Abstract

We propose a new recursive filtering algorithm for wave-front reconstruction in a large-scale adaptive optics system. An embedding step is used in this recursive filtering algorithm to permit fast methods to be used for wave-front reconstruction on an annular aperture. This embedding step can be used alone with a direct residual error updating procedure or used with the preconditioned conjugate-gradient method as a preconditioning step. We derive the Hudgin and Fried filters for spectral-domain filtering, using the eigenvalue decomposition method. Using Monte Carlo simulations, we compare the performance of discrete Fourier transform domain filtering, discrete cosine transform domain filtering, multigrid, and alternative-direction-implicit methods in the embedding step of the recursive filtering algorithm. We also simulate the performance of this recursive filtering in a closed-loop adaptive optics system.

© 2005 Optical Society of America

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References

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  1. R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
    [CrossRef]
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    [CrossRef]
  4. L. A. Poyneer, M. Troy, B. Macintosh, D. T. Gavel, “Experimental validation of Fourier-transform wavefront reconstruction at the Palomar Observatory,” Opt. Lett. 28, 798–800 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  7. D. L. Fried, “Least-squares fitting a wave-front distortion estimate to an array of phase-difference measurements,” J. Opt. Soc. Am. 67, 370–375 (1977).
    [CrossRef]
  8. L. Gilles, C. R. Vogel, B. L. Ellerbroek, “A multigrid preconditioned conjugate gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
    [CrossRef]
  9. L. Gilles, “Order-N sparse minimum-variance open-loop reconstructor for extreme adaptive optics,” Opt. Lett. 28, 1927–1929 (2003).
    [CrossRef] [PubMed]
  10. D. G. MacMartin, “Local, hierachic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).
    [CrossRef]
  11. F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
    [CrossRef]
  12. D. C. Ghiglia, L. A. Romero, “Direct phase estimation from phase differences using fast elliptic partial differential equation solvers,” Opt. Lett. 14, 1107–1109 (1989).
    [CrossRef] [PubMed]
  13. R. H. Hudgin, “Wave-front reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]
  14. D. C. Ghiglia, L. A. Romero, “Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods,” J. Opt. Soc. Am. A 11, 107–117 (1994).
    [CrossRef]
  15. H. Ren, R. Dekany, “Fast wavefront reconstruction by solving the Sylvester equation with the alternating direction implicit method,” Opt. Express 12, 3279–3296 (2004), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  16. R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 170–176 (1985).
    [CrossRef]
  17. G. Rousset, “Wavefront sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]
  18. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1802–1816 (2002).
    [CrossRef]
  19. D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002).
    [CrossRef]
  20. G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).
  21. K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithm, Advantages, Applications (Academic, San Diego, Calif., 1990).
  22. R. G. Lane, A. Glindemann, J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2, 209–224 (1992).
    [CrossRef]
  23. D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

2004 (1)

2003 (3)

2002 (3)

1994 (1)

1992 (1)

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

1991 (1)

1989 (1)

1986 (1)

1978 (1)

1977 (2)

Base, J. M.

Bauman, B.

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Brack, G. L.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Burruss, R. S.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

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]

Dekany, R.

H. Ren, R. Dekany, “Fast wavefront reconstruction by solving the Sylvester equation with the alternating direction implicit method,” Opt. Express 12, 3279–3296 (2004), http://www.opticsexpress.org .
[CrossRef] [PubMed]

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Dekany, R. G.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Ellerbroek, B. L.

L. Gilles, C. R. Vogel, B. L. Ellerbroek, “A multigrid preconditioned conjugate gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. A 19, 1817–1822 (2002).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. A 19, 1802–1816 (2002).
[CrossRef]

Freischlad, K. R.

Fried, D. L.

Gavel, D. T.

Ghiglia, D. C.

Gilles, L.

Glindemann, A.

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

Golub, G. H.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Hudgin, R. H.

Koliopoulos, C. L.

Lane, R. G.

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

Macintosh, B.

MacMartin, D. G.

D. G. MacMartin, “Local, hierachic, and iterative reconstructors for adaptive optics,” J. Opt. Soc. Am. A 20, 1084–1093 (2003).
[CrossRef]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Mooney, J. G.

R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 170–176 (1985).
[CrossRef]

Nelson, J. E.

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

Noll, R. J.

Poyneer, L. A.

Pritt, M. D.

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

Rao, K. R.

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithm, Advantages, Applications (Academic, San Diego, Calif., 1990).

Ren, H.

Roddier, C.

Roddier, F.

Romero, L. A.

