Abstract

Using non-parametric estimation techniques, we have modeled an area of 126 actuators of a micro-electro-mechanical deformable mirror with 1024 actuators. These techniques produce models applicable to open-loop adaptive optics, where the turbulent wavefront is measured before it hits the deformable mirror. The model’s input is the wavefront correction to apply to the mirror and its output is the set of voltages to shape the mirror. Our experiments have achieved positioning errors of 3.1% rms of the peak-to-peak wavefront excursion.

© 2010 OSA

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  1. F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsian, “The FALCON Concept: Multi-Object Spectroscopy Combined with MCAO in Near-IR,” Proc. ESO Workshop (2002)
  2. F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
    [CrossRef]
  3. C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
    [CrossRef]
  4. T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
    [CrossRef]
  5. J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express 14(12), 5558–5570 (2006).
    [CrossRef] [PubMed]
  6. D. Guzmán, F. J. Juez, F. S. Lasheras, R. Myers, and L. Young, “Deformable mirror model for open-loop adaptive optics using multivariate adaptive regression splines,” Opt. Express 18(7), 6492–6505 (2010).
    [CrossRef] [PubMed]
  7. J. B. Stewart, A. Diouf, Y. Zhou, and T. G. Bifano, “Open-loop control of a MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. A 24(12), 3827–3833 (2007).
    [CrossRef]
  8. K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
    [CrossRef]
  9. C. Blain, R. Conan, C. Bradley, and O. Guyon, “Open-loop control demonstration of micro-electro-mechanical-system MEMS deformable mirror,” Opt. Express 18(6), 5433–5448 (2010).
    [CrossRef] [PubMed]
  10. J. Chambers, Software for Data Analysis: Programming with R, (Springer, 2008).
  11. J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. 19(1), 1–67 (1991).
    [CrossRef]
  12. S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992).
    [CrossRef]
  13. K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
    [CrossRef]
  14. J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Trans. Neural Netw. 4(1), 136–141 (1993).
    [CrossRef] [PubMed]
  15. A. W. Minns and M. J. Hall, “Artificial neural networks as rainfall-runoff models / Modelisation pluie-debit par des reseaux neuroneaux artificiels,” Hydrol. Sci. J. 41(3), 399–417 (1996).
    [CrossRef]
  16. R. J. Abrahart and L. See, “Comparing neural network and autoregressive moving average techniques for the provision of continuous river flow forecasts in two contrasting catchments,” Hydrol. Process. 14(11-12), 2157–2172 (2000).
    [CrossRef]
  17. A. Y. Shamseldin, “Application of a neural network technique to rainfall-runoff modelling,” J. Hydrol. (Amst.) 199(3-4), 272–294 (1997).
    [CrossRef]
  18. C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
    [CrossRef]
  19. C. W. Dawson and R. L. Wilby, “Hydrological modelling using artificial neural networks,” Prog. Phys. Geogr. 25(1), 80–108 (2001).
  20. D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
    [CrossRef]
  21. A. J. C. Sharkey, “On combining artificial neural nets,” Connect. Sci. 8(3), 299–314 (1996).
    [CrossRef]
  22. R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
    [CrossRef]
  23. S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
    [CrossRef] [PubMed]
  24. W. Wang, P. H. A. J. M. Van Gelder, and J. K. Vrijling, “Some issues about the generalization of neural networks for time series prediction”. W. Duch, ed., in Artificial Neural Networks: Formal Models and Their Applications, Lecture Notes in Computer Science, Vol. 3697 (2005), pp. 559–564.

2010

2007

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

J. B. Stewart, A. Diouf, Y. Zhou, and T. G. Bifano, “Open-loop control of a MEMS deformable mirror for large-amplitude wavefront control,” J. Opt. Soc. Am. A 24(12), 3827–3833 (2007).
[CrossRef]

F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
[CrossRef]

2006

2004

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

2001

C. W. Dawson and R. L. Wilby, “Hydrological modelling using artificial neural networks,” Prog. Phys. Geogr. 25(1), 80–108 (2001).

