Abstract

For adaptive-optics systems to compensate for atmospheric turbulence effects, the wave-front perturbation must be measured with a wave front sensor (WFS), and key parameters of the atmosphere and the adaptive-optics system must be known. Two parameters of particular interest include the Fried coherence length r 0 and the WFS slope measurement error. Statistics-based optimal techniques, such as the minimum variance phase reconstructor, have been developed to improve the imaging performance of adaptive-optics systems. However, these statistics-based models rely on knowledge of the current state of the key parameters. Neural networks provide nonlinear solutions to adaptive-optics problems while offering the possibility of adapting to changing seeing conditions. We address the use of neural networks for three tasks: (1) to reduce the WFS slope measurement error, (2) to estimate the Fried coherence length r 0, and (3) to estimate the variance of the WFS slope measurement error. All of these tasks are accomplished by using only the noisy WFS measurements as input. Where appropriate, we compare our method with classical statistics-based methods to determine if neural networks offer true benefits in performance. Although a statistics-based method is found to perform better than a neural network in reducing WFS slope measurement error, neural networks perform better in estimating the variance of the WFS slope measurement error, and both methods perform well in estimating r 0.

© 1996 Optical Society of America

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References

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  1. B. Welsh, M. Roggemann, “Evaluating the performance of adaptive optical telescopes,” Adaptive Optics for Astronomy, D. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Norwell, Mass., 1993).
  2. B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [Crossref]
  3. B. Welsh, C. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [Crossref]
  4. 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]
  5. M. Lloyd-Hart, P. Wizinowich, J. Angel, “A neural network wavefront sensor for array telescopes,” presented at the International Commission for Optics, Florence, Italy (1991).
  6. P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).
  7. J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
    [Crossref]
  8. M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.
  9. M. Jorgenson, G. Aitken, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
    [Crossref] [PubMed]
  10. M. Jorgenson, G. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
    [Crossref] [PubMed]
  11. M. Jorgenson, G. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjodi, ed., Proc. SPIE1706 (1992).
  12. M. Jorgenson, G. Aitken, “Wavefront prediction for adaptive optics,” in Proceedings of ICO-16, Active and Adaptive Optics (International Commission for Optics, 1993).
  13. L. Scharf, Statistical Signal Processing (Addison-Wesley, Reading, Mass., 1991).
  14. E. Wallner, “Optimal wave-front correction using slope measurements,” J. Opt. Soc. Am. 73, 1771–1776 (1983).
    [Crossref]
  15. S. Tamura, A. Waibel, “Noise reduction using connectionist models,” IEEE Trans. Acoust. Speech Signal Process. 1, 553–556 (1988).
  16. Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
    [Crossref] [PubMed]
  17. D. Rumelhart, Learning Internal Representations by Error Propagation, Parallel Distributed Processing: Explorations in the Microstructures of Cognition 1 (MIT Press, Cambridge, Mass., 1986).
  18. S. Rogers, M. Kabrisky, An Introduction to Biological and Artificial Neural Networks for Pattern Recognition (SPIE Optical Engineering Press, Bellingham, Wash., 1991).
  19. Y. Pao, Adaptive Pattern Recognition and Neural Networks (Addison-Wesley, Reading, Mass., 1989).
  20. M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
    [Crossref] [PubMed]
  21. S. Troxel, B. Welsh, M. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
    [Crossref]
  22. B. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
    [Crossref]

1995 (1)

M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
[Crossref] [PubMed]

1994 (2)

S. Troxel, B. Welsh, M. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[Crossref]

B. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[Crossref]

1992 (2)

M. Jorgenson, G. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[Crossref] [PubMed]

Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
[Crossref] [PubMed]

1991 (2)

1990 (1)

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

1989 (1)

B. Welsh, C. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

1988 (1)

S. Tamura, A. Waibel, “Noise reduction using connectionist models,” IEEE Trans. Acoust. Speech Signal Process. 1, 553–556 (1988).

1983 (1)

1966 (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]

Aitken, G.

M. Jorgenson, G. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[Crossref] [PubMed]

M. Jorgenson, G. Aitken, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
[Crossref] [PubMed]

M. Jorgenson, G. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjodi, ed., Proc. SPIE1706 (1992).

M. Jorgenson, G. Aitken, “Wavefront prediction for adaptive optics,” in Proceedings of ICO-16, Active and Adaptive Optics (International Commission for Optics, 1993).

