Abstract

The problem of estimating the centroid of an incoherently imaged point with a CCD array is analyzed. An exact analysis is presented that uses the actual short-exposure function at the CCD instead of the traditional Gaussian approximation. The analysis shows that, for Poisson noise, the centroid variance depends on the CCD size and that truncation effects play a significant part in determining the optimum CCD size. The effects of this on a wave-front reconstruction formed by a Shack–Hartmann sensor are described.

© 1999 Optical Society of America

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References

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  1. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  2. J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (1990).
    [CrossRef]
  3. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics system using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 12, 1913–1923 (1989).
    [CrossRef]
  4. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  5. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  6. P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
    [CrossRef]
  7. N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
    [CrossRef]
  8. E. P. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. A 73, 1771–1776 (1983).
    [CrossRef]
  9. N. F. Law, R. G. Lane, “Wavefront estimation at low light levels,” Opt. Commun. 126, 19–24 (1996).
    [CrossRef]
  10. H. W. Sorenson, Parameter Estimation: Principle and Problems (Marcel Dekker, New York, 1980).
  11. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991).
  12. T. Y. Kane, B. M. Welsh, C. S. Gardner, L. A. Thomson, “Wavefront detection optimization for laser guided adaptive telescope,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1990).
    [CrossRef]
  13. K. A. Winick, “Cramer–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
    [CrossRef]
  14. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, T. L. Pennigton, “Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,” Appl. Opt. 34, 4186–4195 (1995).
    [CrossRef] [PubMed]
  15. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,” J. Opt. Soc. Am. A 56, 1372–1379 (1966).
    [CrossRef]
  16. C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
    [CrossRef]

1999 (1)

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)

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

1990 (2)

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

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

1989 (1)

1986 (1)

1983 (1)

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

1980 (1)

1976 (1)

1966 (1)

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

Bakut, P. A.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Dainty, J. C.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Ellerbroek, B. L.

Fontanella, J. C.

J. Primot, G. Rousset, J. C. Fontanella, “Deconvolution from wave-front sensing: a new technique for compensating turbulence-degraded images,” J. Opt. Soc. Am. A 9, 1598–1608 (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. A 56, 1372–1379 (1966).
[CrossRef]

Gardner, C. S.

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics system using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 12, 1913–1923 (1989).
[CrossRef]

T. Y. Kane, B. M. Welsh, C. S. Gardner, L. A. Thomson, “Wavefront detection optimization for laser guided adaptive telescope,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1990).
[CrossRef]

Harding, C. M.

Johnston, R. A.

Kane, T. Y.

T. Y. Kane, B. M. Welsh, C. S. Gardner, L. A. Thomson, “Wavefront detection optimization for laser guided adaptive telescope,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1990).
[CrossRef]

Kirakosyants, V. E.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Lane, R. G.

C. M. Harding, R. A. Johnston, R. G. Lane, “Fast simulation of a Kolmogorov phase screen,” Appl. Opt. 38, 2161–2170 (1999).
[CrossRef]

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

Law, N. F.

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

Loginov, V. A.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Noll, R. J.

Pennigton, T. L.

Primot, J.

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

Roddier, N.

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Roggemann, M. C.

Rousset, G.

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

Scharf, L.

L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991).

Solomon, C. J.

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Sorenson, H. W.

H. W. Sorenson, Parameter Estimation: Principle and Problems (Marcel Dekker, New York, 1980).

Southwell, W. H.

Thomson, L. A.

T. Y. Kane, B. M. Welsh, C. S. Gardner, L. A. Thomson, “Wavefront detection optimization for laser guided adaptive telescope,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1990).
[CrossRef]

Wallner, E. P.

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

Welsh, B.

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

Welsh, B. M.

Winick, K. A.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

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

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

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

K. A. Winick, “Cramer–Rao lower bounds on the performance of charge-coupled-device optical position estimators,” J. Opt. Soc. Am. A 3, 1809–1815 (1986).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics system using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 12, 1913–1923 (1989).
[CrossRef]

Opt. Commun. (2)

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

P. A. Bakut, V. E. Kirakosyants, V. A. Loginov, C. J. Solomon, J. C. Dainty, “Optimal wavefront reconstruction from a Shack–Hartmann sensor by use of a Bayesian algorithm,” Opt. Commun. 109, 10–15 (1994).
[CrossRef]

Opt. Eng. (1)

N. Roddier, “Atmospheric wave-front simulation using Zernike polynomials,” Opt. Eng. 29, 1174–1180 (1990).
[CrossRef]

Other (4)

H. W. Sorenson, Parameter Estimation: Principle and Problems (Marcel Dekker, New York, 1980).

L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991).

