Abstract

Given an interferometric phase image of a surface profile, the task of two-dimensional phase unwrapping is to reconstruct the profile by adding multiples of 2π to the image. Discontinuities in the unwrapped phase must be restricted to areas of noise and true discontinuity in the profile. Such areas can often be identified by their low quality. This suggests that the unwrapped phase should be chosen to minimize a weighted sum of discontinuity magnitudes. An algorithm is presented that computes such an unwrapped phase from any initial guess. The elementary operation of the algorithm is to partition the image into two connected regions, then raise the unwrapped phase by 2π in one of the regions, reducing the weighted sum; this is done repeatedly until no suitable partitions exist. The operations are found by creating paths that follow discontinuity curves and extending them to form complete partitions. The algorithm terminates when no path can be extended. The behavior of the algorithm and the benefits of weighting are illustrated with an example.

© 1997 Optical Society of America

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References

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  1. H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic-aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
    [CrossRef]
  2. C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).
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    [CrossRef]
  4. D. C. Ghiglia, L. A. Romero, “Minimum Lp-norm two-dimensional phaseunwrapping,” J. Opt. Soc. Am. A 13, 1999–2013 (1996).
    [CrossRef]
  5. Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic apertureradar interferometry,” Appl. Opt. 33, 201–208 (1994).
    [CrossRef] [PubMed]
  6. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  7. J. M. Huntley, “Noise-immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  8. 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]
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    [CrossRef] [PubMed]
  10. H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sensing 30, 950–959 (1992).
    [CrossRef]

1996

1995

1994

1992

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

1989

1988

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

1986

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic-aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Bernabeu, E.

Buckland, J. R.

Ghiglia, D. C.

Goldstein, R. M.

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

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic-aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

González-Cano, A.

Huntley, J. M.

Lin, Q.

Quiroga, J. A.

Romero, L. A.

Turner, S. R. E.

Vesecky, J. F.

Vest, C. M.

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

Villasenor, J.

H. A. Zebker, J. Villasenor, “Decorrelation in interferometric radar echoes,” IEEE Trans. Geosci. Remote Sensing 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. A.

Q. Lin, J. F. Vesecky, H. A. Zebker, “Phase unwrapping through fringe-line detection in synthetic apertureradar interferometry,” Appl. Opt. 33, 201–208 (1994).
[CrossRef] [PubMed]

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. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
[CrossRef]

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic-aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sensing

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

J. Geophys. Res.

H. A. Zebker, R. M. Goldstein, “Topographic mapping from interferometric synthetic-aperture radar observations,” J. Geophys. Res. 91, 4993–4999 (1986).
[CrossRef]

J. Opt. Soc. Am. A

Radio Sci.

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

Other

C. M. Vest, Holographic Interferometry (Wiley, New York, 1979).

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Figures (12)

Fig. 1
Fig. 1

Initial unwrapped phase with discontinuities marked. The numbers are the unwrapped phase pixels in cycles. The line segments indicate the location and the magnitude of the discontinuities. The direction marker on each segment points toward the higher unwrapped phase. The discontinuity sum is 11.

Fig. 2
Fig. 2

Result of performing an elementary operation (EO) on the unwrapped phase of Fig. 1. The pixels within the dashed contour have been increased by 1 cycle. The discontinuity markers reflect the changed phases. The discontinuity sum has been reduced to 6.

Fig. 3
Fig. 3

EO of Fig. 2, represented as a loop. The node array, marked with squares, is superimposed on the initial unwrapped phase of Fig. 1. The arrows mark the edges of the loop. Each edge indicates a change of ±1 in its associated jump count.

Fig. 4
Fig. 4

Node array with a possible collection of paths for the initial unwrapped phase of Fig. 1. The nodes are marked with their values. The outline of the EO is shown for reference. The dotted arrow indicates a type 1 revision.

Fig. 5
Fig. 5

Collection of paths after the type 1 revision. The new edge has been added, and the values have been changed accordingly. The dotted arrow indicates a type 2 revision.

Fig. 6
Fig. 6

Collection of paths after the type 2 revision. The new edge has been added, and the existing edge to its destination has been removed. The dotted arrow indicates a type 3 revision.

Fig. 7
Fig. 7

Collection of paths after the type 3 revision. The loop formed by the new edge, and all paths containing loop edges, have been removed. The existing path leading to the loop has been retained.

Fig. 8
Fig. 8

Phase image computed from an ERS-1 interferometric image pair taken at Fort Irwin, California.

Fig. 9
Fig. 9

Thresholded correlation map computed from the ERS-1 image pair. Dark pixels indicate where the correlation is less than 38%.

