,” Appl. Opt. 34, 5100–5108 (1995).
[CrossRef] [PubMed]

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[CrossRef]

1994 (2)

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]

D. C. Ghiglia, 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).
[CrossRef]

1993 (1)

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

1992 (2)

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
[CrossRef]

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

1990 (2)

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

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).
[CrossRef]

1989 (1)

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

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]

1987 (1)

R. M. Goldstein, H. A. Zebker, “Interferometric radar measurements of ocean surface currents,” Nature (London) 328, 707–709 (1987).
[CrossRef]

1986 (1)

H. Zebker, R. Goldstein, “Topographic mapping from interferometric SAR observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

1984 (1)

S. Pallottino, “Shortest-path methods: complexity, interrelations and new propositions,” Networks 14, 257–267 (1984).
[CrossRef]

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 (1998).
[CrossRef]

Ahuja, R. K.

R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1993).

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 (1998).
[CrossRef]

Bruniquel, J.

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

Buckland, J. R.

Carballo, G.

G. Carballo, “Statistically-based multiresolution network flow phase unwrapping for SAR interferometry,” Ph.D. dissertation (Royal Institute of Technology, Stockholm, Sweden, 2000).

Chen, C. W.

Chen, H.

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[CrossRef]

Costantini, M.

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

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 (1998).
[CrossRef]

Flynn, T. J.

Fornaro, G.

Franceschetti, G.

Fung, A. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing, Active and Passive (Addison-Wesley, London, 1981).

Gabriel, A. K.

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

Garey, M. R.

M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, Calif., 1979).

Ghiglia, D. C.

Goering, D. J.

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[CrossRef]

Goldstein, R.

H. Zebker, R. Goldstein, “Topographic mapping from interferometric SAR observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Goldstein, R. M.

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

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

R. M. Goldstein, H. A. Zebker, “Interferometric radar measurements of ocean surface currents,” Nature (London) 328, 707–709 (1987).
[CrossRef]

Guindon, B.

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).
[CrossRef]

Helgason, R. V.

J. L. Kennington, R. V. Helgason, Algorithms for Network Programming (Wiley, New York, 1980).

Hensley, S.

H. A. Zebker, P. A. Rosen, S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
[CrossRef]

Hinzman, L. D.

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[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]

Huntley, J. M.

Johnson, D. S.

M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, Calif., 1979).

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 (1998).
[CrossRef]

Kane, D. L.

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[CrossRef]

Kennington, J. L.

J. L. Kennington, R. V. Helgason, Algorithms for Network Programming (Wiley, New York, 1980).

Lanari, R.

Laur, H.

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

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]

Li, F. K.

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

Lopes, A.

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

Lu, Y.

Magnanti, T. L.

R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1993).

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. M.

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
[CrossRef]

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]

Moore, R. K.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing, Active and Passive (Addison-Wesley, London, 1981).

Nezry, E.

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

Oliver, C. J.

C. J. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).

Orlin, J. B.

R. K. Ahuja, T. L. Magnanti, J. B. Orlin, Network Flows: Theory, Algorithms, and Applications (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Pallottino, S.

S. Pallottino, “Shortest-path methods: complexity, interrelations and new propositions,” Networks 14, 257–267 (1984).
[CrossRef]

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.

C. J. Oliver, S. Quegan, Understanding Synthetic Aperture Radar Images (Artech House, Boston, Mass., 1998).

Rodriguez, E.

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
[CrossRef]

Romero, L. A.

Rosen, P. A.

H. A. Zebker, P. A. Rosen, S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
[CrossRef]

Sansosti, E.

Touzi, R.

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

Turner, S. R. E.

Ulaby, F. T.

F. T. Ulaby, R. K. Moore, A. K. Fung, Microwave Remote Sensing, Active and Passive (Addison-Wesley, London, 1981).

Ulander, L. M. H.

L. M. H. Ulander, “Radiometric slope correction of synthetic aperture radar images,” IEEE Trans. Geosci. Remote Sens. 34, 1115–1122 (1996).
[CrossRef]

Vachon, P. W.

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

Villasenor, J.

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

Werner, C. L.

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.

H. Zebker, R. Goldstein, “Topographic mapping from interferometric SAR observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Zebker, H. A.

