Abstract

In ground-based astronomy, the inverse problem of phase retrieval from speckle images is a means to calibrate static aberrations for correction by active optics. It can also be used to sense turbulent wavefronts. However, the number of local minima drastically increases with the turbulence strength, mainly because of phase wrapping ambiguities. Multifocal phase diversity has been considered to overcome some ambiguities of the phase retrieval problem. We propose an effective algorithm for phase retrieval from a single focused image. Our algorithm makes use of a global optimization strategy and an automatically tuned smoothness prior to overcome local minima and phase degeneracies. We push the limit of Dr0=4 achieved by Irwan and Lane [J. Opt. Soc. Am. A. 15, 2302 (1998 )] up to Dr0=11, which is a major improvement owing to the drastic increase in the problem complexity. We estimate the performances of our approach from consistent simulations for different turbulence strengths and noise levels (down to 1500 photons per image). We also investigate the benefit of temporal correlation.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).
  2. R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 66, 961-964 (1976).
    [CrossRef]
  3. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef]
  4. R. Irwan and R. G. Lane, "Phase retrieval with prior information," J. Opt. Soc. Am. A 15, 2302-2311 (1998).
    [CrossRef]
  5. R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Lett. 21, 829-832 (1982).
  6. A. Tokovinin and S. Heathcote, "Donut: measuring optical aberrations from a single extrafocal image," Publ. Astron. Soc. Pac. 118, 1165-1175 (2006).
    [CrossRef]
  7. F. Roddier, "The effects of atmospheric turbulence in optical astronomy," in Progress in Optics, E.Wolf, ed. (North Holland, 1981), pp. 281-376.
    [CrossRef]
  8. R. G. Lane and M. Tallon, "Wave-front reconstruction using a Shack-Hartmann sensor," Appl. Opt. 31, 6902-6906 (1992).
    [CrossRef] [PubMed]
  9. E. Thiébaut and J.-M. Conan, "Strict a priori constraints for maximum-likelihood blind deconvolution," J. Opt. Soc. Am. A 12, 485-492 (1995).
    [CrossRef]
  10. E. Thiébaut, "Introduction to image reconstruction and inverse problems," in Optics in Astrophysics, R.Foy and F.-C.Foy, eds. (Kluwer Academic, 2005), pp. 397-421.
  11. R. G. Lane, "Methods for maximum-likelihood deconvolution," J. Opt. Soc. Am. A 13, 1992-1998 (1996).
    [CrossRef]
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  13. E. P. Wallner, "Optimal wave-front correction using slope measurements," J. Opt. Soc. Am. 73, 1771-1776 (1983).
    [CrossRef]
  14. J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 2006).
  15. J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 4, 553-572 (1983).
    [CrossRef]
  16. J. Nocedal, "Updating quasi-Newton matrices with limited storage," Math. Comput. 35, 773-782 (1980).
    [CrossRef]
  17. E. Thiébaut, "Optimization issues in blind deconvolution algorithms," Proc. SPIE 4847, 174-183 (2002).
    [CrossRef]
  18. J. Skilling and R. K. Bryan, "Maximum entropy image reconstruction: general algorithm," Mon. Not. R. Astron. Soc. 211, 111-124 (1984).
  19. C. Pichon and E. Thiébaut, "Non-parametric reconstruction of distribution functions from observed galactic discs," Mon. Not. R. Astron. Soc. 301, 419-434 (1998).
    [CrossRef]
  20. C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
    [CrossRef]
  21. R. G. Lane, A. Glindemann, and J. C. Dainty, "Simulation of a Kolmogorov phase screen," Waves Random Media 2, 209-224 (1992).
    [CrossRef]
  22. F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
    [CrossRef]
  23. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

2006

A. Tokovinin and S. Heathcote, "Donut: measuring optical aberrations from a single extrafocal image," Publ. Astron. Soc. Pac. 118, 1165-1175 (2006).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
[CrossRef]

2002

E. Thiébaut, "Optimization issues in blind deconvolution algorithms," Proc. SPIE 4847, 174-183 (2002).
[CrossRef]

1998

R. Irwan and R. G. Lane, "Phase retrieval with prior information," J. Opt. Soc. Am. A 15, 2302-2311 (1998).
[CrossRef]

C. Pichon and E. Thiébaut, "Non-parametric reconstruction of distribution functions from observed galactic discs," Mon. Not. R. Astron. Soc. 301, 419-434 (1998).
[CrossRef]

1996

1995

1992

R. G. Lane and M. Tallon, "Wave-front reconstruction using a Shack-Hartmann sensor," Appl. Opt. 31, 6902-6906 (1992).
[CrossRef] [PubMed]

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

1984

J. Skilling and R. K. Bryan, "Maximum entropy image reconstruction: general algorithm," Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

1983

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 4, 553-572 (1983).
[CrossRef]

E. P. Wallner, "Optimal wave-front correction using slope measurements," J. Opt. Soc. Am. 73, 1771-1776 (1983).
[CrossRef]

1982

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
[CrossRef]

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Lett. 21, 829-832 (1982).