2000

R. J. Abrahart and L. See, “Comparing neural network and autoregressive moving average techniques for the provision of continuous river flow forecasts in two contrasting catchments,” Hydrol. Process. 14(11-12), 2157–2172 (2000).
[CrossRef]

1999

C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
[CrossRef]

1998

R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
[CrossRef]

1997

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

A. Y. Shamseldin, “Application of a neural network technique to rainfall-runoff modelling,” J. Hydrol. (Amst.) 199(3-4), 272–294 (1997).
[CrossRef]

1996

A. W. Minns and M. J. Hall, “Artificial neural networks as rainfall-runoff models / Modelisation pluie-debit par des reseaux neuroneaux artificiels,” Hydrol. Sci. J. 41(3), 399–417 (1996).
[CrossRef]

A. J. C. Sharkey, “On combining artificial neural nets,” Connect. Sci. 8(3), 299–314 (1996).
[CrossRef]

1993

J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Trans. Neural Netw. 4(1), 136–141 (1993).
[CrossRef] [PubMed]

1992

S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992).
[CrossRef]

1991

J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. 19(1), 1–67 (1991).
[CrossRef]

1989

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
[CrossRef]

1986

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[CrossRef]

Abrahart, R. J.

R. J. Abrahart and L. See, “Comparing neural network and autoregressive moving average techniques for the provision of continuous river flow forecasts in two contrasting catchments,” Hydrol. Process. 14(11-12), 2157–2172 (2000).
[CrossRef]

Amari, S.-I.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

Ammons, S.

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

Assémat, F.

F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
[CrossRef]

Barnard, E.

J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Trans. Neural Netw. 4(1), 136–141 (1993).
[CrossRef] [PubMed]

Bierden, P.

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

Bifano, T.

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

Bifano, T. G.

Blain, C.

Braddock, R. D.

R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
[CrossRef]

Bradley, C.

Burn, D. H.

C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
[CrossRef]

Conan, R.

Cornelissen, S.

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

Cuby, J. G.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Dawson, C. W.

C. W. Dawson and R. L. Wilby, “Hydrological modelling using artificial neural networks,” Prog. Phys. Geogr. 25(1), 80–108 (2001).

de Villiers, J.

J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Trans. Neural Netw. 4(1), 136–141 (1993).
[CrossRef] [PubMed]

Dillon, D.

Diouf, A.

Evans, C.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Evans, J. W.

Finke, M.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

Friedman, J.

J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. 19(1), 1–67 (1991).
[CrossRef]

Gavel, D.

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express 14(12), 5558–5570 (2006).
[CrossRef] [PubMed]

Gendron, E.

F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
[CrossRef]

Guyon, O.

Guzmán, D.

Hall, M. J.

A. W. Minns and M. J. Hall, “Artificial neural networks as rainfall-runoff models / Modelisation pluie-debit par des reseaux neuroneaux artificiels,” Hydrol. Sci. J. 41(3), 399–417 (1996).
[CrossRef]

Hammer, F.

F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
[CrossRef]

Harpsoe, K.

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[CrossRef]

Hornik, K.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
[CrossRef]

Juez, F. J.

Kim, J.

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

Kowalski, B. R.

S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992).
[CrossRef]

Kremmer, M. L.

R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
[CrossRef]

Lasheras, F. S.

Lehnert, M.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Macintosh, B.

Minns, A. W.

A. W. Minns and M. J. Hall, “Artificial neural networks as rainfall-runoff models / Modelisation pluie-debit par des reseaux neuroneaux artificiels,” Hydrol. Sci. J. 41(3), 399–417 (1996).
[CrossRef]

Morris, S.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Morzinski, K.

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

J. W. Evans, B. Macintosh, L. Poyneer, K. Morzinski, S. Severson, D. Dillon, D. Gavel, and L. Reza, “Demonstrating sub-nm closed loop MEMS flattening,” Opt. Express 14(12), 5558–5570 (2006).
[CrossRef] [PubMed]

Muller, K.-R.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

Murata, N.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

Myers, R.

Poyneer, L.

Puech, M.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Reza, L.

Rumelhart, D. E.

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[CrossRef]

Sanzogni, L.

R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
[CrossRef]

See, L.

