Abstract

The Green’s formulation for phase unwrapping is generalized to the case of circular phase-support regions. A phase-unwrapping method, believed to be new, is developed in which two forms of the Green’s function are used, one in a closed form and the other in the form of a series of Helmholtz equation eigenfunctions to satisfy homogeneous Neumann boundary conditions in a circular domain. The contribution of the rotational part of the wrapped phase gradient that is due to phase-gradient inconsistencies (residues) is accounted for in the unwrapped phase. Computational results on the reconstruction of a simulated wave front in the presence of aberrations, and on unwrapping real synthetic aperture radar interferograms, show the usefulness and reliability of the method when applied to regions where the conventional rectangular support regions are impractical.

© 2000 Optical Society of America

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References

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  1. 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]
  2. G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0 inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
    [CrossRef] [PubMed]
  3. G. Páez, M. Strojnik, “Phase-shifted interferometry without phase unwrapping: reconstruction of a decentered wave front,” J. Opt. Soc. Am. A 16, 475–480 (1999).
    [CrossRef]
  4. M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
    [CrossRef]
  5. E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
    [CrossRef]
  6. J. M. Huntley, J. R. Buckland, “Characterization of sources of 2π phase discontinuity in speckle interferograms,” J. Opt. Soc. Am. A 12, 1990–1996 (1995).
    [CrossRef]
  7. 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]
  8. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  9. W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
    [CrossRef]
  10. M. Constantini, “A novel phase unwrapping method based on network programming,” IEEE Trans. Geosci. Remote Sens. 36, 813–821 (1998).
    [CrossRef]
  11. G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).
  12. I. Lyuboshenko, H. Maı̂tre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999).
    [CrossRef]
  13. I. Lyuboshenko, A. Maruani, H. Maı̂tre, “Residue influence on phase unwrapping by use of Green–Helmholtz formulation,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. III, pp. 1537–1539.
  14. L. Axel, D. Morton, “Correction of phase wrapping in magnetic resonance imaging,” Med. Phys. 16, 284–287 (1989).
    [CrossRef] [PubMed]
  15. B. D. Bobrov, “Screw dislocations of laser speckle fields in interferograms with a circular line structure,” Sov. J. Quantum Electron. 21, 802–806 (1991).
    [CrossRef]
  16. S. Stramaglia, L. Guerriero, G. Pasquariello, N. Veneziani, “Mean-field annealing for phase unwrapping,” Appl. Opt. 38, 1377–1383 (1999).
    [CrossRef]
  17. P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).
  18. M. Abramowitz, C. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).
  19. M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, New York, 1970).
  20. 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]
  21. U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
    [CrossRef]
  22. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  23. J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

1999

1998

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

1995

1994

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]

1993

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

1991

B. D. Bobrov, “Screw dislocations of laser speckle fields in interferograms with a circular line structure,” Sov. J. Quantum Electron. 21, 802–806 (1991).
[CrossRef]

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0 inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

1989

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

L. Axel, D. Morton, “Correction of phase wrapping in magnetic resonance imaging,” Med. Phys. 16, 284–287 (1989).
[CrossRef] [PubMed]

1988

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

1984

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

Axel, L.

L. Axel, D. Morton, “Correction of phase wrapping in magnetic resonance imaging,” Med. Phys. 16, 284–287 (1989).
[CrossRef] [PubMed]

Bobrov, B. D.

B. D. Bobrov, “Screw dislocations of laser speckle fields in interferograms with a circular line structure,” Sov. J. Quantum Electron. 21, 802–806 (1991).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, New York, 1970).

Buckland, J. R.

Constantini, M.

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

Cumming, I.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Curlander, J. C.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

Feshbach, H.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Fiddy, M. A.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Franceschetti, G.

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Ghiglia, D. C.

Glover, G. H.

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0 inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

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]

Guerriero, L.

Hall, T. J.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Huntley, J. M.

Lanari, R.

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

Lu, Y.

Lyuboshenko, I.

I. Lyuboshenko, H. Maı̂tre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999).
[CrossRef]

I. Lyuboshenko, A. Maruani, H. Maı̂tre, “Residue influence on phase unwrapping by use of Green–Helmholtz formulation,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. III, pp. 1537–1539.

Mai^tre, H.

I. Lyuboshenko, H. Maı̂tre, “Phase unwrapping for interferometric synthetic aperture radar by use of Helmholtz equation eigenfunctions and the first Green’s identity,” J. Opt. Soc. Am. A 16, 378–395 (1999).
[CrossRef]

I. Lyuboshenko, A. Maruani, H. Maı̂tre, “Residue influence on phase unwrapping by use of Green–Helmholtz formulation,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. III, pp. 1537–1539.

