Abstract

An algorithm for phase retrieval with Bayesian statistics is discussed. It is shown how the statistics of Kolmogorov turbulence can be used to compute the likelihood for a particular phase screen. This likelihood is then added to that of the observed data to produce a functional that is maximized directly by use of conjugate gradient maximization. It is shown that although this can significantly improve the quality of the phase estimate, the issue is complicated by local maxima introduced by the possibility of phase wrapping. The causes of the local maxima are analyzed, and a method that increases the likelihood of convergence to the global maximum is presented.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  2. A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from magnitude,” J. Opt. Soc. Am. A 1, 932–943 (1984).
    [CrossRef]
  3. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.
  4. N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
    [CrossRef]
  5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  6. J. Hermann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A 9, 2257–2258 (1992).
    [CrossRef]
  7. C. Schwarz, E. Ribak, S. G. Lipson, “Bimorph adaptive mirrors and curvature sensing,” J. Opt. Soc. Am. A 11, 895–907 (1994).
    [CrossRef]
  8. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  9. J. Phillips, The NAG Library: Programmers Guide (Clarendon, Oxford, 1986).
  10. E. Thiebaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
    [CrossRef]
  11. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  12. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  13. B. R. Hunt, “Matrix formulation of the reconstruction of phase differences from phase differences,” J. Opt. Soc. Am. 69, 393–399 (1979).
    [CrossRef]
  14. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).
  15. W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).
  16. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  17. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
    [CrossRef]
  18. R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
    [CrossRef]
  19. R. G. Lane, R. Irwan, “Phase retrieval as a means of wavefront sensing,” in Proceedings of the International Conference on Image Processing, B. R. Hunt, R. M. Gray, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 242–245.
  20. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
    [CrossRef]
  21. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]

1996 (1)

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (2)

J. Hermann, “Phase variance and Strehl ratio in adaptive optics,” J. Opt. Soc. Am. A 9, 2257–2258 (1992).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

1991 (1)

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

1990 (2)

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
[CrossRef]

R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
[CrossRef]

1984 (1)

1983 (1)

1982 (2)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

1979 (1)

1976 (1)

Conan, J.-M.

Dainty, J. C.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Fienup, J. R.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
[CrossRef]

Glindemann, A.

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Hermann, J.

Hunt, B. R.

Irwan, R.

R. G. Lane, R. Irwan, “Phase retrieval as a means of wavefront sensing,” in Proceedings of the International Conference on Image Processing, B. R. Hunt, R. M. Gray, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 242–245.

Lane, R. G.

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

R. G. Lane, R. Irwan, “Phase retrieval as a means of wavefront sensing,” in Proceedings of the International Conference on Image Processing, B. R. Hunt, R. M. Gray, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 242–245.

Law, N. F.

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

Levi, A.

Lipson, S. G.

Millane, R. P.

Noll, R. J.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

Phillips, J.

J. Phillips, The NAG Library: Programmers Guide (Clarendon, Oxford, 1986).

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
[CrossRef]

Ribak, E.

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wavefront sensing: a new technique for compensating turbulence degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
[CrossRef]

Schulz, T. J.

Schwarz, C.

Stark, H.

Thiebaut, E.

Wallner, E. P.

Appl. Opt. (1)

J. Mod. Opt. (1)

R. G. Lane, “Phase retrieval using conjugate gradient minimization,” J. Mod. Opt. 38, 1797–1813 (1991).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Opt. Commun. (1)

N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
[CrossRef]

Opt. Eng. (Bellingham) (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. (Bellingham) 21, 829–832 (1982).
[CrossRef]

Waves Random Media (1)

R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
[CrossRef]

Other (6)

R. G. Lane, R. Irwan, “Phase retrieval as a means of wavefront sensing,” in Proceedings of the International Conference on Image Processing, B. R. Hunt, R. M. Gray, eds. (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 242–245.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1991).

W. K. Pratt, Digital Image Processing (Wiley, New York, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1981), Vol. 19, pp. 281–376.

J. Phillips, The NAG Library: Programmers Guide (Clarendon, Oxford, 1986).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) Simulated phase screen D/r0=4, (b) the corresponding speckle image for 5000 photons.

Fig. 2
Fig. 2

Phase estimates produced by (a) the conventional ML method, (b) the ML plus a penalty function, and (c) the MAP estimate. In all cases the linear-phase starting point is used, D/r0=4, and there are 5000 photons.

Fig. 3
Fig. 3

St-Sl, the difference between the the Strehl ratio obtained from a linear-phase starting estimate and St obtained from starting at the true phase for (a) ML, (b) ML+, and (c) MAP. Here D/r0=4, and there are 5000 photons. Note that in only a few cases are the differences close to zero, indicating convergence to the same local maximum in the likelihood function.

