Abstract

This paper presents a new method for zonal wavefront reconstruction (WFR) with application to adaptive optics systems. This new method, indicated as Spline based ABerration REconstruction (SABRE), uses bivariate simplex B-spline basis functions to reconstruct the wavefront using local wavefront slope measurements. The SABRE enables WFR on nonrectangular and partly obscured sensor grids and is not subject to the waffle mode. The performance of SABRE is compared to that of the finite difference (FD) method in numerical experiments using data from a simulated Shack–Hartmann lenslet array. The results show that SABRE offers superior reconstruction accuracy and noise rejection capabilities compared to the FD method.

© 2012 Optical Society of America

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  1. J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).
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    [CrossRef]
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    [CrossRef]
  4. F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
    [CrossRef]
  5. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70, 998–1006 (1980).
    [CrossRef]
  6. M. Kissler-Patig, “Overall science goals and top level AO requirements for the E-ELT,” presented at First AO4ELT Conference, Victoria, Canada, B.C., September 25 and 30,2010.
  7. V. Korkiakoski and C. Vérinaud, “Simulations of the extreme adaptive optics system for EPICS,” Proc. SPIE 7736, 773643 (2010).
  8. B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
    [CrossRef]
  9. C. R. Vogel, “Sparse matrix methods for wavefront reconstruction, revisited,” Proc. SPIE5490, 1327–1335 (2004).
  10. L. Gilles, C. R. Vogel, and B. L. Ellerbroek, “Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction,” J. Opt. Soc. Am. 19, 1817–1822 (2002).
    [CrossRef]
  11. C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).
    [CrossRef]
  12. M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. 28, 2132–2138 (2011).
    [CrossRef]
  13. G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. 13, 1218–1225 (1996).
    [CrossRef]
  14. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  15. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
    [CrossRef]
  16. L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
    [CrossRef]
  17. P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
    [CrossRef]
  18. G. Awanou, M. J. Lai, and P. Wenston, “The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations,” in Wavelets and Splines, G. Chen and M. J. Lai, eds. (Nashboro, 2005), pp. 24–75.
  19. C. C. de Visser, “Global nonlinear model identification with multivariate splines,” Ph.D. thesis (Delft University of Technology, 2011).
  20. C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
    [CrossRef]
  21. C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
    [CrossRef]
  22. M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Cambridge University, 2007).
  23. M. D. Oliker, “Sensing waffle in the Fried geometry,” Proc. SPIE 3353, 964–971 (1998).
    [CrossRef]
  24. W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. 23, 2629–2638 (2006).
    [CrossRef]
  25. C. de Boor, “B-form basics,” in Geometric Modeling: Algorithms and New Trends (SIAM, 1987).
  26. X. L. Hu, D. F. Han, and M. J. Lai, “Bivariate splines of various degrees for numerical solution of partial differential equations,” SIAM J. Sci. Comput. 29, 1338–1354 (2007).
    [CrossRef]
  27. M. J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Adv. Comput. Math. 9, 251–279(1998).
    [CrossRef]
  28. M. J. Lai, “Some sufficient conditions for convexity of multivariate Bernstein–Bezier polynomials and box spline surfaces,” Studia Scient. Math. Hung. 28, 363–374 (1990).
    [CrossRef]
  29. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378(1977).
    [CrossRef]
  30. J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
    [CrossRef]
  31. R. Conan, “Mean-square residual error of a wavefront after propagation through atmospheric turbulence and after correction with Zernike polynomials,” J. Opt. Soc. Am. 25, 526–536 (2008).
    [CrossRef]
  32. Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
    [CrossRef]

2011

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. 28, 2132–2138 (2011).
[CrossRef]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[CrossRef]

2010

V. Korkiakoski and C. Vérinaud, “Simulations of the extreme adaptive optics system for EPICS,” Proc. SPIE 7736, 773643 (2010).