C. W. Chen, H. A. Zebker, “Network approaches to two-dimensional phase unwrapping: intractability and two new algorithms,” J. Opt. Soc. Am. A 17, 401–414 (2000).
[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]

H. A. Zebker, P. A. Rosen, S. Hensley, “Atmospheric effects in interferometric synthetic aperture radar surface deformation and topography maps,” J. Geophys. Res. 102, 7547–7563 (1997).
[CrossRef]

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[CrossRef]

A. K. Gabriel, R. M. Goldstein, H. A. Zebker, “Mapping small elevation changes over large areas: differential radar interferometry,” J. Geophys. Res. 94, 9183–9191 (1989).
[CrossRef]

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

R. M. Goldstein, H. A. Zebker, “Interferometric radar measurements of ocean surface currents,” Nature (London) 328, 707–709 (1987).
[CrossRef]

Appl. Opt. (1)

IEE Proc. F, Commun. Radar Signal Process. (1)

E. Rodriguez, J. M. Martin, “Theory and design of interferometric synthetic aperture radars,” IEE Proc. F, Commun. Radar Signal Process. 139, 147–159 (1992).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (10)

R. Touzi, A. Lopes, J. Bruniquel, P. W. Vachon, “Coherence estimation for SAR imagery,” IEEE Trans. Geosci. Remote Sens. 37, 135–149 (1999).
[CrossRef]

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).
[CrossRef]

D. J. Goering, H. Chen, L. D. Hinzman, D. L. Kane, “Removal of terrain effects from SAR satellite imagery of arctic tundra,” IEEE Trans. Geosci. Remote Sens. 33, 185–194 (1995).
[CrossRef]

L. M. H. Ulander, “Radiometric slope correction of synthetic aperture radar images,” IEEE Trans. Geosci. Remote Sens. 34, 1115–1122 (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 (1998).
[CrossRef]

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sens. 30, 950–959 (1992).
[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]

F. K. Li, R. M. Goldstein, “Studies of multibaseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (1990).
[CrossRef]

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

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

Int. J. Remote Sens. (1)

A. Lopes, E. Nezry, R. Touzi, H. Laur, “Structure detection and statistical adaptive speckle filtering in SAR images,” Int. J. Remote Sens. 14, 1735–1758 (1993).
[CrossRef]

J. Geophys. Res. (3)

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Figures (14)

Fig. 1
Fig. 1

Example network equivalent of the phase unwrapping problem. The numbers represent a 2-D array of phase samples (normalized to one cycle). Each 2×2 clockwise loop integral of wrapped phase gradients is a node in the network, and positive and negative residues result in supply and demand nodes. Neighboring nodes are connected by arcs, or possible flow paths.

Fig. 2
Fig. 2

Normalized cost functions for the Lp family of objective functions. The abscissa is the difference in cycles between the unwrapped and the wrapped gradients.

Fig. 3
Fig. 3

Model interferometric phase-noise standard deviation σψ as a function of interferogram coherence ρ for different numbers of independent looks Ni.

Fig. 4
Fig. 4

Facet model used to relate topography to the brightness of a single SAR range-azimuth pixel. Note that Δx depends on both Δr and Δzr.

Fig. 5
Fig. 5

Comparison of (a) actual, normalized SAR image intensity with (b) simulated intensity from a DEM and scattering model.

Fig. 6
Fig. 6

Model intensity as a function of slant-range elevation change for zero azimuth slope. The solid line represents the expected intensity E[I] from Eq. (12) and the dashed line is a piecewise linear approximation to the solid line.

Fig. 7
Fig. 7

Profile of a mountain in layover. The range bins r0r9 represent contours of constant range from the radar. The elevation z and mean intensity E[I] are plotted as they map into slant range for this profile. Because of layover, multiple parts of a surface may map into the same range bin; the solid and open circles represent intersections of the ground surface with the range contours. Unwrapped phase values are assumed to represent elevations at the solid circles, but echoes from the locations of the open circles complicate the topography–intensity relationship when there is layover.

Fig. 8
Fig. 8

Model PDF’s for the topographic component of an unwrapped range gradient, conditional on observed intensity and correlation values. The PDF in the bottom panel is proportional to the product of the curves in the top panel. Note that the dashed curve in the top panel is f(ρ|Δϕtopo(r)) for fixed ρ and variable Δϕtopo(r), not vice versa.