Fig. 10
Fig. 10

Unwrapped phase computed from Fig. 8 by using the preliminary algorithm.

Fig. 11
Fig. 11

Unweighted minimum-discontinuity unwrapped phase computed by using the preliminary solution as an initial guess.

Fig. 12
Fig. 12

Minimum-weighted-discontinuity unwrapped phase computed by using the preliminary solution as an initial guess. Jumps between two high-correlation pixels have weight 128; other jumps have weight 1.

Equations (47)

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ϕ(x, y)=2πh(x, y)H+ν(x, y)mod 2π,
ϕmnϕ(xm, yn),
h^mn=Hϕ^mn/2π,
ϕ^mn=ϕmn+2πcmn.
vmn=ϕ^mn-ϕ^m-1,n+π2πfor(m, n)V,
zmn=ϕ^mn-ϕ^m,n-1+π2πfor(m, n)Z,
vmn=cmn-cm-1,n+ϕmn-ϕm-1,n+π2π,
zmn=cmn-cm,n-1+ϕmn-ϕm,n-1+π2π.
cm0=cm-1,0+vm0-ϕmn-ϕm-1,n+π2π
cmn=cm,n-1+zmn-ϕmn-ϕm,n-1+π2π
E0(c; ϕ)=(m,n)V|vmn|+(m,n)Z|zmn|,
J=(m,n)V|ϕ^mn-ϕ^m-1,n-W(ϕmn-ϕm-1,n)|p+(m,n)Z|ϕ^mn-ϕ^m,n-1-W(ϕmn-ϕm,n-1)|p,
W(ϕ)=ϕ-2πϕ+π2π.
J=(m,n)V|ϕmn-ϕm-1,n+2π(cmn-cm-1,n)-W(ϕmn-ϕm-1,n)|p+(m,n)Z|ϕmn-ϕm,n-1+2π(cmn-cm,n-1)-W(ϕmn-ϕm,n-1)|p,
J=2π(m,n)V|vmn|p+(m,n)Z|zmn|p.
E(c; ϕ)=(m,n)Vwmnv|vmn|+(m,n)Zwmnz|zmn|,
δV(m, n; m, n+1)=-wmnv(|vmn-1|-|vmn|)=wmnv sgn(vmn-1)
for(m, n)V,
δV(m, n+1; m, n)=-wmnv(|vmn+1|-|vmn|)=-wmnv sgn(vmn)
for(m, n)V.
δV(m, n; m+1, n)=-wmnz sgn(zmn)
for(m, n)Z,
δV(m+1, n; m, n)=wmnz sgn(zmn-1)
for(m, n)Z.
k=0L-1δV(mk, nk; mk+1, nk+1).
ΔV=V(m, n)+δV(m, n; m, n)-V(m, n).
E(c¯; ϕ)<E(c; ϕ).
E(c+δc; ϕ)<E(c; ϕ).
cmnk=cmn+min(k, Δcmn).
δckck-ck-1.
ΔEE(c¯; ϕ)-E(c; ϕ)=E(cK; ϕ)-E(c0; ϕ)=k=1K[E(ck; ϕ)-E(ck-1; ϕ)].
δEkE(c+δck; ϕ)-E(c; ϕ)=E(c0+ck-ck-1; ϕ)-E(c0; ϕ).
δEkE(ck; ϕ)-E(ck-1; ϕ)
0E(ck; ϕ)-E(ck-1; ϕ)-E(c0+ck-ck-1; ϕ)+E(c0; ϕ)=Svk+Szk,
Svk=(m,n)Vwmnv(|vmnk|-|vmnk-1|+|vmn0|-|vmn0+vmnk-vmnk-1|),
Szk=(m,n)Zwmnz(|zmnk|-|zmnk-1|+|zmn0|-|zmn0+zmnk-zmnk-1|).
|a-b+c||a|-|b|+|c|
foranya, b, csuchthatabcorabc.
|b|=|γa+(1-γ)c|γ|a|+(1-γ)|c|.
|a-b+c|=|(1-γ)a+γc|(1-γ)|a|+γ|c|=|a|+|c|-[γ|a|+(1-γ)|c|]|a|-|b|+|c|.
E(c+δc; ϕ)<E(c; ϕ)
E(c+c; ϕ)<E(c; ϕ).
V(m, n)+δV(m, n; m, n)V(m, n)
V(mi, ni)+δV(mi, ni; mi+1, ni+1)V(mi+1, ni+1)
i=0L-1V(mi, ni)+i=0L-1δV(mi, ni; mi+1, ni+1)
i=1LV(mi, ni).
i=0L-1δV(mi, ni; mi+1, ni+1)0.

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