2009

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
[CrossRef]

2008

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
[CrossRef]

R. Conan, “Mean-square residual error of a wavefront after propagation through atmospheric turbulence and after correction with Zernike polynomials,” J. Opt. Soc. Am. 25, 526–536 (2008).
[CrossRef]

2007

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

X. L. Hu, D. F. Han, and M. J. Lai, “Bivariate splines of various degrees for numerical solution of partial differential equations,” SIAM J. Sci. Comput. 29, 1338–1354 (2007).
[CrossRef]

L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
[CrossRef]

2006

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. 23, 2629–2638 (2006).
[CrossRef]

C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).
[CrossRef]

2002

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

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

1998

M. J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Adv. Comput. Math. 9, 251–279(1998).
[CrossRef]

M. D. Oliker, “Sensing waffle in the Fried geometry,” Proc. SPIE 3353, 964–971 (1998).
[CrossRef]

1996

G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. 13, 1218–1225 (1996).
[CrossRef]

1995

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
[CrossRef]

1990

M. J. Lai, “Some sufficient conditions for convexity of multivariate Bernstein–Bezier polynomials and box spline surfaces,” Studia Scient. Math. Hung. 28, 363–374 (1990).
[CrossRef]

1988

1980

1977

1976

Agathoklis, P.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
[CrossRef]

Awanou, G.

G. Awanou, M. J. Lai, and P. Wenston, “The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations,” in Wavelets and Splines, G. Chen and M. J. Lai, eds. (Nashboro, 2005), pp. 24–75.

Beckers, J. M.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Beuzit, J. L.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Bradley, C.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
[CrossRef]

Brase, J. M.

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
[CrossRef]

Cheng, X.

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

Chu, Q. P.

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[CrossRef]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
[CrossRef]

Conan, J. M.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
[CrossRef]

Conan, R.

R. Conan, “Mean-square residual error of a wavefront after propagation through atmospheric turbulence and after correction with Zernike polynomials,” J. Opt. Soc. Am. 25, 526–536 (2008).
[CrossRef]

Dai, G. M.

G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. 13, 1218–1225 (1996).
[CrossRef]

Dai, Y.

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

de Boor, C.

C. de Boor, “B-form basics,” in Geometric Modeling: Algorithms and New Trends (SIAM, 1987).

de Visser, C. C.

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[CrossRef]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
[CrossRef]

C. C. de Visser, “Global nonlinear model identification with multivariate splines,” Ph.D. thesis (Delft University of Technology, 2011).

Ellerbroek, B. L.

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

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

Fried, D. L.

Gavel, D. T.

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
[CrossRef]

Gilles, L.

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

Gong, S.

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

Hampton, P. J.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
[CrossRef]

Han, D. F.

X. L. Hu, D. F. Han, and M. J. Lai, “Bivariate splines of various degrees for numerical solution of partial differential equations,” SIAM J. Sci. Comput. 29, 1338–1354 (2007).
[CrossRef]

Herrmann, J.

Hu, X. L.

X. L. Hu, D. F. Han, and M. J. Lai, “Bivariate splines of various degrees for numerical solution of partial differential equations,” SIAM J. Sci. Comput. 29, 1338–1354 (2007).
[CrossRef]

Hudgin, R. H.

Jiang, Z.

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

Kissler-Patig, M.

M. Kissler-Patig, “Overall science goals and top level AO requirements for the E-ELT,” presented at First AO4ELT Conference, Victoria, Canada, B.C., September 25 and 30,2010.

Korkiakoski, V.

V. Korkiakoski and C. Vérinaud, “Simulations of the extreme adaptive optics system for EPICS,” Proc. SPIE 7736, 773643 (2010).

Lai, M. J.

X. L. Hu, D. F. Han, and M. J. Lai, “Bivariate splines of various degrees for numerical solution of partial differential equations,” SIAM J. Sci. Comput. 29, 1338–1354 (2007).
[CrossRef]

M. J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Adv. Comput. Math. 9, 251–279(1998).
[CrossRef]

M. J. Lai, “Some sufficient conditions for convexity of multivariate Bernstein–Bezier polynomials and box spline surfaces,” Studia Scient. Math. Hung. 28, 363–374 (1990).
[CrossRef]

G. Awanou, M. J. Lai, and P. Wenston, “The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations,” in Wavelets and Splines, G. Chen and M. J. Lai, eds. (Nashboro, 2005), pp. 24–75.

M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Cambridge University, 2007).

Lai, O.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Lena, P.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Li, F.

Y. Dai, F. Li, X. Cheng, Z. Jiang, and S. Gong, “Analysis on Shack–Hartmann wave-front sensor with fourier optics,” Opt. Laser Technol. 39, 1374–1379 (2007).
[CrossRef]

Machintosh, B. A.

L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
[CrossRef]

Madec, P. Y.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
[CrossRef]

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Mulder, J. A.

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[CrossRef]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
[CrossRef]

Noll, R. J.

Northcott, M. J.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Oliker, M. D.

