Abstract

The processing required for interferometric synthetic aperture radar (SAR) applications includes a technique for unwrapping the phase in a two-dimensional region. The phase-unwrapping technique is often based on Goldstein’s branch-cut approach. However, this conventional approach leaves holes that are isolated by branch cuts when interferograms have high noise levels. We improve the conventional approach with two new features: (1) we slightly loosen the phase-gradient constraint at the boundary of the already unwrapped area and force growth of the unwrapped area into holes through bridging cuts wherever appropriate; and (2) we recursively fill progressively smaller closed areas by adding rectangular patches while maintaining maximum phase consistency in the overlapped unwrapped areas. The approach is verified with actual SAR data.

© 1999 Optical Society of America

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References

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  1. S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sensing 31, 246–256 (1993).
    [CrossRef]
  2. 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 Sensing 32, 823–836 (1994).
    [CrossRef]
  3. D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
    [CrossRef]
  4. H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).
  5. B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
    [CrossRef]
  6. R. M. Goldstein, H. A. Zebker, C. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radioscience 23, 713–720 (1988).
  7. B. R. Hunt, “Matrix formulation of the reconstruction of phase values from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  8. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  9. 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]
  10. M. D. Pritt, “Phase unwrapping by means of multigrid techniques for interferometric SAR,” IEEE Trans. Geosci. Remote Sensing 34, 728–738 (1996).
    [CrossRef]
  11. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  12. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  13. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sensing 34, 720–727 (1996).
    [CrossRef]
  14. G. Fornaro, G. Franceschetti, R. Lanari, A. Sansosti, “Robust phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 13, 2355–2366 (1996).
    [CrossRef]
  15. H. A. Zebker, Y. Lu, “Phase unwrapping algorithm for radar interferometry: residue-cut, least-squares, and synthesis algorithms,” J. Opt. Soc. Am. A 15, 586–598 (1998).
    [CrossRef]
  16. H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
    [CrossRef]
  17. Q. Lin, J. F. Vesecky, H. A. Zebker, “Registration of interferometric SAR images,” in Proceedings of the International Geosciences and Remote Sensing Systems 1992 Symposium, R. Williamson, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1579–1581.
  18. C. Prati, F. Rocca, “Limits to the resolution of elevation maps from stereo SAR images,” Int. J. Remote Sensing 11, 2215–2235 (1990).
    [CrossRef]
  19. N. Marechal, “Tomographic formulation of interferometric SAR for terrain elevation mapping,” IEEE Trans. Geosci. Remote Sensing 33, 726–739 (1995).
    [CrossRef]

1998 (1)

1996 (4)

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

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

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

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

1995 (1)

N. Marechal, “Tomographic formulation of interferometric SAR for terrain elevation mapping,” IEEE Trans. Geosci. Remote Sensing 33, 726–739 (1995).
[CrossRef]

1994 (3)

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]

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 Sensing 32, 823–836 (1994).
[CrossRef]

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

1993 (2)

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

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

1992 (1)

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

1990 (1)

C. Prati, F. Rocca, “Limits to the resolution of elevation maps from stereo SAR images,” Int. J. Remote Sensing 11, 2215–2235 (1990).
[CrossRef]

1989 (2)

1988 (2)

H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
[CrossRef]

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

1979 (1)

Achache, J.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Adragna, F.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Armijo, R.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Briole, P.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Carmona, C.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

de Chabalier, J. B.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Delacourt, C.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Feigl, K.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Fornaro, G.

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

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

Franceschetti, G.

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

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

Gabriel, A.

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

Ghiglia, D. C.

Goldstein, R. M.

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

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

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 Sensing 32, 823–836 (1994).
[CrossRef]

Hunt, B. R.

Huntley, J. M.

Lanari, R.

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

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

Lin, Q.

Q. Lin, J. F. Vesecky, H. A. Zebker, “Registration of interferometric SAR images,” in Proceedings of the International Geosciences and Remote Sensing Systems 1992 Symposium, R. Williamson, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1579–1581.

Lu, Y.

Madsen, S. N.

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

Marechal, N.

N. Marechal, “Tomographic formulation of interferometric SAR for terrain elevation mapping,” IEEE Trans. Geosci. Remote Sensing 33, 726–739 (1995).
[CrossRef]

Martin, J.

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

Massonnet, D.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Meyer, B.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Papanastassiou, D.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Peltzer, G.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Prati, C.

C. Prati, F. Rocca, “Limits to the resolution of elevation maps from stereo SAR images,” Int. J. Remote Sensing 11, 2215–2235 (1990).
[CrossRef]

Pritt, M. D.

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

Rabaute, T.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Rocca, F.

C. Prati, F. Rocca, “Limits to the resolution of elevation maps from stereo SAR images,” Int. J. Remote Sensing 11, 2215–2235 (1990).
[CrossRef]

Romero, L. A.

Rosen, P.

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

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 Sensing 32, 823–836 (1994).
[CrossRef]

Rosi, M.

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Ruegg, J. C.

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

Sansosti, A.

Takahashi, T.

Takajo, H.

Vesecky, J. F.

Q. Lin, J. F. Vesecky, H. A. Zebker, “Registration of interferometric SAR images,” in Proceedings of the International Geosciences and Remote Sensing Systems 1992 Symposium, R. Williamson, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1579–1581.

Villasenor, J.