Maruani, A.

I. Lyuboshenko, A. Maruani, H. Maı̂tre, “Residue influence on phase unwrapping by use of Green–Helmholtz formulation,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. III, pp. 1537–1539.

McDonough, R. N.

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

Morse, P. M.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

Morton, D.

L. Axel, D. Morton, “Correction of phase wrapping in magnetic resonance imaging,” Med. Phys. 16, 284–287 (1989).
[CrossRef] [PubMed]

Páez, G.

Pasquariello, G.

Pritt, M. D.

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]

Romero, L. A.

Schneider, E.

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0 inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

Scivier, M. S.

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

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]

Spagnolini, U.

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

Stramaglia, S.

Strojnik, M.

Takahashi, T.

Takajo, H.

Veneziani, N.

Volostnikov, V. G.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[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]

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, New York, 1970).

Xu, W.

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

Zebker, H. A.

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]

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.

Geophysics

U. Spagnolini, “2-D phase unwrapping and phase aliasing,” Geophysics 58, 1324–1334 (1993).
[CrossRef]

IEEE Trans. Geosci. Remote Sens.

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]

W. Xu, I. Cumming, “A region-growing algorithm for InSAR phase unwrapping,” IEEE Trans. Geosci. Remote Sens. 37, 124–134 (1999).
[CrossRef]

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

J. Opt. Soc. Am. A

Magn. Reson. Med.

G. H. Glover, E. Schneider, “Three-point Dixon technique for true water/fat decomposition with B0 inhomogeneity correction,” Magn. Reson. Med. 18, 371–383 (1991).
[CrossRef] [PubMed]

Med. Phys.

L. Axel, D. Morton, “Correction of phase wrapping in magnetic resonance imaging,” Med. Phys. 16, 284–287 (1989).
[CrossRef] [PubMed]

Opt. Acta

M. S. Scivier, T. J. Hall, M. A. Fiddy, “Phase unwrapping using the complex zeros of a band-limited function and the presence of ambiguities in two dimensions,” Opt. Acta 31, 619–623 (1984).
[CrossRef]

Opt. Commun.

E. G. Abramochkin, V. G. Volostnikov, “Relationship between two-dimensional intensity and phase in a Fresnel diffraction zone,” Opt. Commun. 74, 144–148 (1989).
[CrossRef]

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]

Sov. J. Quantum Electron.

B. D. Bobrov, “Screw dislocations of laser speckle fields in interferograms with a circular line structure,” Sov. J. Quantum Electron. 21, 802–806 (1991).
[CrossRef]

Other

I. Lyuboshenko, A. Maruani, H. Maı̂tre, “Residue influence on phase unwrapping by use of Green–Helmholtz formulation,” in Proceedings of International Geoscience and Remote Sensing Symposium (Institute of Electrical and Electronics Engineers, New York, 1999), Vol. III, pp. 1537–1539.

P. M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953).

M. Abramowitz, C. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1972).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, New York, 1970).

G. Franceschetti, R. Lanari, Synthetic Aperture Radar Processing (CRC Press, Boca Raton, Fla., 1999).

J. C. Curlander, R. N. McDonough, Synthetic Aperture Radar: Systems and Signal Processing (Wiley, New York, 1991).

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Figures (8)

Fig. 1
Fig. 1

SAR image of New Caledonia area (image size is 512 × 512 pixels). Dashed circles, boundaries of support regions for the corresponding interferograms shown in Fig. 6, below. Image spans the area of ∼20 km (in ground range) ×20.5 km (in azimuth). Resolutions in the radar range, ground range, and azimuth are 10, 25, and 6 m, respectively.

Fig. 2
Fig. 2

Sample configuration of vectors a, r, and r′ for derivation of the Green’s function in Eq. (6) satisfying Neumann boundary conditions on the boundary C of the support region S (filled with gray), example of integration path C 0 used for computation of a residue sign in Eq. (13), and example of integration path C′ (dashed curve in the second quarter) to obtain the hidden phase at point P′S according to Eq. (25).

Fig. 3
Fig. 3

(a) Simulated wave front, (b) corresponding phase surface. Image size, 128 × 128 pixels. The phase is affected by all primary aberrations, which render problematic its unambiguous unwrapping in the presence of undersampling.

