Abstract

A study has been made of a Gram–Charlier matched filter for Shack–Hartmann sensing of wavefront slopes. The method is based on modeling the point-spread function by an expansion in terms of Gauss–Hermite polynomials. We present results for several subaperture/coherence area sizes both with and without CCD read noise. A more accurate estimation of the local slopes can be achieved at low light levels in this way than with the standard first-moment estimator.

© 1998 Optical Society of America

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References

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  1. N. Dillon, Gemini A & G System Tech. Rep. (Royal Observatories, Cambridge, UK, 1995).
  2. B. F. Alexander and K. C. Ng, Opt. Eng. 30, 1320 (1991).
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    [CrossRef]
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    [CrossRef]
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  14. T. O. Lewis and P. L. Odell, Estimation in Linear Models (Prentice-Hall, Engelwood Cliffs, N. J., 1971).
  15. D. L. Ash and C. J. Solomon, Proc. SPIE 3126, 300 (1997).
    [CrossRef]

1997

A. Patwardhan, J. Microsc. 186, 246 (1997).
[CrossRef]

D. L. Ash and C. J. Solomon, Proc. SPIE 3126, 300 (1997).
[CrossRef]

S. A. Sallberg, B. M. Welsh, and M. C. Roggemann, J. Opt. Soc. Am. A 14, 1347 (1997).
[CrossRef]

1996

1995

C. J. Solomon, N. J. Wooder, and J. C. Dainty, Opt. Rev. 2, 217 (1995).
[CrossRef]

I. E. Abdou, Opt. Eng. 34, 3347 (1995).

M. Chemloul and V. Comparat, Nucl. Instrum. Meth. A 367, 290 (1995).
[CrossRef]

1993

M. P. Wrenet and A. Pline, Exp. Fluids 15, 295 (1993).

1991

B. F. Alexander and K. C. Ng, Opt. Eng. 30, 1320 (1991).
[CrossRef]

R. Baribeau and M. Rioux, Appl. Opt. 30, 3752 (1991).
[CrossRef] [PubMed]

1986

1976

R. J. Noll, J. Opt. Soc. Am. A 66, 207 (1976).
[CrossRef]

Abdou, I. E.

I. E. Abdou, Opt. Eng. 34, 3347 (1995).

Alexander, B. F.

B. F. Alexander and K. C. Ng, Opt. Eng. 30, 1320 (1991).
[CrossRef]

Ash, D. L.

D. L. Ash and C. J. Solomon, Proc. SPIE 3126, 300 (1997).
[CrossRef]

Baribeau, R.

Chemloul, M.

M. Chemloul and V. Comparat, Nucl. Instrum. Meth. A 367, 290 (1995).
[CrossRef]

Comparat, V.

M. Chemloul and V. Comparat, Nucl. Instrum. Meth. A 367, 290 (1995).
[CrossRef]

Dainty, J. C.

C. J. Solomon, N. J. Wooder, and J. C. Dainty, Opt. Rev. 2, 217 (1995).
[CrossRef]

Dillon, N.

N. Dillon, Gemini A & G System Tech. Rep. (Royal Observatories, Cambridge, UK, 1995).

Lewis, T. O.

T. O. Lewis and P. L. Odell, Estimation in Linear Models (Prentice-Hall, Engelwood Cliffs, N. J., 1971).

Montera, D. A.

Ng, K. C.

B. F. Alexander and K. C. Ng, Opt. Eng. 30, 1320 (1991).
[CrossRef]

Noll, R. J.

R. J. Noll, J. Opt. Soc. Am. A 66, 207 (1976).
[CrossRef]

Odell, P. L.

T. O. Lewis and P. L. Odell, Estimation in Linear Models (Prentice-Hall, Engelwood Cliffs, N. J., 1971).

Patwardhan, A.

A. Patwardhan, J. Microsc. 186, 246 (1997).
[CrossRef]

Pline, A.

M. P. Wrenet and A. Pline, Exp. Fluids 15, 295 (1993).

Rioux, M.

