Abstract

The advent of interferometric synthetic aperature radar for geophysical studies has resulted in the need for accurate, efficient methods of two-dimensional phase unwrapping. Inference of the lost integral number of cycles in phase measurements is critical for three-pass surface deformation studies as well as topographic mapping and can result in an order of magnitude increase in sensitivity for two-pass deformation analysis. While phase unwrapping algorithms have proliferated over the past ten years, two main approaches are currently in use. Each is most useful only for certain restricted applications. All these algorithms begin with the measured gradient of the phase field, which is subsequently integrated to recover the unwrapped phases. The earliest approaches in interferometric applications incorporated residue identification and cuts to limit the possible integration paths, while a second class using least-squares techniques was developed in the early 1990’s. We compare the approaches and find that the residue-cut algorithms are quite accurate but do not produce estimates in regions of moderate phase noise. The least-squares methods yield complete coverage but at the cost of distortion in the recovered phase field. A new synthesis approach, combining the cuts from the first class with a least-squares solution, offers greater spatial coverage with less distortion in many instances.

© 1998 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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1997 (1)

1996 (3)

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

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

1995 (2)

D. Massonnet, K. L. Feigl, “Discrimination of geophysical phenomena in satellite radar interferograms,” Geophys. Res. Lett. 22, 1537–1540 (1995).
[CrossRef]

S. N. Madsen, J. Martin, H. A. Zebker, “Analysis and evaluation of the NASA/JPL TOPSAR interferometric SAR system,” IEEE Trans. Geosci. Remote Sens. 33, 383–391 (1995).
[CrossRef]

1994 (6)

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

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

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, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

1993 (2)

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

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

1992 (2)

E. Rodriguez, J. Martin, “Theory and design of interferometric SARs,” Proc. IEEE 139, 147–159 (1992).

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

1991 (1)

1990 (1)

F. Li, R. M. Goldstein, “Studies of multi-baseline spaceborne interferometric synthetic aperture radars,” IEEE Trans. Geosci. Remote Sens. 28, 88–97 (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. 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 measurement 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]

1979 (1)

Adragna, F.

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

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

Bone, D. J.

Carmona, C.

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

Collaro, A.

Dixon, T. H.

H. A. Zebker, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

Engelhardt, H.

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Farr, T. G.

H. A. Zebker, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

Feigl, K.

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

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

Feigl, K. L.

D. Massonnet, K. L. Feigl, “Discrimination of geophysical phenomena in satellite radar interferograms,” Geophys. Res. Lett. 22, 1537–1540 (1995).
[CrossRef]

Ferreiro, M. S.

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansoti, “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 Sens. 34, 720–727 (1996).
[CrossRef]

Franceschetti, G.

Frolich, R. M.

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Gabriel, A.

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

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]

Ghiglia, D. C.

Goldstein, R.

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

Goldstein, R. M.

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

F. Li, R. M. Goldstein, “Studies of multi-baseline 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. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

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

Hensley, S.

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

Hunt, B. R.

Kamb, B.

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansoti, “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 Sens. 34, 720–727 (1996).
[CrossRef]

Li, F.

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

Madsen, S. N.

S. N. Madsen, J. Martin, H. A. Zebker, “Analysis and evaluation of the NASA/JPL TOPSAR interferometric SAR system,” IEEE Trans. Geosci. Remote Sens. 33, 383–391 (1995).
[CrossRef]

Martin, J.

S. N. Madsen, J. Martin, H. A. Zebker, “Analysis and evaluation of the NASA/JPL TOPSAR interferometric SAR system,” IEEE Trans. Geosci. Remote Sens. 33, 383–391 (1995).
[CrossRef]

E. Rodriguez, J. Martin, “Theory and design of interferometric SARs,” Proc. IEEE 139, 147–159 (1992).

Massonnet, D.

D. Massonnet, K. L. Feigl, “Discrimination of geophysical phenomena in satellite radar interferograms,” Geophys. Res. Lett. 22, 1537–1540 (1995).
[CrossRef]

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

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

Palmieri, F.

Peltzer, G.

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

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]

Rabaute, T.

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

Rodriguez, E.

E. Rodriguez, J. Martin, “Theory and design of interferometric SARs,” Proc. IEEE 139, 147–159 (1992).

Romero, L. A.

Rosen, P.

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

Rosen, P. A.

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

Rossi, M.

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

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

Salazar, R. P.

H. A. Zebker, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

Sansoti, E.

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]

Villasenor, J.

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

Werner, C.

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

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

Werner, C. L.

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

Zebker, H.

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

Zebker, H. A.

S. N. Madsen, J. Martin, H. A. Zebker, “Analysis and evaluation of the NASA/JPL TOPSAR interferometric SAR system,” IEEE Trans. Geosci. Remote Sens. 33, 383–391 (1995).
[CrossRef]

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

H. A. Zebker, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[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. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

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

Appl. Opt. (1)

Geophys. Res. Lett. (1)

D. Massonnet, K. L. Feigl, “Discrimination of geophysical phenomena in satellite radar interferograms,” Geophys. Res. Lett. 22, 1537–1540 (1995).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (7)

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

S. N. Madsen, J. Martin, H. A. Zebker, “Analysis and evaluation of the NASA/JPL TOPSAR interferometric SAR system,” IEEE Trans. Geosci. Remote Sens. 33, 383–391 (1995).
[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]