T. Y. Kane, B. M. Welsh, C. S. Gardner, L. A. Thomson, “Wavefront detection optimization for laser guided adaptive telescope,” in Active Telescope Systems, F. J. Roddier, ed., Proc. SPIE1114, 160–171 (1990).
[CrossRef]

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

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Figures (8)

Fig. 1
Fig. 1

Displacement of the spot on the detector due to the presence of atmospheric turbulence. The tilt of the wave front is proportional to the displacement of the spot formed.

Fig. 2
Fig. 2

Poisson noise corrupted diffraction-limited image. The number of photons is equal to 500.

Fig. 3
Fig. 3

Difference in the noise variance inherent in the centroid calculation with the Gaussian approximation (dashed line) and the diffraction-limited spot (solid curve). The subaperture size is set to d = 25 cm, and the number of photons n = 1000.

Fig. 4
Fig. 4

OTF for the unaberrated system. The abscissa is normalized spatial frequency.

Fig. 5
Fig. 5

Geometry of the error measurement on a detector array due to truncation. The diffraction-limited spot is held at the origin, and the detector is allowed to move at a variable distance ζ x .

Fig. 6
Fig. 6

Illustration of the relative importance of the Poisson and the truncation noise for a diffraction-limited spot. The centroid variances corresponding to 10 photons (dotted–dashed curve) and 100 (dashed curve) are shown. Solid curve, variance due to the truncation.

Fig. 7
Fig. 7

Comparison of simulation and analytical results of truncation error, σ t 2, as a function of the detection radius, ρ. Solid curves, theoretical results; dashed curves, results for d/ r 0 = 1 (lower curve), d/ r 0 = 2 (middle curve), and d/ r 0 = 4 (upper curve).

Fig. 8
Fig. 8

Centroid variances due to Poisson noise for aberrated systems with d/r 0 = 1 (lower solid curve) and d/ r 0 = 4 (upper solid curve). Circles, results of the simulations for d/ r0 = 1. Stars, results of the simulations for d/ r 0 = 4. The subaperture size d is assumed to be 25 cm, and n = 1000 photons.

Tables (1)

Tables Icon

Table 1 Entries for the Noise Matrix Na

Equations (24)

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ϕu, v=n=2J anZnu, v,
m=Θa+n,
N=nnT,
aˆ=ΘTN-1Θ+C-1-1ΘTN-1m=CΘTΘCΘT+N-1m,
Ω=CΘTΘCΘT+N-1,
Δ=a-aˆa-aˆT=I-ΩΘCI-ΩΘT+ΩNΩT,
σc2=λd22n-- x2hx, ydxdy  rad2,
-- hx, ydxdy=1.
σα2=2πλ2σc2 rad/m2.
βc=1.2222λd rad
βs=1.0522λd rad
σαc=0.86πnd rad/m
σαs=0.74πnd rad/m
hx, y=1π2sinxsinyxy2.
K1-ρρsin2xdx,
u=λfLfd,
bζx, ρ=x=-ρ+ζxρ+ζxy=0ρ2-x-ζx21/2 xhx, ydxdyx=-ρ+ζxρ+ζxy=0ρ2-x-ζx21/2 hx, ydxdy.
σt2ρ=2πλ2-bζx, ρ212πσζ1/2×exp-ζx22πσζ2dζx rad/m2.
uSE=2πcos-1u-u1-u21/2×exp-3.44λ|u|r05/31-λ|u|d1/3
uSE=2πcos-1u-u1-u21/2×exp-3.44λ|u|r05/3.
K2=- uLEdu- uSEdu,
σα=K20.86πnd rad/m.
hrSE=2π 01 ηSEJ02πrηηdη,
σζ2=0.36λd2dr05/3 rad2.

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