Angel, J.

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

M. Lloyd-Hart, P. Wizinowich, J. Angel, “A neural network wavefront sensor for array telescopes,” presented at the International Commission for Optics, Florence, Italy (1991).

Angel, R.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

Colucci, D.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

Dekany, R.

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

Ellerbroek, B.

M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
[Crossref] [PubMed]

B. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[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]

Gardner, C.

B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[Crossref]

B. Welsh, C. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

Hu, Y.

Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
[Crossref] [PubMed]

Hulburd, B.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

Jorgenson, M.

M. Jorgenson, G. Aitken, “Prediction of atmospherically induced wave-front degradations,” Opt. Lett. 17, 466–468 (1992).
[Crossref] [PubMed]

M. Jorgenson, G. Aitken, “Evidence of a chaotic attractor in star-wander data,” Opt. Lett. 16, 64–66 (1991).
[Crossref] [PubMed]

M. Jorgenson, G. Aitken, “Wavefront prediction for adaptive optics,” in Proceedings of ICO-16, Active and Adaptive Optics (International Commission for Optics, 1993).

M. Jorgenson, G. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjodi, ed., Proc. SPIE1706 (1992).

Kabrisky, M.

S. Rogers, M. Kabrisky, An Introduction to Biological and Artificial Neural Networks for Pattern Recognition (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Lloyd-Hart, M.

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

M. Lloyd-Hart, P. Wizinowich, J. Angel, “A neural network wavefront sensor for array telescopes,” presented at the International Commission for Optics, Florence, Italy (1991).

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

McCarthy, D.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

McLeod, B.

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

Pao, Y.

Y. Pao, Adaptive Pattern Recognition and Neural Networks (Addison-Wesley, Reading, Mass., 1989).

Rhoadarmer, T.

M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
[Crossref] [PubMed]

Rogers, S.

S. Rogers, M. Kabrisky, An Introduction to Biological and Artificial Neural Networks for Pattern Recognition (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Roggemann, M.

M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
[Crossref] [PubMed]

S. Troxel, B. Welsh, M. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[Crossref]

B. Welsh, M. Roggemann, “Evaluating the performance of adaptive optical telescopes,” Adaptive Optics for Astronomy, D. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Norwell, Mass., 1993).

Rumelhart, D.

D. Rumelhart, Learning Internal Representations by Error Propagation, Parallel Distributed Processing: Explorations in the Microstructures of Cognition 1 (MIT Press, Cambridge, Mass., 1986).

Sandler, D.

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

Scharf, L.

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

Tamura, S.

S. Tamura, A. Waibel, “Noise reduction using connectionist models,” IEEE Trans. Acoust. Speech Signal Process. 1, 553–556 (1988).

Tompkins, W.

Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
[Crossref] [PubMed]

Troxel, S.

Waibel, A.

S. Tamura, A. Waibel, “Noise reduction using connectionist models,” IEEE Trans. Acoust. Speech Signal Process. 1, 553–556 (1988).

Wallner, E.

Welsh, B.

S. Troxel, B. Welsh, M. Roggemann, “Off-axis optical transfer function calculations in an adaptive-optics system by means of a diffraction calculation for weak index fluctuations,” J. Opt. Soc. Am. A 11, 2100–2111 (1994).
[Crossref]

B. Welsh, C. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[Crossref]

B. Welsh, C. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

B. Welsh, M. Roggemann, “Evaluating the performance of adaptive optical telescopes,” Adaptive Optics for Astronomy, D. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Norwell, Mass., 1993).

Wittman, D.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

Wizinowich, P.

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, J. Angel, “A neural network wavefront sensor for array telescopes,” presented at the International Commission for Optics, Florence, Italy (1991).

Xue, Q.

Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
[Crossref] [PubMed]

Appl. Opt. (1)

M. Roggemann, B. Ellerbroek, T. Rhoadarmer, “Widening the effective field-of-view of adaptive-optics telescopes using deconvolution from wave-front sensing: average and signal-to-noise ratio performance,” Appl. Opt. 34, 1432–1444 (1995).
[Crossref] [PubMed]

IEEE Trans. Acoust. Speech Signal Process. (1)

S. Tamura, A. Waibel, “Noise reduction using connectionist models,” IEEE Trans. Acoust. Speech Signal Process. 1, 553–556 (1988).