Fig. 4
Fig. 4

Illustration of the wrapping effects when a zero-phase starting point is used. (a) ML+ estimate, (b) MAP estimate. D/r0=4, and there are 5000 photons.

Fig. 5
Fig. 5

St-Sz plotted as a function of LLt-LLz. (a) ML, (b) ML+, and (c) MAP. Note in particular the difference in scales of the two graphs. Also note the large number of cases in the ML algorithm when either St-Sz or LLt-LLz is negative, indicating that the global maximum of the likelihood is displaced from the true phase.

Fig. 6
Fig. 6

Plot of the attained Strehl ratio as a function of the mse of the starting phase estimate (a) ML, 500 photons; (b) ML+, 500 photons; and (c) MAP, 500 photons.

Fig. 7
Fig. 7

Performance of the MAP algorithm with a variable weighting on the prior probability. The dashed curve shows 1000 trials of the variable weighting scheme sorted in ascending order of Strehl ratio obtained. The solid curve shows the Strehl ratio obtained from the true phase, for comparison.

Tables (3)

Tables Icon

Table 1 Mean Strehl Ratios Obtained for Phase Retrieval for Speckles Formed with D/r0=4a

Tables Icon

Table 2 Standard Deviations of the Mean Strehl Ratio Values Reported in Table 1

Tables Icon

Table 3 Performance in Terms of Strehl Ratio for MAP Phase Retrieval with an Annealed Weighting on the Priora

Equations (28)

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

A(u, v)=M(u, v)exp[iϕ(u, v)].
μ(x, y)=|a(x, y)|2,
A(u, v)=F[a(x, y)]=-+-+a(x, y)exp[-i2π(ux+vy)]dxdy.
d(x, y)=μ(x, y)+ζ(x, y)+n(x, y),
S=μˆϕ(u, v)-ϕˆ(u, v)(xm, ym)μˆ0(xm, ym).
ϕˆmaxPr[ϕˆ(u, v)|d(x, y)],
Pr[ϕˆ(u, v)|d(x, y)]=Pr[d(x, y)|ϕˆ(u, v)]Pr[ϕˆ(u, v)]Pr[d(x, y)];
log{Pr[ϕˆ(u, v)|d(x, y)]}=log{Pr[d(x, y)|ϕˆ(u, v)]}+log{Pr[ϕˆ(u, v)]}-log{Pr[d(x, y)]}.
LLϕˆ(u, v)=LLaˆ(x, y)aˆ(x, y)Aˆ(u, v)Aˆ(u, v)ϕˆ(u, v),
Pr[d(x, y)|ϕˆ(u, v)]=x, y[μˆ(x, y)+ζ(x, y)]d(x, y) exp[-μˆ(x, y)-ζ(x, y)]d(x, y)!,
LL=x, yd(x, y)log[μˆ(x, y)+ζ(x, y)]-μˆ(x, y)-ζ(x, y).
LLϕˆ(u, v)=ImFd(x, y)-μˆ(x, y)-ζ(x, y)μˆ(x, y)+ζ(x, y)×a(x, y)A*(u, v),
-+-+WA(u, v)dudv=1
D(u1, v1, u2, v2)=6.88[|(u1, v1)-(u2, v2)|/r0]5/3,
ψ(u, v)=ϕ(u, v)--+-+ϕ(u, v)WA(u, v)dudv.
ψ(u1, v1)ψ(u2, v2)=C(u1, v1, u2, v2),
C(u1, v1, u2, v2)=-12D(u1, v1, u2, v2)+g(u1, v1)+g(u2, v2)-b,
g(u, v)=12-+-+WA(u, v)D(u, v, u, v)dudv,
b=12-+-+-+-+WA(u, v)WA(u, v)×D(u, v, u, v)dudvdudv.
log{Pr[ϕˆ(u, v)]}=12ΦˆTC-1Φˆ.
 log{Pr[ϕˆ(u, v)]}ϕˆ(u, v)=R-1(ΦˆTC-1),
LLp=LL+γΓ(u, v),
Γ(u, v)=--2ϕˆ(u, v)dudv,
l=0-10-14-10-10.
LLp=LL+γu,v[ϕˆ(u, v)  l]2,
LLpϕˆ(u, v)=LLϕˆ(u, v)+2γ[ϕˆ(u, v)  l]*l,
ϕ(u, v)=jcjrjvj(u, v),
1.31(D/r0)(5/3),

Metrics