Abstract

A new phase unwrapping algorithm is presented that implements phase retrieval without any need for preliminary cuts or weights of the wrapped phase diagram. Excellent results on simulated as well as real data validate the proposed technique.

© 1998 Optical Society of America

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References

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  1. G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
    [CrossRef]
  2. 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]
  3. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  4. H. Takajo, T. Takahashi, “Noniterative method for obtaining the exact solution for the normal equation in least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 1818–1827 (1988).
    [CrossRef]
  5. M. D. Prit, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
    [CrossRef]
  6. G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
    [CrossRef]
  7. Notice that the problem of choosing a constant Φ0 for the reconstruction is common to all phase unwrapping algorithms. Thus the reconstructed phase pattern is always affected by a common bias that must be resolved by other means.
  8. To guarantee a perfect reconstruction, the summation path must be made only of horizontal and vertical segments. This is a direct consequence of the definition of Eq. (8) for the finite difference.
  9. Some of these paths may even generate a phase value corresponding to k=0 when different critical region crossings compensate each other.
  10. G. Rudolph, “Convergence analysis of canonical genetic algorithms,” IEEE Trans. Neural Netw. 5, 96–101 (1994).
    [CrossRef] [PubMed]
  11. X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: Analysis of the diversification role of crossover,” IEEE Trans. Neural Netw. 5, 120–129 (1994).
    [CrossRef] [PubMed]
  12. X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
    [CrossRef]
  13. D. Just, R. Bamler, “Phase statistics of interferogram with application to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
    [CrossRef] [PubMed]
  14. The difference between the wrapped phase pattern and its filtered version is saved, wrapped, and added to the final reconstructed unwrapped phase.
  15. At variance with the simulated case, the filtered noise is no longer added to the final result, a usual technique to smooth the final image.

1997

1996

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
[CrossRef]

1994

G. Rudolph, “Convergence analysis of canonical genetic algorithms,” IEEE Trans. Neural Netw. 5, 96–101 (1994).
[CrossRef] [PubMed]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: Analysis of the diversification role of crossover,” IEEE Trans. Neural Netw. 5, 120–129 (1994).
[CrossRef] [PubMed]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

M. D. Prit, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (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]

D. Just, R. Bamler, “Phase statistics of interferogram with application to synthetic aperture radar,” Appl. Opt. 33, 4361–4368 (1994).
[CrossRef] [PubMed]

1988

Bamler, R.

Fornaro, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
[CrossRef]

Franceschetti, G.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
[CrossRef]

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]

Just, D.

Lanari, R.

G. Fornaro, G. Franceschetti, R. Lanari, E. Sansosti, M. Tesauro, “Global and local phase unwrapping techniques: a comparison,” J. Opt. Soc. Am. A 14, 2702–2708 (1997).
[CrossRef]

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
[CrossRef]

Palmieri, F.

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: Analysis of the diversification role of crossover,” IEEE Trans. Neural Netw. 5, 120–129 (1994).
[CrossRef] [PubMed]

Prit, M. D.

M. D. Prit, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Qi, X. F.

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[CrossRef]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: Analysis of the diversification role of crossover,” IEEE Trans. Neural Netw. 5, 120–129 (1994).
[CrossRef] [PubMed]

Romero, L. A.

Rudolph, G.

G. Rudolph, “Convergence analysis of canonical genetic algorithms,” IEEE Trans. Neural Netw. 5, 96–101 (1994).
[CrossRef] [PubMed]

Sansosti, E.

Shipman, J. S.

M. D. Prit, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

Takahashi, T.

Takajo, H.

Tesauro, M.

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.

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

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

G. Fornaro, G. Franceschetti, R. Lanari, “Interferometric SAR phase unwrapping using Green’s formulation,” IEEE Trans. Geosci. Remote Sens. 34, 1–8 (1996).
[CrossRef]

M. D. Prit, J. S. Shipman, “Least-squares two-dimensional phase unwrapping using FFT’s,” IEEE Trans. Geosci. Remote Sens. 32, 706–708 (1994).
[CrossRef]

IEEE Trans. Neural Netw.

G. Rudolph, “Convergence analysis of canonical genetic algorithms,” IEEE Trans. Neural Netw. 5, 96–101 (1994).
[CrossRef] [PubMed]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part II: Analysis of the diversification role of crossover,” IEEE Trans. Neural Netw. 5, 120–129 (1994).
[CrossRef] [PubMed]

X. F. Qi, F. Palmieri, “Theoretical analysis of evolutionary algorithms with an infinite population size in continuous space. Part I: Basic properties of selection and mutation,” IEEE Trans. Neural Netw. 5, 102–119 (1994).
[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

Notice that the problem of choosing a constant Φ0 for the reconstruction is common to all phase unwrapping algorithms. Thus the reconstructed phase pattern is always affected by a common bias that must be resolved by other means.

To guarantee a perfect reconstruction, the summation path must be made only of horizontal and vertical segments. This is a direct consequence of the definition of Eq. (8) for the finite difference.