IEEE Trans. Biomed. Eng. (1)

Q. Xue, Y. Hu, W. Tompkins, “Neural-network-based adaptive matched filtering for QRS detection,” IEEE Trans. Biomed. Eng. 39, 317–329 (1992).
[Crossref] [PubMed]

J. Opt. Soc. Am. (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]

J. Opt. Soc. Am. A (1)

B. Ellerbroek, “First-order performance evaluation of adaptive-optics systems for atmospheric-turbulence compensation in extended-field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[Crossref]

J. Opt. Soc. Am. A (1)

B. Welsh, C. Gardner, “Performance analysis of adaptive optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (London) (1)

J. Angel, P. Wizinowich, M. Lloyd-Hart, D. Sandler, “Adaptive optics for array telescopes using neural network techniques,” Nature (London) 358, 221–224 (1990).
[Crossref]

Opt. Lett. (2)

Other (10)

M. Jorgenson, G. Aitken, “Neural network prediction of turbulence induced wavefront degradations with applications to adaptive optics,” in Adaptive and Learning Systems, F. A. Sadjodi, ed., Proc. SPIE1706 (1992).

M. Jorgenson, G. Aitken, “Wavefront prediction for adaptive optics,” in Proceedings of ICO-16, Active and Adaptive Optics (International Commission for Optics, 1993).

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

B. Welsh, M. Roggemann, “Evaluating the performance of adaptive optical telescopes,” Adaptive Optics for Astronomy, D. Alloin, J.-M. Mariotti, eds. (Kluwer Academic, Norwell, Mass., 1993).

M. Lloyd-Hart, P. Wizinowich, J. Angel, “A neural network wavefront sensor for array telescopes,” presented at the International Commission for Optics, Florence, Italy (1991).

P. Wizinowich, M. Lloyd-Hart, B. McLeod, D. Colucci, R. Dekany, D. Wittman, R. Angel, D. McCarthy, B. Hulburd, D. Sandler, “Neural network adaptive optics for the Multiple Mirror Telescope,” in Active and Adaptive Optical Systems, M. A. Ealey, ed., Proc. SPIE1542, 148–158 (1991).

M. Lloyd-Hart, P. Wizinowich, B. McLeod, D. Wittman, D. Colucci, R. Dekany, D. McCarthy, J. Angel, D. Sandler, “First results of an on-line optics system with atmospheric wavefront sensing by an artificial neural network,” Astrophys. J. Lett., to be published.

D. Rumelhart, Learning Internal Representations by Error Propagation, Parallel Distributed Processing: Explorations in the Microstructures of Cognition 1 (MIT Press, Cambridge, Mass., 1986).

S. Rogers, M. Kabrisky, An Introduction to Biological and Artificial Neural Networks for Pattern Recognition (SPIE Optical Engineering Press, Bellingham, Wash., 1991).

Y. Pao, Adaptive Pattern Recognition and Neural Networks (Addison-Wesley, Reading, Mass., 1989).

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Figures (14)

Fig. 1
Fig. 1

WFS subaperture geometry. Numbering depicts placement of each of the 50 slope measurements in the 50-element slope vector.

Fig. 2
Fig. 2

Comparison of the neural network and variance-based technique for a noise-free D/r 0 estimation. The rms error is plotted versus M, the number of frames averaged per D/r 0 value. The rms error is averaged over 1000 randomly selected values of D/r 0.

Fig. 3
Fig. 3

Comparison of the neural network and variance-based estimation of D/r 0 with the WFS slope measurement error present. The rms error is plotted versus M, the number of frames averaged per D/r 0 value. The rms error is averaged over 1000 randomly selected values of D/r 0. The WFS mean-square slope measurement error is 0–25%.

Fig. 4
Fig. 4

Comparison of the neural network and variance-based estimation of D/r 0 on global-tilt-removed data with the WFS slope measurement error present. The rms error is plotted versus M, the number of frames averaged per D/r 0 value. The rms error is averaged over 1000 randomly selected values of D/r 0. The WFS mean-square slope measurement error is 0–25%.

Fig. 5
Fig. 5

Comparison of slope covariance with and without global tilt. Plotted as C s (s 1, s j ) versus j [see Fig. 1 and Eq. (3)] where j indexes all 50 elements of the slope vector. The slope covariance is shown for D/r 0 = 1.0.