Fig. 9
Fig. 9

Model conditional PDF for an unwrapped range gradient, with both topographic and noise components included.  

Fig. 10
Fig. 10

Example cost functions for unwrapped topographic range (top) and azimuth (bottom) gradients in the presence (solid curves) and absence (dashed curves) of layover. Note that the abscissa is the unwrapped gradient Δϕ itself, not Δϕ-Δψ. The model parameters are based on the physical observables as shown and differ throughout the interferogram.

Fig. 11
Fig. 11

Example cost functions for unwrapped differential phase gradients when a discontinuity is expected (solid curve) and not expected (dashed curve).

Fig. 12
Fig. 12

Topographic test data: (a) interferogram with wrapped phase in color and magnitude in gray scale, (b) biased coherence magnitude, (c) reference DEM with elevation in color and shaded relief in gray scale.

Fig. 13
Fig. 13

Algorithm results on the topographic test interferogram of Fig. 12: (a) SNAPHU, (b) reside-cut, (c) MCF, (d) least-squares. The gray scale depicts the interferogram magnitude, and the color represents relative unwrapped phase error with reference to the DEM. Unwrapping errors are manifest as jumps of 2π rad.

Fig. 14
Fig. 14

Results of our algorithm on a differential interferogram: (a) interferogram with wrapped phase in color and magnitude in gray scale, (b) biased coherence magnitude, (c) unwrapped solution from our algorithm, rewrapped modulo 40 rad (6.37 cycles) for display.

Equations (33)

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minimizeijgi, j(r)(Δϕi, j(r), Δψi, j(r))+ijgi, j(a)(Δϕi, j(a), Δψi, j(a)),
g(Δϕ, Δψ)=w|Δϕ-Δψ|p.
f(ΔΦ|ΔΨ)=kf(Δϕk|Δψk).
minimize-klog[f(Δϕk|Δψk)].
gk(Δϕk, Δψk)=-log[ f(Δϕk|Δψk)].
f(Δϕ|Δψ)
=fΔϕ(Δϕ)m=-fΔϕ(Δψ+m2π)ifΔϕ=Δψ+n2π0otherwise,
Δϕ=Δϕtopo+Δϕnoise.
f(Δϕ|I, ρ)=f(Δϕtopo|I, ρ)*f(Δϕnoise|ρ).
ρˆ=k=1Ns1ks2k*k=1N|s1k|2k=1N|s2k|2,
ϕtopo=-4πBλr sin θ z,
f(Δϕtopo|I, ρ)=f(Δϕtopo|I)f(ρ|Δϕtopo)f(ρ|I),
E[I]=Cσ0A,
Δx=Δrsin θ+Δzrtan θ.
cos θi=(Δzr/Δx)sin θ+cos θ[(Δzr/Δx)2+(Δza/Δy)2+1]1/2
A=[(ΔyΔzr)2+(ΔxΔza)2+(ΔxΔy)2]1/2.
σ0=kdscos2 θi+cosn 2θicos θiifcos 2θi>0kdscos2 θiotherwise.
ρs=max0, 1-2|B|Rrλr|tan θi|.
Δϕmax=min{Δϕlay, Δϕp}.
σΔϕ2=σΔψ2+σmeas2.
σΔϕ2=σΔψ2+σmeas2+σlay2.
g(r)(Δϕ)
=Δϕ2σΔϕ2ifΔϕΔϕcrit(r)glay(r)ifΔϕcrit(r)<ΔϕΔϕmax(Δϕ-Δϕmax)2cσΔϕ2+glay(r)ifΔϕ>Δϕmax,
 
g(r)(Δϕ)=(Δϕ-ΔϕI)2σΔϕ2 .
g(a)(Δϕ)
=Δϕ2σΔϕ2if|Δϕ|Δϕcrit(a)glay(a)ifΔϕcrit(a)<|Δϕ|Δϕmax(|Δϕ|-Δϕmax)2cσΔϕ2+glay(a)if|Δϕ|>Δϕmax.
 
g(a)(Δϕ)=Δϕ2σΔϕ2,
f(Δϕ|I, ρ)=f(Δϕdefo|I, ρ)*f(Δϕnoise|ρ).
ϕdefo=-4πdrλ,
ca(x0, δ)=g(x0+δ)-g(x0),
(πout(p)-πout(m))+(πin(q)-πin(m))+ca<0.

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