F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
[CrossRef]

1980

J. Nocedal, "Updating quasi-Newton matrices with limited storage," Math. Comput. 35, 773-782 (1980).
[CrossRef]

1976

1972

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Béchet, C.

C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Bryan, R. K.

J. Skilling and R. K. Bryan, "Maximum entropy image reconstruction: general algorithm," Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

Conan, J.-M.

Dainty, J. C.

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

Fienup, J. R.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Gilli, J. M.

F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
[CrossRef]

Glindemann, A.

R. G. Lane, A. Glindemann, and 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. Lett. 21, 829-832 (1982).

R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am. 66, 961-964 (1976).
[CrossRef]

Heathcote, S.

A. Tokovinin and S. Heathcote, "Donut: measuring optical aberrations from a single extrafocal image," Publ. Astron. Soc. Pac. 118, 1165-1175 (2006).
[CrossRef]

Irwan, R.

Lane, R. G.

Lund, G.

F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
[CrossRef]

Moré, J. J.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 4, 553-572 (1983).
[CrossRef]

Nocedal, J.

J. Nocedal, "Updating quasi-Newton matrices with limited storage," Math. Comput. 35, 773-782 (1980).
[CrossRef]

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 2006).

Papoulis, A.

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

Pichon, C.

C. Pichon and E. Thiébaut, "Non-parametric reconstruction of distribution functions from observed galactic discs," Mon. Not. R. Astron. Soc. 301, 419-434 (1998).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Roddier, F.

F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
[CrossRef]

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," in Progress in Optics, E.Wolf, ed. (North Holland, 1981), pp. 281-376.
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Skilling, J.

J. Skilling and R. K. Bryan, "Maximum entropy image reconstruction: general algorithm," Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

Sorensen, D. C.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 4, 553-572 (1983).
[CrossRef]

Tallon, M.

C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
[CrossRef]

R. G. Lane and M. Tallon, "Wave-front reconstruction using a Shack-Hartmann sensor," Appl. Opt. 31, 6902-6906 (1992).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Thiébaut, E.

C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
[CrossRef]

E. Thiébaut, "Optimization issues in blind deconvolution algorithms," Proc. SPIE 4847, 174-183 (2002).
[CrossRef]

C. Pichon and E. Thiébaut, "Non-parametric reconstruction of distribution functions from observed galactic discs," Mon. Not. R. Astron. Soc. 301, 419-434 (1998).
[CrossRef]

E. Thiébaut and J.-M. Conan, "Strict a priori constraints for maximum-likelihood blind deconvolution," J. Opt. Soc. Am. A 12, 485-492 (1995).
[CrossRef]

E. Thiébaut, "Introduction to image reconstruction and inverse problems," in Optics in Astrophysics, R.Foy and F.-C.Foy, eds. (Kluwer Academic, 2005), pp. 397-421.

Tokovinin, A.

A. Tokovinin and S. Heathcote, "Donut: measuring optical aberrations from a single extrafocal image," Publ. Astron. Soc. Pac. 118, 1165-1175 (2006).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

Wallner, E. P.

Wright, S. J.

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 2006).

Appl. Opt.

J. Opt.

F. Roddier, J. M. Gilli, and G. Lund, "On the origin of speckle boiling and its effects in stellar speckle interferometry," J. Opt. 13, 263-271 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Math. Comput.

J. Nocedal, "Updating quasi-Newton matrices with limited storage," Math. Comput. 35, 773-782 (1980).
[CrossRef]

Mon. Not. R. Astron. Soc.

J. Skilling and R. K. Bryan, "Maximum entropy image reconstruction: general algorithm," Mon. Not. R. Astron. Soc. 211, 111-124 (1984).

C. Pichon and E. Thiébaut, "Non-parametric reconstruction of distribution functions from observed galactic discs," Mon. Not. R. Astron. Soc. 301, 419-434 (1998).
[CrossRef]

Opt. Lett.