Fig. 6
Fig. 6

WFS slope-measurement-error reduction, low-measurement-error case. The MSE is plotted as a function of the WFS slope index (see Fig. 1). The parameters are fixed at D/r 0 = 1.0, yielding a signal variance of 0.43 rad2/m2, and the noise variance is set at 0.043 rad2/m2.

Fig. 7
Fig. 7

WFS slope-measurement-error reduction, high-measurement-error case. The MSE is plotted as a function of the WFS slope index (see Fig. 1). The parameters are fixed at D/r 0 = 1.0, yielding a signal variance of 0.43 rad2/m2, and the noise variance is set at 1.00 rad2/m2.

Fig. 8
Fig. 8

Comparison of a neural network with a Bayes optimal solution for WFS slope-measurement-error reduction. WFS slope MSE plotted versus the measurement-error variance for (a) D/r 0 = 5.0, (b) D/r 0 = 7.5, and (c) D/r 0 = 10.0.

Fig. 9
Fig. 9

Comparison of a neural network, an optimized Bayes solution, and a suboptimized Bayes solution (optimized for D/r 0 = 7.5 and a WFS slope measurement error of 12.5%) for WFS slope-measurement-error reduction. The WFS slope MSE plotted versus the measurement-error variance for D/r 0 = 10.0.

Fig. 10
Fig. 10

Average WFS slope mean-square error plotted as a function of the slope measurement index (see Fig. 1) for D/r 0 ranging between 5 and 10 and the WFS slope-measurement-error levels ranging between 0% and 25%. The three curves represent the original WFS slope-measurement-error level, the slope-measurement-error level after neural-network processing, and the slope-measurement-error level after processing with a suboptimized Bayes solution (optimized for D/r 0 = 7.5 and a WFS slope-measurement-error level of 12.5%).

Fig. 11
Fig. 11

Comparison of a neural network with a Bayes optimal solution for WFS slope-measurement-error reduction on global-tilt-removed data. The WFS slope MSE plotted versus the measurement-error variance for (a) D/r 0 = 5.0, (b) D/r 0 = 7.5, and (c) D/r 0 = 10.0.

Fig. 12
Fig. 12

Average WFS slope mean-square error plotted as a function of the slope-measurement index (see Fig. 1) on the global-tilt-removed data for D/r 0 ranging between 5 and 10 and the WFS slope-measurement-error levels ranging between 0% and 25%. The three curves represent the original WFS slope-measurement-error level, the slope-measurement-error level after neural-network processing, and the slope-measurement-error level after processing with a suboptimized Bayes solution (optimized for D/r 0 = 7.5 and a WFS slope-measurement-error level of 12.5%).

Fig. 13
Fig. 13

Comparison of a neural network and variance-based estimators for a WFS slope-measurement-error estimation. The rms error plotted versus M, the number of frames averaged per the estimation-error level. The rms error is averaged over 1000 randomly selected error levels. The error level from 0% to 100% for D/r 0 between 5 and 20.

Fig. 14
Fig. 14

Comparison of a neural network and variance-based estimators for a WFS slope-measurement-error estimation of data with the global tilt removed. The rms error plotted versus M, the number of frames averaged per the estimation-error level. The rms error is averaged over 1000 randomly selected error levels. The error level is from 0% to 100% for D/r 0 between 5 and 20.

Equations (31)

