Abstract

Use of mean-field annealing theory is proposed for solving the phase-unwrapping (PU) problem. PU is formulated as a constrained optimization problem for the field of integer corrections to be added to the wrapped gradient field. A deterministic algorithm is described to provide an approximation of the average of the correction field over the global minima of the cost function. The proposed algorithm can be applied for any choice of the cost function. Using a cost function based on second-order differences, we obtain results close to those from simulated annealing and spend less computational time.

© 1999 Optical Society of America

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References

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  1. D. L. Fried, “Least-squares fitting of wave-front distortion estimate to an array of phase-difference measurement,” J. Opt. Soc. Am. A 67, 370–375 (1977).
  2. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  3. R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. A 68, 139–140 (1980).
  4. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast trasforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  5. R. M. Goldstein, H. A. Zebker, C. L. Werner, “Satellite radar interferometry: two-dimensional phase unwrapping,” Radio Sci. 23, 713–720 (1988).
    [CrossRef]
  6. J. M. Huntley, “Noise immune phase unwrapping algorithm,” Appl. Opt. 28, 3268–3270 (1989).
    [CrossRef] [PubMed]
  7. J. A. Quiroga, A. Gonzáles-Cano, E. Bernabeu, “Stable-marriages algorithm for preprocessing phase maps with discontinuity sources,” Appl. Opt. 34, 5029–5038 (1995).
    [CrossRef] [PubMed]
  8. D. C. Ghiglia, G. A. Mastin, L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A 4, 267–280 (1987).
    [CrossRef]
  9. A. Collaro, G. Franceschetti, F. Palmieri, M. S. Ferreiro, “Phase unwrapping by means of genetic algorithms,” J. Opt. Soc. Am. A 15, 407–418 (1998).
    [CrossRef]
  10. J. L. Marroquin, M. Rivera, “Quadratic regularization functionals for phase unwrapping,” J. Opt. Soc. Am. A 12, 2393–2400 (1995).
    [CrossRef]
  11. J. L. Marroquin, M. Tapia, R. Rodriguez-Vera, M. Servin, “Parallel algorithms for phase unwrapping based on Markov random field models,” J. Opt. Soc. Am. A 12, 2578–2585 (1995).
    [CrossRef]
  12. L. Guerriero, G. Nico, G. Pasquariello, S. Stramaglia, “New regularization scheme for phase unwrapping,” Appl. Opt. 37, 3053–3058 (1998).
    [CrossRef]
  13. G. Parisi, Statistical Field Theory (Addison-Wesley, Reading, Mass., 1988).
  14. J. Zhang, “Mean field theory in EM procedures for MRF’s,” IEEE Trans. Signal Process. 40, 2570–2583 (1992).
    [CrossRef]
  15. S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
    [CrossRef] [PubMed]
  16. A. L. Yuille, J. J. Kosowsky, “Statistical physics algorithms that converge,” Neural Comput. 6, 341–356 (1994).
    [CrossRef]
  17. D. J. Bone, “Fourier fringe analysis: the two-dimensional phase unwrapping problem,” Appl. Opt. 30, 3627–3632 (1991).
    [CrossRef] [PubMed]
  18. In the limit of high temperature, all the m’s vanish except for those whose location is close to residua. Hence the best starting point for iterations is {m = 0} and {λ = 0}.
  19. We iterated Eqs. (19) asynchronously.
  20. One can easily adapt to this task the classical least-squares algorithm,1–4 originally designed to find the scalar field whose discrete gradient is closer to A.
  21. E. Wasserstrom, “Numerical solutions by the continuation method,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 89–119 (1973).

1998 (2)

1995 (3)

1994 (2)

1992 (1)

J. Zhang, “Mean field theory in EM procedures for MRF’s,” IEEE Trans. Signal Process. 40, 2570–2583 (1992).
[CrossRef]

1991 (1)

1989 (1)

1988 (1)

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

1987 (1)

1984 (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

1980 (1)

R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. A 68, 139–140 (1980).

1977 (2)

D. L. Fried, “Least-squares fitting of wave-front distortion estimate to an array of phase-difference measurement,” J. Opt. Soc. Am. A 67, 370–375 (1977).

R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
[CrossRef]

1973 (1)

E. Wasserstrom, “Numerical solutions by the continuation method,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 89–119 (1973).

Bernabeu, E.

Bone, D. J.

Collaro, A.

Ferreiro, M. S.

Franceschetti, G.

Fried, D. L.

D. L. Fried, “Least-squares fitting of wave-front distortion estimate to an array of phase-difference measurement,” J. Opt. Soc. Am. A 67, 370–375 (1977).

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

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]

Gonzáles-Cano, A.

Guerriero, L.

Hudgin, R. H.

Huntley, J. M.

Kosowsky, J. J.

A. L. Yuille, J. J. Kosowsky, “Statistical physics algorithms that converge,” Neural Comput. 6, 341–356 (1994).
[CrossRef]

Marroquin, J. L.

Mastin, G. A.

Nico, G.

Noll, R. J.

R. J. Noll, “Phase estimates from slope-type wave-front sensors,” J. Opt. Soc. Am. A 68, 139–140 (1980).

Palmieri, F.

Parisi, G.

G. Parisi, Statistical Field Theory (Addison-Wesley, Reading, Mass., 1988).

Pasquariello, G.

Quiroga, J. A.

Rivera, M.

Rodriguez-Vera, R.

Romero, L. A.

Servin, M.

Stramaglia, S.

Tapia, M.

Wasserstrom, E.

E. Wasserstrom, “Numerical solutions by the continuation method,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 89–119 (1973).

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]

Yuille, A. L.