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

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

F. Li, R. M. Goldstein, “Studies of multi-baseline 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]

J. Geophys. Res. (3)

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

H. A. Zebker, P. A. Rosen, R. M. Goldstein, A. Gabriel, C. Werner, “On the derivation of coseismic displacement fields using differential radar interferometry: the Landers earthquake,” J. Geophys. Res. 99, 19617–19634 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Nature (London) (3)

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

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

D. Massonnet, K. Feigl, M. Rossi, F. Adragna, “Radar interferometric mapping of deformation in the year after the Landers earthquake,” Nature (London) 369, 227–230 (1994).
[CrossRef]

Proc. IEEE (2)

E. Rodriguez, J. Martin, “Theory and design of interferometric SARs,” Proc. IEEE 139, 147–159 (1992).

H. A. Zebker, T. G. Farr, R. P. Salazar, T. H. Dixon, “Mapping the world’s topography using radar interferometry: the TOPSAT mission,” Proc. IEEE 82, 1774–1786 (1994).
[CrossRef]

Radio Sci. (1)

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

Science (1)

R. M. Goldstein, H. Engelhardt, B. Kamb, R. M. Frolich, “Satellite radar interferometry for monitoring ice sheet motion: application to an Antarctic ice stream,” Science 262, 1525–1530 (1993).
[CrossRef] [PubMed]

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Fig. 1
Fig. 1

Test data sets for algorithm intercomparison, in wrapped phase (raw interferogram) form. Two sets, (a) and (b), are simulated data. One contains simple geometric shapes designed to illustrate the importance of residue distributions and proper tree locations, and one, derived from topographic data, includes layover and thermal noise. The box outlined in the lower left of the scene depicts added noise. Actual data comprise two more scenes: (c) a very high signal-to-noise interferogram of a fairly flat area in Hawaii and (d) rugged terrain in central California. Radar brightness appears as the intensity at each point and the phase as color, with one color cycle corresponding to one fringe. (a) The simple geometrical target scene includes a pyramid structure that exhibits no phase discontinuities at its edges, a two-sided ramp structure that is continuous with the background on the left and right edges but discontinuous along the top and bottom, and a slanted wedge structure with exactly 2π phase change along its length. (b) The simulated topographic interferogram is most rugged in the upper left-hand corner, leading to significant layover in this area and less elsewhere. Scene (b) possesses a very high fringe rate, scene (c) a moderate rate, and in scene (d) the background flat-Earth fringes have been removed.

Fig. 2
Fig. 2

Phase errors after application of four unwrapping algorithms to the geometric-shape data set. (a) Results from the Goldstein et al.1 residue-cut tree algorithm; only a one-cycle error on half of the wedge is generated. The rest of the image unwraps perfectly. Both the unweighted [(b)] and the weighted [(c)] least-squares algorithms exhibit large distortion fields associated with discontinuities at the top and bottom of the ramp structure and from the half-wedge that showed the error in the tree case [(a)]. These results are identical, as the weights assigned were unity everywhere (see text). The synthesis algorithm [(d)] produces the same result as the residue-cut algorithm here.

Fig. 3
Fig. 3

Phase errors after application of four unwrapping algorithms to the synthetic topography data set. (a) Results from the Goldstein et al.1 residue-cut tree algorithm. The algorithm fails to unwrap the heavily laid-over region in the upper-left and part of the noisy inset and also generates a few local one-cycle (2π) error regions. (b) The unweighted least-squares algorithm unwraps the image completely but exhibits distortion associated with residues from layover and thermal noise. It also underestimates the overall slope of the scene from left to right. (c) The results from the weighted least-squares algorithm differs in detail but is similar. (d) The synthesis algorithm produces a complete result much closer to the actual answer than the traditional least-squares approach, although errors associated with layover in the upper left and the noise in the box remain.

Fig. 4
Fig. 4

Error signature of a dipole formed by a positive and negative residue pair.

Fig. 5
Fig. 5

Unwrapping of an easy scene, the Hawaii SIR-C data, by each of the four algorithms: (a) residue-cut trees, (b) unweighted least-squares, (c) weighted least-squares, and (d) synthesis. All unwrap completely and get nearly the same result.

Fig. 6
Fig. 6

Results of applying the four algorithms [(a) residue-cut trees, (b) unweighted least-squares, (c) weighted least-squares, and (d) synthesis] to a more difficult unwrapping problem, ERS-1 data over Parkfield, Calif. The residue-cut algorithm leaves significant gaps. Both of the least-squares algorithms produce subtly different results that significantly underestimate the tilt from upper left to lower right in the scene. The synthesis algorithm gives a complete answer and recovers the global tilt accurately. There are undoubtedly errors of 2π in magnitude in this solution, but they are not visible at this scale.

Tables (3)

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Table 1 Rms Errors from Four Algorithms

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Table 2 Computational Efficiencies of the Algorithms

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Table 3 Summary of Algorithm Intercomparisons

Equations (11)

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s=Pϕ+n
ϕ=(PTP)-1PTs.
i=0M-2j=0N-1(ϕi+1,j-ϕi,j-Δi,jx)2
+i=0M-1j=0N-2(ϕi,j+1-ϕi,j-Δi,jy)2,
Δi,jx=φi+1,j-φi,j,
Δi,jy=φi,j+1-φi,j,
(ϕi+1,j-2ϕi,j+ϕi-1,j)+(ϕi,j+1-2ϕi,j+ϕi,j-1)
=(Δi,jx-Δi-1,jx)+(Δi,jy-Δi,j-1y).
Ws=WPϕ+n.
ϕ=(PTWTWP)-1PTWTWs.
alli,j|ϕi,jk+1-ϕi,jk|2alli,j|ϕi,jk+1|2,

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