Some of these paths may even generate a phase value corresponding to k=0 when different critical region crossings compensate each other.

The difference between the wrapped phase pattern and its filtered version is saved, wrapped, and added to the final reconstructed unwrapped phase.

At variance with the simulated case, the filtered noise is no longer added to the final result, a usual technique to smooth the final image.

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Figures (14)

Fig. 1
Fig. 1

Local integration method in a domain with a turbulent region.

Fig. 2
Fig. 2

Global integration method in a domain with a turbulent region.

Fig. 3
Fig. 3

Family of possible integration paths between P0(n0, m0) and P(n, m).

Fig. 4
Fig. 4

Number of iterations as a function of b parameterized by ξ.

Fig. 5
Fig. 5

Three-dimensional representation of the test phase pattern (normalized to 2π) with a ledge. The corresponding wrapped phase pattern is also shown.

Fig. 6
Fig. 6

Three-dimensional representation of the reconstructed (normalized to 2π) phase pattern retrieved from the data of Fig. 5.

Fig. 7
Fig. 7

Three-dimensional representation of the test (normalized to 2π) phase pattern with ledge and added noise with coherence that is spatially variable as detailed in Fig. 8 below.

Fig. 8
Fig. 8

Coherency maps and wrapped phase patterns without [(A) and (B), respectively] and with [(C) and (D), respectively] filter implementation.

Fig. 9
Fig. 9

Three-dimensional representation of the reconstructed (normalized) phase pattern retrieved from the data of Fig. 8.

Fig. 10
Fig. 10

Plane cut of difference between the reference and reconstructed phase patterns with the use of genetic (dotted–dashed curve) and weighted least-squares (solid curve) algorithms.

Fig. 11
Fig. 11

Wrapped phase patterns and coherency maps without [(A) and (B) respectively] and with [(C) and (D), respectively] filter implementation for a real site (Mount Etna, Sicily).

Fig. 12
Fig. 12

Normalized unwrapped phase pattern corresponding to Fig. 11 and reconstructed with use of the genetic algorithm.

Fig. 13
Fig. 13

Example of a phase point, a pivot point, and four quadrants for implementation of the integration path.

Fig. 14
Fig. 14

Four integration paths for a pivot point belonging to quadrant II.

Equations (31)

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s=Φw.
sΦ,×s0
Φˆ(r)=Φˆ0+0rdc s·c,Φˆ0=Φˆ(0).
L(Φ)=s-Φˆ2.
Φˆ(r)=-SdS s(r)·g(r-r)+Cdc Φˆ(rc) g(r-rc)n
{Φ(n, m),n=0, , N-1,m=0,  , M-1}
Φw(n, m)=Φ(n, m)+2kπ,-π<Φw<π,
s(n, m)=[s1(n, m),s2(n, m)]=[Φw(n, m)-Φw(n-1, m), Φw(n, m)-Φw(n, m-1)],
si(n, m)=si(n, m)+2πifsi<-πsi(n, m)if|si|<πsi(n, m)-2πifsi>π,i=1,2.
Φˆ(n, m)=Φ(n,m)+2kπ.
α1+β1+γ1+=1.
αq+1=[αq+(1-αq)bq]ξ=αq(1-bq)ξ+bqξ.
αq+1=αq(1-b)ξ+bξ,α0=0,
αq+1=bξi=0q[(1-b)ξ]i=bξ 1-[(1-b)ξ]q+11-[(1-b)ξ].
Q=log{(ξ-1)/bξ}log{(1-b)ξ}-1,
(N, 1)(nν, 1)(nν, mν)(n, mν)(n, m).
(N, 1)(nν, 1)(nν, mν)(n+[(n-nν)mod(N-n)], mν)(n+[(n-nν)mod(N-n)], m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m-[(mν-m)mod m])(n, m-[(mν-m)mod m])(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m)(n, m);
(N, 1)(nν, 1)(nν, mν)(n, mν)(n, m);
(N, 1)(nν, 1)(nν, mν)(n+[(n-nν)mod(N-n)], mν)(n+[(n-nν)mod(N-n)], m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m+[(m-mν)mod(M-m)])(n, m+[(m-mν)mod(M-m)])(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m+[(m-mν)mod(M-m)])(n, m+[(m-mν)mod(M-m)])(n, m);
(N, 1)(nν, 1)(nν, mν)(n-[(nν-n)mod n], mν)(n-[(nν-n)mod n], m)(n, m);
(N, 1)(nν, 1)(nν, mν)(nν, m)(n, m);
(N, 1)(N, mν)(nν, mν)(n, mν)(n, m);
(N, 1)(nν, 1)(nν, mν)(n-[(nν-n)mod n], mν)(n-[(nν-n)mod n], m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m)(n, m);
(N, 1)(N, mν)(nν, mν)(nν, m-[(mν-m)mod m])(n, m-[(mν-m)mod m])(n, m).

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