R. J. Abrahart and L. See, “Comparing neural network and autoregressive moving average techniques for the provision of continuous river flow forecasts in two contrasting catchments,” Hydrol. Process. 14(11-12), 2157–2172 (2000).
[CrossRef]

Sekulic, S.

S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992).
[CrossRef]

Severson, S.

Shamseldin, A. Y.

A. Y. Shamseldin, “Application of a neural network technique to rainfall-runoff modelling,” J. Hydrol. (Amst.) 199(3-4), 272–294 (1997).
[CrossRef]

Sharkey, A. J. C.

A. J. C. Sharkey, “On combining artificial neural nets,” Connect. Sci. 8(3), 299–314 (1996).
[CrossRef]

Simonovic, S. P.

C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
[CrossRef]

Stewart, J. B.

Stinchcombe, M.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
[CrossRef]

Swinbank, M.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

White, H.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
[CrossRef]

Wilby, R. L.

C. W. Dawson and R. L. Wilby, “Hydrological modelling using artificial neural networks,” Prog. Phys. Geogr. 25(1), 80–108 (2001).

Williams, R. J.

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[CrossRef]

Yang, H. H.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

Young, L.

Zealand, C. M.

C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
[CrossRef]

Zhou, Y.

Zhu, H.

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

Ann. Stat.

J. Friedman, “Multivariate adaptive regression splines,” Ann. Stat. 19(1), 1–67 (1991).
[CrossRef]

Astron. Geophys.

C. Evans, S. Morris, M. Swinbank, J. G. Cuby, M. Lehnert, and M. Puech, “EAGLE: galaxy evolution with the E-ELT,” Astron. Geophys. 51(2), 2.17–2.21 (2010).
[CrossRef]

Connect. Sci.

A. J. C. Sharkey, “On combining artificial neural nets,” Connect. Sci. 8(3), 299–314 (1996).
[CrossRef]

Environmetrics

R. D. Braddock, M. L. Kremmer, and L. Sanzogni, “Feed-forward artificial neural network model for forecasting rainfall run-off,” Environmetrics 9(4), 419–432 (1998).
[CrossRef]

Hydrol. Process.

R. J. Abrahart and L. See, “Comparing neural network and autoregressive moving average techniques for the provision of continuous river flow forecasts in two contrasting catchments,” Hydrol. Process. 14(11-12), 2157–2172 (2000).
[CrossRef]

Hydrol. Sci. J.

A. W. Minns and M. J. Hall, “Artificial neural networks as rainfall-runoff models / Modelisation pluie-debit par des reseaux neuroneaux artificiels,” Hydrol. Sci. J. 41(3), 399–417 (1996).
[CrossRef]

IEEE Trans. Neural Netw.

S.-I. Amari, N. Murata, K.-R. Muller, M. Finke, and H. H. Yang, “Asymptotic statistical theory of overtraining and cross-validation,” IEEE Trans. Neural Netw. 8(5), 985–996 (1997).
[CrossRef] [PubMed]

J. de Villiers and E. Barnard, “Backpropagation neural nets with one and two hidden layers,” IEEE Trans. Neural Netw. 4(1), 136–141 (1993).
[CrossRef] [PubMed]

J. Chemometr.

S. Sekulic and B. R. Kowalski, “MARS: a tutorial,” J. Chemometr. 6(4), 199–216 (1992).
[CrossRef]

J. Hydrol. (Amst.)

A. Y. Shamseldin, “Application of a neural network technique to rainfall-runoff modelling,” J. Hydrol. (Amst.) 199(3-4), 272–294 (1997).
[CrossRef]

C. M. Zealand, D. H. Burn, and S. P. Simonovic, “Short term streamflow forecasting using artificial neural networks,” J. Hydrol. (Amst.) 214(1-4), 32–48 (1999).
[CrossRef]

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

F. Assémat, E. Gendron, and F. Hammer, “The FALCON concept: multi-object adaptive optics and atmospheric tomography for integral field spectroscopy - principles and performance on an 8-m telescope,” Mon. Not. R. Astron. Soc. 376(1), 287–312 (2007).
[CrossRef]