Rousset, G.

G. Rousset, “Wavefront sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Sasiela, R. J.

R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 170–176 (1985).
[CrossRef]

Shi, F.

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

Troy, M.

L. A. Poyneer, M. Troy, B. Macintosh, D. T. Gavel, “Experimental validation of Fourier-transform wavefront reconstruction at the Palomar Observatory,” Opt. Lett. 28, 798–800 (2003).
[CrossRef] [PubMed]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

van Loan, C. F.

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

Vogel, C. R.

Watkins, D. S.

D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002).
[CrossRef]

Yip, P.

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithm, Advantages, Applications (Academic, San Diego, Calif., 1990).

Appl. Opt. (1)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (6)

Opt. Express (1)

Opt. Lett. (3)

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 (8)

D. C. Ghiglia, M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, New York, 1998).

D. S. Watkins, Fundamentals of Matrix Computations, 2nd ed. (Wiley, New York, 2002).
[CrossRef]

G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).

K. R. Rao, P. Yip, Discrete Cosine Transform: Algorithm, Advantages, Applications (Academic, San Diego, Calif., 1990).

R. Dekany, J. E. Nelson, B. Bauman, “Design considerations for CELT adaptive optics,” in Optical Design, Materials, Fabrication, and Maintenance, P. Dierickx, ed., Proc. SPIE4003, 212–225 (2000).
[CrossRef]

F. Shi, D. G. MacMartin, M. Troy, G. L. Brack, R. S. Burruss, R. G. Dekany, “Sparse matrix wavefront reconstruction: simulations and experiments,” in Adaptive Optical System Technologies II, P. Wizinowich, ed., Proc. SPIE4839, 1035–1044 (2002).
[CrossRef]

R. J. Sasiela, J. G. Mooney, “An optical phase reconstructor based on using a multiplier-accumulator approach,” in Adaptive Optics, J. E. Ludman, ed., Proc. SPIE551, 170–176 (1985).
[CrossRef]

G. Rousset, “Wavefront sensors,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

(a), (b) Hudgin and Fried filters, respectively, in the DCT domain; (c), (d) the corresponding filters in the DFT domain.

Fig. 2
Fig. 2

Flow chart of the RF algorithm for closed-loop AO simulations.

Fig. 3
Fig. 3

rTRMS PRE curves of the DFTDF and DCTDF methods used with the CG residual error updating procedure.

Fig. 4
Fig. 4

rTRMS PRE curves of the DFTDF and DCTDF methods used with the direct residual error updating procedure.

Fig. 5
Fig. 5

rTRMS PRE curves of the DCTDF method when the Fried filters are used with the direct residual error updating procedure. The curvature and Tikhonov regularized Fried filters are compared with filters with no regularization.

Fig. 6
Fig. 6

rTRMS PRE curves of the DCTDF method when the Fried filters are used with the direct residual error updating procedure and with no masking operation used in the embedding step. The curvature and Tikhonov regularized Fried filters are compared with filters with no regularization.

Fig. 7
Fig. 7

rTRMS PRE curves of the DCTDF method when the Fried filters are used with the CG residual updating procedure and with no masking operation used in the embedding step. The curvature and Tikhonov regularized Fried filters are compared with the filters with no regularization.

Fig. 8
Fig. 8

rTRMS PRE curves of the DCTDF method when the non-regularized Hudgin and Fried filters are used with the CG residual error updating procedure.

Fig. 9
Fig. 9

rTRMS PRE curves of the DCTDF method when the non-regularized Hudgin and Fried filters are used with the direct residual error updating procedure.

Fig. 10
Fig. 10

rTRMS PRE curves of the DCTDF and multigrid methods when they are used with the CG residual error updating procedure. The nonregularized Hudgin filter is used in the DCTDF method.

Fig. 11
Fig. 11

rTRMS PRE curves of the DCTDF and multigrid methods when they are used with the direct residual error updating procedure. The nonregularized Hudgin filter is used in the DCTDF method.

Fig. 12
Fig. 12

rTRMS PRE curves of the DFTDF, DCTDF, multigrid, and ADI methods when they are used in the embedding step of the closed-loop RF algorithm and the residual error updating step is implemented by the PSD method.

Fig. 13
Fig. 13

rTRMS PRE curves of the DFTDF, DCTDF, multigrid, and ADI methods when they are used in the embedding step of the closed-loop RF algorithm and the residual error updating step is implemented by the direct updating procedure.