Roggeman, M. C.

Roggemann, M. C.

Ruck, D. W.

Sallberg, S. A.

Solomon, C. J.

D. L. Ash and C. J. Solomon, Proc. SPIE 3126, 300 (1997).
[CrossRef]

C. J. Solomon, N. J. Wooder, and J. C. Dainty, Opt. Rev. 2, 217 (1995).
[CrossRef]

Welsh, B. M.

Winick, K. A.

Wooder, N. J.

C. J. Solomon, N. J. Wooder, and J. C. Dainty, Opt. Rev. 2, 217 (1995).
[CrossRef]

Wrenet, M. P.

M. P. Wrenet and A. Pline, Exp. Fluids 15, 295 (1993).

Appl. Opt.

Exp. Fluids

M. P. Wrenet and A. Pline, Exp. Fluids 15, 295 (1993).

J. Microsc.

A. Patwardhan, J. Microsc. 186, 246 (1997).
[CrossRef]

J. Opt. Soc. Am. A

Nucl. Instrum. Meth. A

M. Chemloul and V. Comparat, Nucl. Instrum. Meth. A 367, 290 (1995).
[CrossRef]

Opt. Eng.

B. F. Alexander and K. C. Ng, Opt. Eng. 30, 1320 (1991).
[CrossRef]

I. E. Abdou, Opt. Eng. 34, 3347 (1995).

Opt. Rev.

C. J. Solomon, N. J. Wooder, and J. C. Dainty, Opt. Rev. 2, 217 (1995).
[CrossRef]

Proc. SPIE

D. L. Ash and C. J. Solomon, Proc. SPIE 3126, 300 (1997).
[CrossRef]

Other

T. O. Lewis and P. L. Odell, Estimation in Linear Models (Prentice-Hall, Engelwood Cliffs, N. J., 1971).

B. R. Frieden, ed., Probability, Statistical Optics, and Data Testing (Springer-Verlag, Berlin, 1991).
[CrossRef]

N. Dillon, Gemini A & G System Tech. Rep. (Royal Observatories, Cambridge, UK, 1995).

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Figures (2)

Fig. 1
Fig. 1

Mean-square (MS) error in the centroid estimate with the first-moment estimator and for different orders (0 to 6) of the Gram–Charlier fit. The flux level is 30 photoelectrons/subaperture. No read noise is present, and l/r0 varies from 1 to 3.

Fig. 2
Fig. 2

Mean-square (MS) error in the centroid estimate with the first-moment estimator and for different orders (0 to 6) of the Gram–Charlier fit. The flux level is 30 photoelectrons/subaperture. Poisson read noise is added, and l/r0 varies from 1 to 3.

Tables (1)

Tables Icon

Table 1 Root-Mean-Square Wave-Front Reconstruction Error for the First-Moment and Gram–Charlier Estimatorsa

Equations (11)

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

Ixi, yj=I0xi, yj+Nxi, yj i=1,,N; j=1,N,
Pk=e-mmkk!.
cx=i=1Nj=1NxiIxi, yji=1Nj=1NIxi, yj, cy=i=1Nj=1NyiIxi, yji=1Nj=1NIxi, yj.
Ix, y=G0x, y1+i=1maij,kAij,kx, yj!i-j!,
G0x, y=αexp-x-u22σx2+y-v22σy2.
χ2α, u, v, σx, σy=i=1Nj=1NIxi, yj-G0xi, yj2.
Aij,kx, y=HjxHky j+k=i, 0ji, 0ki,
Hj,Hk=12π R exp-x2/2Hjxdx =j!δjk,
Aij,k,Aij,k=RRexp-x2+y2/22πHjxHjx×HkyHkydxdy=j!k!δj-jδk-k.
aij,k=RRexp-x2+y2/22πIx, yG0x, y-1×Aijkx, ydxdy,
aij,k=l=1Np=1Nexp[-(xl2+yp2/2]2πIxl, ypG0xl, yp-1×Aijkxl, yp.

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