Nature

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 323(6088), 533–536 (1986).
[CrossRef]

Neural Netw.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Netw. 2(5), 359–366 (1989).
[CrossRef]

Opt. Express

Proc. SPIE

T. Bifano, P. Bierden, H. Zhu, S. Cornelissen, and J. Kim, “Megapixel wavefront correctors,” Proc. SPIE 5490, 1472–1481 (2004).
[CrossRef]

K. Morzinski, K. Harpsoe, D. Gavel, and S. Ammons, “The open-loop control of MEMS: Modeling and experimental results,” Proc. SPIE 6467, 64670G–1 (2007).
[CrossRef]

Prog. Phys. Geogr.

C. W. Dawson and R. L. Wilby, “Hydrological modelling using artificial neural networks,” Prog. Phys. Geogr. 25(1), 80–108 (2001).

Other

J. Chambers, Software for Data Analysis: Programming with R, (Springer, 2008).

W. Wang, P. H. A. J. M. Van Gelder, and J. K. Vrijling, “Some issues about the generalization of neural networks for time series prediction”. W. Duch, ed., in Artificial Neural Networks: Formal Models and Their Applications, Lecture Notes in Computer Science, Vol. 3697 (2005), pp. 559–564.

F. Hammer, F. Sayede, E. Gendron, T. Fusco, D. Burgarella, V. Cayatte, J. M. Conan, F. Courbin, H. Flores, I. Guinouard, L. Jocou, A. Lancon, G. Monnet, M. Mouhcine, F. Rigaud, D. Rouan, G. Rousset, V. Buat, and F. Zamkotsian, “The FALCON Concept: Multi-Object Spectroscopy Combined with MCAO in Near-IR,” Proc. ESO Workshop (2002)

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Figures (9)

Fig. 1
Fig. 1

Topology of feed forward multi-layer perceptron back-propagation ANN.

Fig. 2
Fig. 2

Topology of DM actuators and the cloning structure of ANNb, with actuators named from 1 to 126. The actuators in purple represent overlapping actuators.

Fig. 3
Fig. 3

The actuators structure is illustrated for 6 cases, with a different 6 x 5 DM actuators sector in each one. The seventh actuator is the one at the middle of the region (illustrated as the white box in Fig. 2).

Fig. 4
Fig. 4

Training and estimation processes.

Fig. 5
Fig. 5

Optical setup for measuring the DM surface with our interferometer.

Fig. 6
Fig. 6

phase map of our MEMS DM in nanometers, taken with the Fisba interferometer, for a random phase in the modeled actuators. Actuator locations are indicated with ‘ + ’ marks, while plane samples are indicated with ‘x’ marks.

Fig. 7
Fig. 7

Slice in the X axis for a Zernike focus term from each of the three models implemented. Actuator locations indicated by diamond marks.

Fig. 8
Fig. 8

Slice in the Y axis for a Zernike focus term from each of the three models implemented. Actuator locations indicated by diamond marks.

Fig. 9
Fig. 9

Residual errors for each of the models, expressed in terms of both figures of merit. Left panels use Eq. (9) and right panels Eq. (10). Top panels come from MARS model; central panels ANNs model and lower panels ANNb model.

Tables (2)

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Table 1 Residual errors when commanding a Zernike focus terms, for the three different models

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Table 2 Average peak-to-valley and RMS of the residuals, for the three different models

Equations (10)

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y = a 0 + m = 1 M a m B m ( x )
b q ( x t ) = [ ( x t ) ] q = { ( t x ) q i f     x < t 0 o t h e r w i s e
b q + ( x t ) = [ + ( x t ) ] q = { ( t x ) q i f     x > t 0 o t h e r w i s e
n e t j = i = 0 n w i , j y i ( i = 0 , 1 , ... , n ; j = 1 , ... , m )
z j = f H ( n e t j ) ( j = 1 , 2 , ... , m )
f H ( x ) = 1 1 + e x
O = f o ( j = 0 m w j , k z j ) ( j = 1 , 2 , ... , m )
E = 1 2 ( O y t ) 2
Re s i d u a l R M S D e s i r e d P V
Re s i d u a l R M S D e s i r e d R M S

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