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

Werner, C.

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

Werner, C. L.

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

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 Sensing 32, 823–836 (1994).
[CrossRef]

Zebker, H. A.

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

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

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 Sensing 32, 823–836 (1994).
[CrossRef]

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

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

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

Q. Lin, J. F. Vesecky, H. A. Zebker, “Registration of interferometric SAR images,” in Proceedings of the International Geosciences and Remote Sensing Systems 1992 Symposium, R. Williamson, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1579–1581.

Appl. Opt. (1)

Geophys. Res. Lett. (1)

B. Meyer, R. Armijo, D. Massonnet, J. B. de Chabalier, C. Delacourt, J. C. Ruegg, J. Achache, P. Briole, D. Papanastassiou, “The 1995 Grevena (Northern Greece) earthquake: fault model constrained with tectonic observations and SAR interferometry,” Geophys. Res. Lett. 23, 2677–2680 (1996).
[CrossRef]

IEEE Trans. Geosci. Remote Sensing (6)

S. N. Madsen, H. A. Zebker, J. Martin, “Topographic mapping using radar interferometry: processing techniques,” IEEE Trans. Geosci. Remote Sensing 31, 246–256 (1993).
[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 Sensing 32, 823–836 (1994).
[CrossRef]

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

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

N. Marechal, “Tomographic formulation of interferometric SAR for terrain elevation mapping,” IEEE Trans. Geosci. Remote Sensing 33, 726–739 (1995).
[CrossRef]

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

Int. J. Remote Sensing (1)

C. Prati, F. Rocca, “Limits to the resolution of elevation maps from stereo SAR images,” Int. J. Remote Sensing 11, 2215–2235 (1990).
[CrossRef]

J. Geophys. Res. (1)

H. A. Zebker, P. Rosen, R. M. Goldstein, A. Gabriel, C. L. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99(B10), 617–619, 634 (1994).

J. Opt. Soc. Am. (1)

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

Nature (1)

D. Massonnet, M. Rosi, C. Carmona, F. Adragna, G. Peltzer, K. Feigl, T. Rabaute, “The displacement field of the Landers earthquake mapped by radar interferometry,” Nature 364, 138–142 (1993).
[CrossRef]

Opt. Lett. (1)

Radioscience (1)

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

Other (1)

Q. Lin, J. F. Vesecky, H. A. Zebker, “Registration of interferometric SAR images,” in Proceedings of the International Geosciences and Remote Sensing Systems 1992 Symposium, R. Williamson, ed. (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 1579–1581.

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Figures (5)

Fig. 1
Fig. 1

Diagram of the definitions of marginal points and crossing points. Inside a hole there exist marginal points 1, 2, … , 5 that connect to points a, b, … , i in the previously unwrapped area through crossing points A, B, … , H. Wrapped phase values at individual marginal points can be calculated from an initial crossing route a–A–1 and the following phase unwrapping inside the hole. These values can also be calculated directly from the nearby crossing routes a–A–1, b–A–1, b–B–1, … , i–H–5. Differences of the phase values at individual marginal points 1–5 that were derived from the initial route and from the direct routes can then be evaluated to form a deviation set. The resultant deviation set is compared with Eqs. (2) and (4) to determine whether the hole can be filled without violating the criteria of smoothness and uniqueness.

Fig. 2
Fig. 2

Flowchart showing the sequence of operations involved in filling holes through a cut-bridging procedure.

Fig. 3
Fig. 3

Example of phase unwrapping for a relatively clean phase field: (a) interferogram 222 and the unwrapped phase images derived from various methods, including (b) branch cutting; (c) branch cutting plus cut bridging; (d) combination of branch cutting, cut bridging, and window patching; (e) weighted least squares; and (f) synthesis least squares. All methods yield similar results.

Fig. 4
Fig. 4

Example of phase unwrapping for a relatively noisy phase field: (a) interferogram 21I and the unwrapped phase images derived from various methods, including (b) branch cutting; (c) branch cutting plus cut bridging; (d) combination of branch cutting, cut bridging, and window patching; (e) weighted least squares; and (f) synthesis least squares. The resultant unwrapped phase images show that the least-squares methods exaggerate the elevation in region A while underestimating the terrain relief in the same region where the phase-noise level is high. The white lines represent the transects along which the topographic profiles were derived from the DEM’s that resulted from the corresponding unwrapped phase images.

Fig. 5
Fig. 5

Comparison of the topographic profiles derived from the unwrapped phase images in Fig. 4 and the U.S. Geological Survey topographic maps: (a) Direct comparison of topographic profiles derived from the weighted least-squares method (top curve), the synthesis least-squares method (middle curve), and the new method and the topographic maps (bottom two curves). Because the topographic profile derived from the new method is so close to the one extracted from the topographic maps, the two curves appear as a single curve. (b) Differences in elevation along the topographic transect among the profiles derived from different unwrapping methods and the one extracted from the topographic maps. The pixel spacing in both (a) and (b) is 40 m.

Tables (1)

Tables Icon

Table 1 List of Interferograms Created for the Study Area

Equations (4)

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

ΔΦ  0, 2π, -2π,
ΔΦ=0, 2π, -2π,
ΔΦ=2π, -2π,
ΔΦ=0, 2π, 4π,ΔΦ=0, 4π,ΔΦ=0, -2π, -4π,ΔΦ=0, -4π,

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