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Lett. 21, 829-832 (1982).

Optik (Stuttgart)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik (Stuttgart) 35, 237-246 (1972).

Proc. SPIE

E. Thiébaut, "Optimization issues in blind deconvolution algorithms," Proc. SPIE 4847, 174-183 (2002).
[CrossRef]

C. Béchet, M. Tallon, and E. Thiébaut, "FRIM: minimum-variance reconstructor with a fractal iterative method," Proc. SPIE 6272, 62722U (2006).
[CrossRef]

Publ. Astron. Soc. Pac.

A. Tokovinin and S. Heathcote, "Donut: measuring optical aberrations from a single extrafocal image," Publ. Astron. Soc. Pac. 118, 1165-1175 (2006).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput.

J. J. Moré and D. C. Sorensen, "Computing a trust region step," SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 4, 553-572 (1983).
[CrossRef]

Waves Random Media

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

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C (Cambridge U. Press, 1992).

E. Thiébaut, "Introduction to image reconstruction and inverse problems," in Optics in Astrophysics, R.Foy and F.-C.Foy, eds. (Kluwer Academic, 2005), pp. 397-421.

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

F. Roddier, "The effects of atmospheric turbulence in optical astronomy," in Progress in Optics, E.Wolf, ed. (North Holland, 1981), pp. 281-376.
[CrossRef]

J. Nocedal and S. J. Wright, Numerical Optimization (Springer-Verlag, 2006).

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 (5)

Fig. 1
Fig. 1

Example of phase screens (left) and speckle images (right). The top panels show the true wavefront and the corresponding simulated image; the bottom panels show the recovered phase and corresponding model image. The phase scales are given in radians. The conditions of the simulation were D r 0 = 11 and N ph = 50,000 photons.

Fig. 2
Fig. 2

Quartiles and average for uncorrelated wavefronts of the best phase rms errors achieved with respect to the number of random phase starts. Different turbulence conditions and mean photon counts are considered. The bottom panel shows the importance of model sampling for the success of the optimization for the same noise and turbulence levels. Indeed, finer sampling of the model enables better prior and reparametrization smoothness regarding the modulo- 2 π phase wrapping ambiguity.

Fig. 3
Fig. 3

Cumulative distribution functions Pr f ( x X ) of the phase retrieval rms errors, in radian units, for different turbulence and noise conditions.

Fig. 4
Fig. 4

Values of the ML penalty after 70 iterations of the local optimization algorithm versus the phase standard deviation error (rms error) with respect to the true phase. Different turbulence conditions and mean photon counts are considered. The threshold likelihood f ML min is indicated for each case. The correlation between the best (smallest) penalties and the smallest phase errors is clearly seen.

Fig. 5
Fig. 5

Cumulative distribution functions of the phase retrieval rms errors (radian units) for time-correlated wavefronts.

Tables (1)

Tables Icon

Table 1 Parameters Used in Our Different Simulations

Equations (68)