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P = R s [ R s + R n ] 1 ,
α ( lower noise ) = P α ( noisy ) ,
C s [ s i ( t 1 ) , s j ( t 2 ) ] = E [ s i ( t 1 ) s j ( t 2 ) ] = 3 . 44 ( D r 0 ) 5 / 3 k = 1 N f ( z k ) × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) × | x 1 x 2 ( t 1 t 2 ) v ( z k ) | 5 / 3 .
w i s ( x ) = w i s ( x , y ) = 1 L x L y rect ( x x i L x , y y i L y ) × [ δ ( x x i L x 2 ) δ ( x x i + L x 2 ) ] ,
w i s ( x ) = w i s ( x , y ) = 1 L x L y rect ( x x i L x , y y i L y ) × [ δ ( y y i L y 2 ) δ ( y y i + L y 2 ) ] ,
C s = R R T .
σ s 2 ( calculated from measured data for unknown D / r 0 ) σ s 2 [ calculated from Eq . ( 3 ) with D / r 0 = 1 . 0 ] = ( D / r 0 ) 5 / 3 ,
2 = 1 1000 j = 1 1000 [ ( D ˆ r 0 ) j ( D r 0 ) j ] 2 ,
D ˆ r 0 = 1 M k = 1 M ( D ˆ r 0 ) k ,
signal variance = ( measured variance ) / ( 1 + estimated percent error ) .
i 2 = 1 10 , 000 j = 1 10 , 000 ( υ ˆ i , j υ i , j ) 2 ,
E [ u i u j ] = 0 . 00969 ( 2 π λ ) 2 d x 1 d x 2 υ i ( x 1 ) υ j ( x 2 ) × 1 2 w i ( τ 1 ) w j ( τ 2 ) j = 1 N C n 2 ( z j ) Δ z × { d k | k | 11 / 3 exp [ j 2 π k · ( x 1 x 2 ) ] × exp  [ j 2 π ( τ 1 τ 2 ) k · v ( z j ) ] + c } ,
w k ( τ ) = δ ( t k τ ) ,
υ k ( x ) = { W ( x ) [ δ ( x k x ) 1 ] for phase measurements w i s ( x ) for slope measurements ,
C s [ s i ( t 1 ) , s j ( t 2 ) ] = E [ s i ( t 1 ) s j ( t 2 ) ] = 0 . 00969 ( 2 π λ ) 2 d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) × 1 2 δ ( τ 1 t 1 ) δ ( τ 2 t 2 ) j = 1 N C n 2 ( z j ) Δ z × { d k | k | 11 / 3 exp [ j 2 π k · ( x 1 x 2 ) ] × exp  [ j 2 π ( τ 1 τ 2 ) k · v ( z j ) ] + c } .
C s [ s i ( t 1 ) , s j ( t 2 ) ] = 0 . 00969 ( 2 π λ ) 2 d x 1 d x 2 w i s ( x 1 ) × w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × [ d k k 11 / 3 exp ( j k · β ) + c ] .
k · β = k β cos θ,
d k = k d k .
C s [ s i ( t 1 ) , s j ( t 2 ) ] = 0 . 00969 ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × [ d k k 8 / 3 exp ( j k β cos θ ) + c ] .
0 2 π exp  [ j a cos  ( θ ϕ ) ] = 2 π J 0 ( a )
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 2 π ) ( 0 . 00969 ) ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × [ d k k 8 / 3 J 0 ( k β ) + c ] .
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 2 π ) ( 0 . 00969 ) ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × d k [ 1 J 0 ( k β ) ] k 8 / 3 .
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 2 π ) ( 0 . 00969 ) ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × d k β 8 / 3 [ 1 J 0 ( k β ) ] ( k β ) 8 / 3 .
[ 1 J 0 ( x ) ] x p d x = π { 2 p Γ 2 ( p + 1 2 ) sin  [ π ( p 1 ) 2 ] } 1
1 < p < 3
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 2 π ) ( 0 . 00969 ) ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) j = 1 N C n 2 ( z j ) Δ z × β 5 / 3 π { 2 8 / 3 Γ 2 ( 11 6 ) sin [ π ( 5 ) 6 ] } 1 ,
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 0 . 068 ) ( 2 π λ ) 2 × d x 1 d x 2 w i s ( x 1 ) w j s ( x 2 ) × j = 1 N C n 2 ( z j ) Δ z β 5 / 3 .
C s [ s i ( t 1 ) , s j ( t 2 ) ] = ( 0 . 068 ) ( 2 π λ ) 2 j = 1 N C n 2 ( z j ) Δ z d x 1 d x 2 w i s ( x 1 ) × w j s ( x 2 ) [ 2 π | x 1 x 2 + ( t 1 t 2 ) v ( z j ) | ] 5 / 3 .
C n 2 ( z j ) Δ z = 6 . 88 2 . 91 ( 2 π λ ) 2 r 0 ( z j ) 5 / 3 ,
f ( z j ) = [ r 0 r 0 ( z j ) ] 5 / 3 ,
C s [ s i ( t 1 ) , s j ( t 2 ) ] = E [ s i ( t 1 ) s j ( t 2 ) ] = 3 . 44 ( D r 0 ) 5 / 3 k = 1 N f ( z k ) d x 1 d x 2 w i s ( x 1 ) × w j s ( x 2 ) | x 1 x 2 ( t 1 t 2 ) v ( z k ) | 5 / 3 .

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