Abstract

Current methods for estimating the wave-front slope at the pupil of a telescope by using a Shack–Hartmann wave-front sensor (SH–WFS) are based on a simple centroid calculation of the irradiance distributions (spots) recorded in each subaperture. The centroid calculation does not utilize knowledge concerning the correlation properties of the slopes over the subapertures or the amount of light collected by the SH–WFS. We present the derivation of a maximum a posteriori (MAP) estimation of the irradiance centroids by incorporating statistical knowledge of the wave-front tilts. Information concerning the light level in each subaperture and the relative spot size is also employed by the estimator. The MAP centroid estimator is found to be unbiased, and the mean square error performance is upper bounded by that exhibited by the classical centroid technique. This error performance is demonstrated by using Kolmogorov wave-front slope statistics for various light levels.

© 1997 Optical Society of America

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References

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  1. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  2. J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
    [CrossRef]
  3. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  4. D. L. Fried, “Post-detection wavefront distortion compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. SPIE828, 127–133 (1987).
    [CrossRef]
  5. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensat-ing turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
    [CrossRef]
  6. M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).
  7. T. P. McGarty, “The estimation of the ‘center of gravity’ of a photon density profile in noise,” IEEE Trans. Aeros. Electron. Syst. AES-5, 947–979 (1969).
  8. K. A. Winick, “Cramér–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  9. R. C. Cannon, “Global wave-front reconstruction using Shack–Hartmann sensors,” J. Opt. Soc. Am. A 12, 2031–2039 (1995).
    [CrossRef]
  10. T. J. Schulz, “Estimation–theoretic approach to the deconvolution of atmospherically degraded images with wave-front sensor measurements,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 311–320 (1993).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  12. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  13. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Inc., Fort Worth, Texas, 1988).
  14. A. Graham, Kronecker Products and Matrix Calculus with Applications (Halsted, New York, 1981).
  15. S. A. Sallberg, “Maximum likelihood estimation of wave front slopes using a Hartmann-type sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1995].
  16. T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
    [CrossRef]
  17. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
    [CrossRef] [PubMed]
  18. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [CrossRef]
  19. G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (The Johns Hopkins University Press, Baltimore, Md., 1989).
  20. D. A. Montera, B. M. Welsh, M. C. Roggemann, D. W. Ruck, “Processing wave-front-sensor slope measurements using artificial neural networks,” Appl. Opt. 35, 4238–4251 (1996).
    [CrossRef] [PubMed]
  21. T. Pennington, “Performance comparison of shearing interferometer and Hartmann wave front sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1993].

1996 (1)

1995 (2)

1992 (1)

1990 (1)

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensat-ing turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

1986 (1)

1983 (1)

1978 (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1970 (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

1969 (1)

T. P. McGarty, “The estimation of the ‘center of gravity’ of a photon density profile in noise,” IEEE Trans. Aeros. Electron. Syst. AES-5, 947–979 (1969).

1966 (1)

Cannon, R. C.

Ellerbroek, B. L.

Fienup, J. R.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensat-ing turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Fried, D. L.

D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
[CrossRef]

D. L. Fried, “Post-detection wavefront distortion compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. SPIE828, 127–133 (1987).
[CrossRef]

Gardner, C. S.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (The Johns Hopkins University Press, Baltimore, Md., 1989).

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Graham, A.

A. Graham, Kronecker Products and Matrix Calculus with Applications (Halsted, New York, 1981).

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Kane, T. J.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

McGarty, T. P.

T. P. McGarty, “The estimation of the ‘center of gravity’ of a photon density profile in noise,” IEEE Trans. Aeros. Electron. Syst. AES-5, 947–979 (1969).

Montera, D. A.

Paxman, R. G.

Pennington, T.

T. Pennington, “Performance comparison of shearing interferometer and Hartmann wave front sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1993].

Pennington, T. L.

Primot, J.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensat-ing turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensat-ing turbulence-degraded images,” J. Opt. Soc. Am. A 7, 1589–1608 (1990).
[CrossRef]

Ruck, D. W.

Sallberg, S. A.

S. A. Sallberg, “Maximum likelihood estimation of wave front slopes using a Hartmann-type sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1995].

Schulz, T. J.

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

T. J. Schulz, “Estimation–theoretic approach to the deconvolution of atmospherically degraded images with wave-front sensor measurements,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 311–320 (1993).
[CrossRef]

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Inc., Fort Worth, Texas, 1988).

Thompson, L. A.

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (The Johns Hopkins University Press, Baltimore, Md., 1989).

Wallner, E. P.

Welsh, B. M.