M. D. Oliker, “Sensing waffle in the Fried geometry,” Proc. SPIE 3353, 964–971 (1998).
[CrossRef]

Poyneer, L. A.

L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
[CrossRef]

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
[CrossRef]

Rigaut, F.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Roddier, F.

F. Roddier, “Curvature sensing and compensation: a new concept in adaptive optics,” Appl. Opt. 27, 1223–1225 (1988).
[CrossRef]

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Rolland, J. P.

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. 23, 2629–2638 (2006).
[CrossRef]

Rosensteiner, M.

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. 28, 2132–2138 (2011).
[CrossRef]

Rousset, G.

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
[CrossRef]

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Sandler, D. G.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Schumaker, L. L.

M. J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Adv. Comput. Math. 9, 251–279(1998).
[CrossRef]

M. J. Lai and L. L. Schumaker, Spline Functions on Triangulations (Cambridge University, 2007).

Séchaud, M.

J. M. Beckers, P. Lena, O. Lai, P. Y. Madec, G. Rousset, M. Séchaud, M. J. Northcott, F. Roddier, J. L. Beuzit, F. Rigaut, and D. G. Sandler, Adaptive Optics in Astronomy (Cambridge University, 1999).

Southwell, W. H.

Veran, J. P.

L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
[CrossRef]

Vérinaud, C.

V. Korkiakoski and C. Vérinaud, “Simulations of the extreme adaptive optics system for EPICS,” Proc. SPIE 7736, 773643 (2010).

Vogel, C. R.

C. R. Vogel and Q. Yang, “Multigrid algorithm for least-squares wavefront reconstruction,” Appl. Opt. 45, 705–715 (2006).
[CrossRef]

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

C. R. Vogel, “Sparse matrix methods for wavefront reconstruction, revisited,” Proc. SPIE5490, 1327–1335 (2004).

Wenston, P.

G. Awanou, M. J. Lai, and P. Wenston, “The multivariate spline method for scattered data fitting and numerical solutions of partial differential equations,” in Wavelets and Splines, G. Chen and M. J. Lai, eds. (Nashboro, 2005), pp. 24–75.

Yang, Q.

Zou, W.

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. 23, 2629–2638 (2006).
[CrossRef]

Adv. Comput. Math.

M. J. Lai and L. L. Schumaker, “On the approximation power of bivariate splines,” Adv. Comput. Math. 9, 251–279(1998).
[CrossRef]

Appl. Opt.

Automatica

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “A new approach to linear regression with multivariate splines,” Automatica 45, 2903–2909 (2009).
[CrossRef]

C. C. de Visser, Q. P. Chu, and J. A. Mulder, “Differential constraints for bounded recursive identification with multivariate splines,” Automatica 47, 2059–2066 (2011).
[CrossRef]

IEEE J. Select. Topics Signal Process.

P. J. Hampton, P. Agathoklis, and C. Bradley, “A new wave-front reconstruction method for adaptive optics systems using wavelets,” IEEE J. Select. Topics Signal Process. 2, 781–792 (2008).
[CrossRef]

J. Opt. Soc. Am.

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

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. 19, 2100–2111(2002).
[CrossRef]

L. A. Poyneer, B. A. Machintosh, and J. P. Veran, “Fourier transform wavefront control with adaptive prediction of the atmosphere,” J. Opt. Soc. Am. 24, 2645–2660 (2007).
[CrossRef]

B. L. Ellerbroek, “Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques,” J. Opt. Soc. Am. 19, 1803–1816 (2002).
[CrossRef]

W. Zou and J. P. Rolland, “Quantifications of error propagation in slope-based wavefront estimations,” J. Opt. Soc. Am. 23, 2629–2638 (2006).
[CrossRef]

M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. 28, 2132–2138 (2011).
[CrossRef]

G. M. Dai, “Modal wave-front reconstruction with Zernike polynomials and Karhunen–Loève functions,” J. Opt. Soc. Am. 13, 1218–1225 (1996).
[CrossRef]

J. M. Conan, G. Rousset, and P. Y. Madec, “Wave-front temporal spectra in high-resolution imaging through turbulence,” J. Opt. Soc. Am. 12, 1559–1570 (1995).
[CrossRef]

R. Conan, “Mean-square residual error of a wavefront after propagation through atmospheric turbulence and after correction with Zernike polynomials,” J. Opt. Soc. Am. 25, 526–536 (2008).
[CrossRef]

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
[CrossRef]

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