Fig. 4
Fig. 4

Comparison between wave fronts reconstructed by (a) developed Green’s function method and (b) conventional LMS reconstruction algorithms proposed in Refs. 7 and 20. Reconstruction error of Green’s function solution remains localized in the undersampled area (left-hand side), whereas the LMS-unwrapping error has also a marked destructive effect on the other areas of the reconstructed wave front.

Fig. 5
Fig. 5

(a) Residue map associated with the phase in Fig. 3(b). Bright pixels denote positions of positive residues, whereas dark pixels indicate positions of negative ones. (b) Corresponding harmonic term computed with Eq. (19). Image size, 128 × 128 pixels.

Fig. 6
Fig. 6

Interferograms of New Caledonia and associated residue maps corresponding to Regions 1 [(a) and (b)] and 2 [(c) and (d)], respectively, in Fig. 1. The boundaries of the support regions are shown as dashed white circles. Image size, 128 × 128 pixels.

Fig. 7
Fig. 7

(a) Phase unwrapped by use of the developed method [Eqs. (4), (19), and (25)] and (b) mod 2π unwrapping error image corresponding to the interferogram in Fig. 6(a). Unwrapped phase is superimposed with the corresponding amplitude values from Fig. 1. Remaining shears in the unwrapping error image are due to unmatched residues. Image size, 128 × 128 pixels.

Fig. 8
Fig. 8

(a) Phase unwrapped by use of the developed method [Eqs. (4), (19), and (25)] and (b) mod 2π unwrapping error image corresponding to the interferogram in Fig. 6(c). Unwrapped phase is superimposed with the corresponding amplitude values from Fig. 1. Remaining shears in the unwrapping error image are due to unmatched residues. Image size, 128 × 128 pixels.

Equations (27)

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ϕr+cmod 2π=θr,  θ  -π, π],r=(x, y  S,
Fr=limδ0θr+δxˆ-θrmod 2πδxˆ+θr+δyˆ-θrmod 2πδyˆ,
gr, rnSrC=0,
ϕr=SdSFr·gr, r.
ϕr=SdSFr·gr, r-Cdcϕrgr, rnS,
gr, r=2π-1 lnr-r 2a-r-r
gr, r=m,n=0vmn20fmnrfmnrvmn2,
fmnrnSrC=0.
fmnρ, φ=cosmφ+sinmφJmπαmnρ/a,
fmn2=02πdφ 0a ρdρfmn2ρ, φ=a2/παmn2π2αmn2-m2Jm2παmn.
gr, r=1/πm,n=0 m×cosmφ-φJmπαmnρ/aJmπαmnρ/aπ2αmn2-m2Jm2παmn,
ϕρ, φ=A000  piston term+1/2A0406ρ4-6ρ2+1  spherical aberration+A05110ρ5-12ρ3+3ρcosφ  high-order coma+A022ρ22 cos2φ-1  astigmatism+1/2A1202ρ2-1  curvature of the field+A111ρ cosφ  distortion,
ωiri=C0 Fr·dr=±2π,
χˆr=-i ωiri SdSgr, ri×gr, rz,
χˆr=-i ωiri02πdφ 0adρ-gφr, rigρr, r+gρr, rigφr, r,
χˆr=02πdφ 0adρρχr·gr, r,
χρr, ri=-ρ-1i-ωirigφr, ri,ρ-1χφr, ri=i-ωirigρr, ri,
χˆr=02πdφ 0adρρχρr, rigρr, r+ρ-1χφr, rigφr, r.
χˆr=4π-1i ωirim=1 m sinmφi-φ×smρismρ,
smρ=n=0Jmπαmnρ/aπ2αmn2-m2Jmπαmn.
ψr=02πdφ 0adρρψr·gr, r,
χr=C χr·dc,
χr=i-ωiriC ρgρr, ridφ-ρ-1gφr, ridρ.
χr=i ωiriaρ ρ-1gφρ, φ; ridρ.
χr=-2π-1i ωiriχiρ, φ-χia, φ,
χiρ, φ=tan-1ρ-ρi cosφ-φiρi sinφ-φi+ρi2-2aρi cosφ-φi4a2+ρi2-4aρi cosφ-φitan-1×2a-ρ-ρi cosφ-φiρi sinφ-φi+aρi sinφ-φi4a2+ρi2-4aρi cosφ-φi×lnρ2ρ-2a2+ρi2+2ρ-2aρi cosφ-φi-1.
Φr=φr-χˆr+χr,

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