Equations (53)

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s = P ϕ + η ,
Ω = ( s P ϕ ) T C η 1 ( s P ϕ ) + β ϕ T K T K ϕ .
ϕ ̂ = R s ,
R = ( P T C η 1 P + β K T K ) 1 P T C η 1 .
R = F P T ,
F = ( W + σ 2 β K T K ) 1 .
Q = V Λ V 1 = V Λ V T ,
Λ F = ( Λ W + σ 2 β Λ K 2 ) 1 ,
W = I A + A I ,
A = [ 1 1 1 2 1 1 2 1 1 1 ] .
V = U U ,
Λ W = I Λ A + Λ A I ,
( A λ m I ) u m = 0 ( m = 0 , 1 , , N 1 ) .
u m , n 1 + ( 2 λ m ) u m , n u m , n + 1 = 0 ( n = 0 , 1 , , N 1 ) ,
u m , 1 = u m , 0 , u m , N = u m , N 1 .
r 2 + ( 2 λ m ) r 1 = 0 .
u m , n = w m { a cos [ ( n + 1 ) θ m ] + b sin [ ( n + 1 ) θ m ] } ,
[ 1 cos θ m sin θ m cos [ ( N + 1 ) θ m ] cos ( N θ m ) sin [ ( N + 1 ) θ m ] sin ( N θ m ) ] ( a b ) = 0 .
( cos θ m 1 ) sin ( N θ m ) = 0 .
u m , n = w m sin ( m π 2 N ) cos [ m ( 2 n + 1 ) π 2 N ] ,
w m = sin 1 ( m π 2 N ) ( n = 0 N 1 cos 2 [ m ( 2 n + 1 ) π 2 N ] ) 1 / 2 .
u m , n = 2 N κ m cos [ m ( 2 n + 1 ) π 2 N ] ( n = 0 , 1 , , N 1 ) ,
λ m = 4 sin 2 ( m π 2 N ) ( m = 0 , 1 , , N 1 ) .
W = H A + A H ,
H = 1 4 [ 1 1 1 2 1 1 2 1 1 1 ] ,
u m , n 1 + ( 2 τ m ) u m , n + u m , n + 1 = 0 , ( n = 0 , 1 , , N 1 ) ,
u m , 1 = u m , 0 , u m , N = u m , N 1 ,
Λ W Λ H Λ A + Λ A Λ H ,
H DCT = 1 4 [ 3 1 1 2 1 1 2 1 1 3 ] .
τ m = cos 2 ( m π 2 N ) ( m = 0 , 1 , , N 1 ) .
Y m , n = ( S m , n + σ 2 β ) 1 ,
Y m , n = { S m , n + 16 σ 2 β [ sin 2 ( π m 2 N ) + sin 2 ( π n 2 N ) ] 2 } 1 ,
S m , n = 4 [ sin 2 ( π m 2 N ) + sin 2 ( π n 2 N ) ] ,
S m , n = 4 [ sin 2 ( π m 2 N ) cos 2 ( π n 2 N ) + sin 2 ( π n 2 N ) cos 2 ( π m 2 N ) ] ,
u m , 1 = u m , N 1 , u m , N = u m , 1 ,
A DFT = [ 2 1 1 1 2 1 1 2 1 1 1 2 ] .
f m = 4 sin 2 ( m π N ) ( m = 0 , 1 , , N 1 ) .
H DFT = 1 4 [ 2 1 1 1 2 1 1 2 1 1 1 2 ] .
h m = cos 2 ( m π N ) ( m = 0 , 1 , , N 1 ) .
S m , n F = 4 [ sin 2 ( π m N ) + sin 2 ( π n N ) ] ,
S m , n F = 4 [ sin 2 ( π m N ) cos 2 ( π n N ) + sin 2 ( π n N ) cos 2 ( π m N ) ] ,
Y m , n F = ( S m , n F + σ 2 β ) 1 ,
Y m , n F = { S m , n F + 16 σ 2 β [ sin 2 ( π m N ) + sin 2 ( π n N ) ] 2 } 1 .
X = IDCT [ Y × DCT ( B ) ] ,
X = real { IDFT Y F × DFT ( B ) } ,
NPC = 1 N 2 Trace ( R R T ) .
NPC DCTDF = 1 N 2 m , n = 0 N 1 Y m , n 2 S m , n .
NPC DCTDF = 0.31481 + 0.08042 ln N 2 ,
NPC DCTDF = 0.15597 + 0.15925 ln N 2 .
C x ̂ = b ,
C = W i + σ 2 β K T K ,
F 1 x ̂ = b .
s k = G ( φ k x k ) + η k .

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