D. A. Montera, B. M. Welsh, M. C. Roggemann, D. W. Ruck, “Processing wave-front-sensor slope measurements using artificial neural networks,” Appl. Opt. 35, 4238–4251 (1996).
[CrossRef] [PubMed]

B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennington, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
[CrossRef] [PubMed]

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
[CrossRef]

Winick, K. A.

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analysing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

IEEE Trans. Aeros. Electron. Syst. (1)

T. P. McGarty, “The estimation of the ‘center of gravity’ of a photon density profile in noise,” IEEE Trans. Aeros. Electron. Syst. AES-5, 947–979 (1969).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Proc. IEEE (1)

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Other (10)

M. C. Roggemann, B. M. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

D. L. Fried, “Post-detection wavefront distortion compensation,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. SPIE828, 127–133 (1987).
[CrossRef]

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovich, Inc., Fort Worth, Texas, 1988).

A. Graham, Kronecker Products and Matrix Calculus with Applications (Halsted, New York, 1981).

S. A. Sallberg, “Maximum likelihood estimation of wave front slopes using a Hartmann-type sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1995].

T. J. Kane, B. M. Welsh, C. S. Gardner, L. A. Thompson, “Wave front detector optimization for laser guided adaptive telescopes,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1989).
[CrossRef]

T. J. Schulz, “Estimation–theoretic approach to the deconvolution of atmospherically degraded images with wave-front sensor measurements,” in Digital Image Recovery and Synthesis II, P. S. Idell, ed., Proc. SPIE2029, 311–320 (1993).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

G. H. Golub, C. F. Van Loan, Matrix Computations, 2nd ed. (The Johns Hopkins University Press, Baltimore, Md., 1989).

T. Pennington, “Performance comparison of shearing interferometer and Hartmann wave front sensor,” master’s thesis [Graduate School of Engineering, Air Force Institute of Technology (AETC), Wright-Patterson Air Force Base, Ohio, 1993].

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Figures (4)

Fig. 1
Fig. 1

Factor 1-1/(1+K¯jσs2/σp2) versus K¯jσs2/σp2.

Fig. 2
Fig. 2

Ratio of the mean square errors (MSE's) for the MAP and the conventional centroid estimator as a function of average subaperture photon count, K¯. The wave-front slope correlation statistics match those that would be expected from Kolmogorov atmospheric statistics. The individual curves correspond to a particular value of the ratio of the mean square spot motion, σs2 to the mean square spot size, σp2.

Fig. 3
Fig. 3

Ratio of the MSE's for the MAP and the conventional centroid estimator as a function of average subaperture photon count, K¯. The wave-front slope correlation statistics match those that would be expected from Kolmogorov atmospheric statistics with global tilt removed. The individual curves correspond to a particular value of the ratio of the mean square spot motion, σs2, to the mean square spot size, σp2.

Fig. 4
Fig. 4

Ratio of the average subaperture MSE to the mean square spot size, σp2, as a function of average subaperture photon count, K¯. The wave-front slope correlation statistics match those that would be expected from Kolmogorov atmospheric statistics. The curves are plotted for σs2/σp2=0.5.

Equations (56)