A. L. Yuille, J. J. Kosowsky, “Statistical physics algorithms that converge,” Neural Comput. 6, 341–356 (1994).
[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]

Zhang, J.

J. Zhang, “Mean field theory in EM procedures for MRF’s,” IEEE Trans. Signal Process. 40, 2570–2583 (1992).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. 6, 721–741 (1984).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

J. Zhang, “Mean field theory in EM procedures for MRF’s,” IEEE Trans. Signal Process. 40, 2570–2583 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Neural Comput. (1)

A. L. Yuille, J. J. Kosowsky, “Statistical physics algorithms that converge,” Neural Comput. 6, 341–356 (1994).
[CrossRef]

Radio Sci. (1)

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

SIAM (Soc. Ind. Appl. Math.) Rev. (1)

E. Wasserstrom, “Numerical solutions by the continuation method,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 89–119 (1973).

Other (4)

G. Parisi, Statistical Field Theory (Addison-Wesley, Reading, Mass., 1988).

In the limit of high temperature, all the m’s vanish except for those whose location is close to residua. Hence the best starting point for iterations is {m = 0} and {λ = 0}.

We iterated Eqs. (19) asynchronously.

One can easily adapt to this task the classical least-squares algorithm,1–4 originally designed to find the scalar field whose discrete gradient is closer to A.

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Figures (3)

Fig. 1
Fig. 1

(a) Synthetic smooth surface (256 × 256 pixels), corresponding to the function ϕ(x, y) = 50 exp{-[(x - 70)2 + (y - 70)2]/150} + 20 exp{-[(x - 150)2 + (y - 150)2]/300} + 0.15(x + y), 1 ≤ x ≤ 256, 1 ≤ y ≤ 256. Units on the vertical axis are in radians. (b) Residua map displaying inconsistencies due to undersampling. (c) Unwrapped phase surface reconstructed with our algorithm.

Fig. 2
Fig. 2

(a) Synthetic surface (256 × 256 pixels) obtained from the surface depicted in Fig. 1(a) by adding Gaussian noise with variance 0.8. Units on the vertical axis are in radians. (b) Residua map displaying inconsistencies due to both undersampling and noise. (c) Unwrapped phase surface reconstructed with our algorithm.

Fig. 3
Fig. 3

(a) Wrapped phase pattern (128 × 128 pixels) for a real site (Senerchia, Campania, Italy). (b) Residua map. (c) Unwrapped phase surface reconstructed MFA algorithm. The units on the vertical axis are in radians. (d) Phase field obtained by wrapping the MFA’s output, to be compared with the input wrapped phase field in Fig. 3(a).

Equations (30)

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

gx, y=Wfx, y=modπ+fx, y, 2π-π,
Ax, y=Wgx, y,
xgx, y=gx+1, y-gx, y,  ygx, y=gx, y+1-gx, y.
|fx, y|<π
fx, y=Ax, y
fx, y=Ax, y+2πkx, y,
×Ax, y+2πkx, y=0.
Ix, y=1/2πAxx, y+Ayx+1, y-Axx, y+1-Ayx, y.
Pβk=exp-βRkk exp-βRk,
Pk= pxx,ykxx, y pyx,ykyx, y.
mxx, y=kxx, y,  myx, y=kyx, y,  Qxx, y=kx2x, y,  Qyx, y=ky2x, y,
pxx,ykxx, y=0=1-Qxx, y,  pxx,ykxx, y=±1=Qxx, y±mxx, y2.
Um, Q=Rk.
Sm, Q=- 1-Qxlog1-Qx+Qx+mx2×logQx+mx2+Qx-mx2 logQx-mx2 - 1-Qylog1-Qy+Qy+my2×logQy+my2+Qy-my2 logQy-my2,
Fm, Q=Um, Q-TSm, Q+x,y λx, ymxx, y+myx+1, y-mxx, y+1-myx, y+Ix, y.
Fmxx, y=0,  Fmyx, y=0,
FQxx, y=0,  FQyx, y=0.
Qxx, y=αx-0.5αx+mx2x, y0.25-αx1/2αx-0.25,  Qyx, y=αy-0.5αy+my2x, y0.25-αy1/2αy-0.25,
mxx, y=αx-0.5αx+mx2x, y0.25-αx1/2αx-0.25×tanh-β Umxx, y-λx, y+λx, y-1,  myx, y=αy-0.5αy+my2x, y0.25-αy1/2αy-0.25×tanh-β Umyx, y+λx, y-λx-1, y.
λx, y=λx, y-bmxx, y+myx+1, y-mxx, y+1-myx, y+Ix, y,
fest=A+2πmout
|×fest|<
R=14π2  xf2+yf2,
R= kx2+Axπ kx+ky2+Ayπ ky,
U=R= Qx+Axπ mx+Qy+Ayπ my,
Umxx, y=Axx, yπ,  Umyx, y=Ayx, yπ,  UQxx, y=1,  UQyx, y=1,
R=14π2  xfx+1, y-xfx, y2+14π2  yfx, y+1-yfx, y2+14π2  xfx, y+1-xfx, y2+14π2  yfx+1, y-yfx, y2,
UQxx, y=4,  UQyx, y=4,
Umxx, y=-2mxx-1, y+1πAxx, y-Axx-1, y+-2mxx+1, y+1πAxx, y-Axx+1, y+-2mxx, y-1+1πAxx, y-Axx, y-1+-2mxx, y+1+1πAxx, y-Axx, y+1.
Umyx, y=-2myx, y-1+1πAyx, y-Ayx, y-1+-2myx, y+1+1πAyx, y-Ayx, y+1+-2myx+1, y+1πAyx, y-Ayx+1, y+-2myx-1, y+1πAyx, y-Ayx-1, y.

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