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

φ ( r ) = 2 π λ δ ( r ) ,
A ( r ) = P ( r ) e i φ ( r ) ,
m ( x ) = α a ( x ) 2 ,
a ( x ) = A ( λ u ) e + 2 i π x u d 2 u ,
m j = def m ( x j ) ,
m j = α a j 2 = α k F j , k A k 2 ,
A k = def A ( λ u k ) = P k e i φ k ,
P k = def P ( λ u k ) ,
φ k = def φ ( λ u k ) ,
φ k = K k , ψ ,
φ = K ψ ,
d = m + n ,
{ α ML , ψ ML } = arg max α , ψ Pr ( d m ( α , ψ ) )
= arg min α , ψ f ML ( α , ψ ) ,
f ML ( α , ψ ) = log Pr ( d m ( α , ψ ) ) ,
{ α MAP , ψ MAP } = arg max α , ψ Pr ( d m ( α , ψ ) ) Pr ( m ( α , ψ ) )
= arg min α , ψ [ f ML ( α , ψ ) + f prior ( ψ ) ] ,
f prior ( ψ ) = log [ Pr ( q ( ψ ) ) ] ,
f ( ψ ) = f ML ( ψ ) + μ f prior ( ψ ) ,
f ML ( α , ψ ) = 1 2 ( d m ) T W ML ( d m ) ,
f ML ( ψ ) = 1 2 [ d α + ( ψ ) q ( ψ ) ] T W ML [ d α + ( ψ ) q ( ψ ) ] ,
α + ( ψ ) = q ( ψ ) T W ML d q ( ψ ) T W ML q ( ψ )
f ML ( ψ ) ψ k = f ML ( α , ψ ) ψ k α = α + ( ψ ) + α + ( ψ ) ψ k f ML ( α , ψ ) α α = α + ( ψ ) .
f ML ( α , ψ ) α = 0 for α = α + ( ψ ) .
f ML ( ψ ) ψ k = f ML ( α , ψ ) ψ k α = α + ( ψ ) ,
f prior ( φ ) = 1 2 φ T C φ 1 φ ,
f prior ( ψ ) = 1 2 ψ T K T C φ 1 K ψ .
D φ ( r i , r j ) 6.88 × ( r i r j r 0 ) 5 3 ,
ψ ( n ) = ψ ( n 1 ) + δ ψ ( n 1 ) .
δ f ( ψ , δ ψ ) = def f ( ψ + δ ψ ) f ( ψ )
= g ( ψ ) T δ ψ + 1 2 δ ψ T H ( ψ ) δ ψ + o ( δ ψ 2 ) ,
δ f quad ( ψ , δ ψ ) = def g ( ψ ) T δ ψ + 1 2 δ ψ T H ( ψ ) δ ψ
δ ψ quad ( ψ ) = arg min δ ψ δ f quad ( ψ , δ ψ )
= H ( ψ ) 1 g ( ψ ) ,
δ ψ TR ( ψ ) = arg min δ ψ δ f quad ( ψ , δ ψ ) s.t. δ ψ Δ ,
δ ψ TR ( ψ ) = arg min δ ψ { δ f quad ( ψ , δ ψ ) + λ 2 δ ψ T Q δ ψ } ,
= [ H ( ψ ) + λ Q ] 1 g ( ψ ) ,
δ ψ = S β = i β i s i ,
δ f quad ( ψ , β , μ ) = 1 2 β T A ( ψ , μ ) β b ( ψ , μ ) T β ,
A ( ψ , μ ) = S T H ML ( ψ ) S + μ S T H prior ( ψ ) S ,
b ( ψ , μ ) = S T g ML ( ψ ) μ S T g prior ( ψ ) .
δ ψ ( μ , λ ) = S [ A ML + μ A prior + λ S T Q S ] 1 [ b ML + μ b prior ] ,
δ ψ QN = B g ( ψ )
= B g ML ( ψ ) μ B g prior ( ψ ) .
B = 1 μ H prior 1 ,
δ ψ QN = 1 μ H prior 1 g ML ( ψ ) H prior 1 g prior ( ψ ) .
s 1 = H prior 1 g ML ( ψ ) ,
s 2 = H prior 1 g prior ( ψ ) ,
g ( ψ + δ ψ ) g ( ψ ) + H ( ψ ) δ ψ ,
δ ψ QN next = B g ( ψ + δ ψ QN )
δ ψ QN B H ( ψ ) δ ψ QN ,
s 3 = H prior 1 H ML s 1 ,
s 4 = H prior 1 H ML s 3 .
s 5 = δ ψ ( n 1 ) .
μ s.t. δ f ML quad [ ψ , δ ψ ( μ ) ] = ε δ f ML quad [ ψ , δ ψ ( μ = 0 ) ] ,
SNR = N ph N sp N ph N sp + σ CCD 2 × s sp ,
f ( ψ ) = 1 2 e ( ψ ) T W e ( ψ ) ,
e ML ( ψ ) = d m ( ψ ) ; W ML = n n T 1 ,
e prior ( ψ ) = ψ ; W prior = K T C φ 1 K .
g ( ψ ) = def f ( ψ ) = J ( ψ ) T W e ( ψ ) ,
H ( ψ ) = def 2 f ( ψ ) J ( ψ ) T W J ( ψ ) ,
g ML = J ML T ( W ML ( d m ) ) ,
A i , j ML = ( J ML s i ) T W ML ( J ML s j ) .
s 3 , 4 = μ × H prior 1 J ML T ( J ML s 1 , 3 ) .
J j , ML = m j ( ψ ) p ,
= 2 α λ Im ( a λ , j k F j , k A λ , k φ λ , k p ) ,
( J ML q ) j = 2 α λ Im ( a λ , j k F j , k A λ , k φ λ , k p q ) ,
( J ML T q ) = 2 α λ k Im ( A λ , k j F k , j a λ , j q j ) φ λ , k p ,

Metrics