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idlx-λavgfl2π a,
fd|λ{d(x)|λ(x)}=λ(x)d(x) exp{-λ(x)}d(x)!,xS,
λ(x)=idl(x-xs),
fd|xs(d|xs)=i=1I [idl(xi-xs)]d(xi) exp{-idl(xi-xs)}d(xi)!,
fd,xs(d, xs)=fd|xs(d|xs)fxs(xs),
fxs(xs)=12πσs2 exp-|xs|22σs2,
fd,xs(d, xs)=fd|xs(d|xs)fxs(xs),
fd|xs(d|xs)
=j=1Ji=1I [idl(xi-xsj)]dj(xi) exp{-idl(xi-xsj)}dj(xi)!.
fxs(xs)=(2π)-J|R|-1/2 exp-12 xsTR-1xs,
L(d, xs)=j=1Ji=1Idj(xi)ln{idl(xi-xsj)}-idl(xi-xsj)-ln{dj(xi)!}+ln{(2π)-J}+ln{|R|-1/2}-12 xsTR-1xs.
xs L(d, xs)=xs j=1Ji=1Idj(xi)ln{idl(xi-xsj)}-idl(xi-xsj)-xs 12 xsTR-1xs.
1=[1, , 1]T,
xi1=[xi, , xi]T,
idl(xi1-xs)=[idl(xi-xs1), , idl(xi-xsJ)]T,
d(xi)=[d1(xi), , dJ(xi)]T,
xs L(d, xs)=i=1Ixs ln{[idl(xi1-xs)]T}d(xi)-xs [idl(xi1-xs)]T1-xs 12 xsTR-1xs.
i=1I xs [idl(xi1-xs)]T(ΛD-1d(xi)-1)-R-1xs=0,
idl(xi-xsj)=l2K¯j2πσp2 exp-|xi-xsj|22σp2,
K¯j=Ei=1Idj(xi)=E{djT1}=[idl(x)]T1,
σs2=σθ2fl2,
σθ2=12.803b1/3r05/3k2,
F/#=fldb1/3r05/31.2972λ2.
F/#=flbb2λ.
[idl(x-xsj1)]TxK¯j=xsjforallj.
i=1I xs [idl(xi1-xs)]T1
Cxs=m,
C=K¯100K¯J+σp2R-1=K+σp2R-1
m=d1TxdJTx=[d1, , dJ]Tx=DTx,
x^s=C-1m.
m˘=d1TxK¯1dJTxK¯J=K-1m=K-1DTx.
x^s=C-1KK-1m=C˘-1m˘,
C˘=(C-1K)-1=K-1C=I+σp2K-1R-1.
C˘-1=I-1-I-1(σp2K-1)[I+R-1I-1×(σp2K-1)]-1R-1I-1,=I-σp2K-1(I+σp2R-1K-1)-1R-1,=I-I+1σp2 RK-1.
x^s=I-I+1σp2 RK-1m˘=m˘-I+1σp2 RK-1m˘.
I+1σp2 RK-1=0,
x^s=(I-0)m˘=m˘.
I+1σp2 RK-1=I,
x^s=(I-I)m˘=0.
x^sj=1-11+K¯jσs2/σp2m˘j,
MSE(x^s)=tr(E{[x^s-xs][x^s-xs]T})=tr(E{x^sx^sT}-2E{x^sxsT}+E{xsxsT})=tr(E{x^sx^sT})-2 tr(E{x^sxsT})+tr(R),
MSE(x^s)=tr(E{[C˘-1m˘][C˘-1m˘]T})-2 tr(E{[C˘-1m˘]xsT})+tr(R)=tr(C˘-1E{m˘m˘T}C˘-1)-2 tr(C˘-1E{m˘xsT})+tr(R)=tr(C˘-1MC˘-1)-2 tr(C˘-1N)+tr(R),
N=E{K-1mxsT}=E{K-1DTxxsT}=Exs{Ed|xs(K-1DTx)xsT}=R.
M=E{m˘m˘T}=Exs{Ed|xs(m˘m˘T)}.
M=ExsEd|xs(m˘)Ed|xs(m˘T)-diag[Ed|xs(m˘1)Ed|xs(m˘1T), , Ed|xs(m˘J)Ed|xs(m˘JT)]+diag[Ed|xs(m˘1m˘1T), , Ed|xs(m˘Jm˘JT)].
M=ExsxsxsT-diag[xs1xs1T, , xsJxsJT]+diag[Ed|xs(m˘1m˘1T), , Ed|xs(m˘Jm˘JT)],
=R+Exsdiag[Ed|xs(m˘1m˘1T-xs1xs1T), ,Ed|xs(m˘Jm˘JT-xsJxsJT)].
Exs{Ed|xs(m˘jm˘jT-xsjxsjT)}.
Exs1K¯j2 Ed|xs{djT[(x-1xsj)xˆ]}2,
Exs1K¯j2 Ed|xs{djT[(x-1xsj)xˆ]}2
=1K¯j2 Exs{[(x-1xsj)xˆ]T×Ed|xs(djdjT)[(x-1xsj)xˆ]}.
1K¯j2 Exs{Ed|xs({djT[(x-1xsj)xˆ]}2)}
=1K¯j2 Exs{[(x-1xsj)xˆ]T    [idl(x-1xsj)]×[idl(x-1xsj)]T+diag[idl(x-1xsj)]×[(x-1xsj)xˆ]}=1K¯j2 [(x-1xsj)xˆ]T diag[idl(x-1xsj)]×[(x-1xsj)xˆ]=σp2K¯j,
M=R+diagσp2K¯1, ,σp2K¯J=R+σp2K-1.
MSE(x^s)=tr(σp2K-1)-2 trI+1σp2 RK-1σp2K-1+trI+1σp2 RK-1(R+σp2K-1)×I+1σp2 RK-1=tr(σp2K-1)-trI+1σp2 RK-1σp2K-1=trI-I+1σp2 RK-1σp2K-1.
σs2σp2=(flσθ)2(0.37flλ/b)2=2.37br05/3,

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