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]
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    [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.

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]

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

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

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.

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).
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Figures (10)

Fig. 1.
Fig. 1.

Principle of the multivariate simplex spline: A 5th degree spline function with C1 continuity defined on four triangles with (left) the four individual spline pieces [p1(b), p2(b), p3(b), and p4(b)] and (right) the global spline function p(b) formed by combining the four spline pieces.

Fig. 2.
Fig. 2.

B-net for a 4th degree spline function on a triangulation consisting of the three triangles ti, tj, and tk.

Fig. 3.
Fig. 3.

Southwell geometry (top left) and Fried (top right) sensor geometries compared with four different SABRE geometries (middle and bottom rows). Black dots are the phase point locations and horizontal and vertical lines are the slope measurements in the x and y directions, respectively. The open circles are the locations of the slope measurements. Gray lines in the SABRE geometries are the triangle edges, while the shaded area inside the triangulations is the area in which ϕ(x,y) is defined.

Fig. 4.
Fig. 4.

General simplex Type-IB geometry (left) and simplex Type-IG geometry (right). Open circles are the lenslet apertures.

Fig. 5.
Fig. 5.

Simplex Type-I sensor geometry (left) and the triangulation and B-net for a linear simplex B-spline function (right).

Fig. 6.
Fig. 6.

SH array image for a turbulence phase screen realization used in the numerical experiment.

Fig. 7.
Fig. 7.

Four SABRE sensor geometries used in the numerical experiments: A Type II geometry (top left) containing 1024 triangles, Type-IIG geometry (top right) containing 256 triangles, Type-IIB geometry containing 816 triangles (bottom left), and a Type-IIBG geometry containing 208 triangles (bottom right). The open circles are the locations of the SH-lenslets.

Fig. 8.
Fig. 8.

Linear Fried FD reconstruction (bottom left) and a quadratic SABRE reconstruction (bottom right) of a turbulence phase screen.

Fig. 9.
Fig. 9.

Average normalized residual E¯R as a function of signal to noise ratio for a number of different linear and nonlinear SABRE models compared with the Southwell and Fried reconstructors using wavefront slope and curvature measurements. A total of 100 turbulence realizations and 50 noise realizations are used per SNR magnitude.

Fig. 10.
Fig. 10.

Investigation of the influence of sensor geometry on the average normalized residual E¯R for SABRE models of equal degree and continuity order. A total of 100 turbulence realizations and 50 noise realizations are used for each noise setting.

Tables (2)

Tables Icon

Table 1. Properties of SABRE Geometries

Tables Icon

Table 2. Results of the Zero-Noise Experiments

Equations (84)

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t[v0xv0y],[v1xv1y],[v2xv2y]R2.
[x1x2]=[v0xv1xv2xv0yv1yv2y][b0b1b2],b0+b1+b2=1.
V[(v1xv0x)(v2xv0x)(v1yv0y)(v2yv0y)].
[b1b2]=V1[x1x2]b0=1b1b2.
b(x)(b0,b1,b2)R3,xR2.
Ti=1Jti,titj{,t˜},ti,tjT
(b0+b1+b2)d=κ0+κ1+κ2=dd!κ0!κ1!κ2!b0κ0b1κ1b2κ2
|κ|=κ0+κ1+κ2=d,κ00,κ10,κ20.
Bκd(b(x)){d!κ0!κ1!κ2!b0κ0b1κ1b2κ2,xt0,xt.
p(b(x)){|κ|=dcκtBκd(b(x)),xt0,xt
d^(d+2)!2d!.
p(b(x)){Bd(b(x))·ct,xt0,xt
Bd(b(x))[Bd,0,0d(b(x))Bd1,1,0d(b(x))B0,1,d1d(b(x))B0,0,dd(b(x))]R1×d^.
ct=[cd,0,0cd1,1,0c0,1,d1c0,0,d]Rd^×1.
p(b(x))=B1(b(x))·ct=[b01b10b20b00b11b20b00b10b21][c1,0,0tc0,1,0tc0,0,1t].
sdr(b(x))Bd·cR,xTJ,
Bd[Bt1d(b(x))Bt2d(b(x))BtJd(b(x))]R1×J·d^.
c[ct1ct2ctJ]RJ·d^×1
Sdr(T){sdrCr(T):sdr|tPd,tT}
tiv0,v1,w,tjv0,v1,v2.
t˜titj=v0,v1.
c(κ0,κ1,m)ti+|γ|=mc(κ0,κ1,0)+γtjBγm(b(w))=0,0mr,
Qm=0r(dm+1).
Hc=0.
ab(v)b(w)R3
Dump(b(x))=d!(dm)!Bdm(b(x))Pd,dm(a)·ct
Du1p(b(x))=B0(b(x))P1,0(a)·ct,
Du1p(b(x))={[a0a1a2]·ct,xt,0,xt.
Pud,dmdiag(Pjd,dm(au))j=1JRJ(dm+2)!2(dm)!×Jd^,
Dumsdr(b(x))=d!(dm)!BdmPud,dm·c.
σx(x,y)=ϕ(x,y)x,σy(x,y)=ϕ(x,y)y
σx(i,j)=[(ϕ(i+1,j)ϕ(i,j))+(ϕ(i+1,j+1)ϕ(i,j+1))]/(2h)+nx(i,j)σy(i,j)=[(ϕ(i,j+1)ϕ(i,j))+(ϕ(i+1,j+1)ϕ(i+1,j))]/(2h)+ny(i,j)
Ω(M,N)R2,M,NN.
σ=Gϕ+n
ϕ^FD=G+σ
ϕ(x,y)sdr(b(x,y))=Bd(b(x,y))·c,d1,(x,y)t.
σx(x,y)=d!(d1)!Bd1(b(x,y))Pd,d1(ax)·ct+νx(x,y),σy(x,y)=d!(d1)!Bd1(b(x,y))Pd,d1(ay)·ct+νy(x,y)
TDu1sdr(b(x))du=TdBd1Pd,dmcdu,=Bdc+k
sdr(b(x))=Bdc+k.1
sdr(b(x))=Bd(c+k·1)
c=[cd,0,0t1+kc˜+k·1]
k=cd,0,0t1.
sdr(b(x))=Bd·[0c˜cd,0,0t1·1].
h·[(cd,0,0t1+k)(c˜+k·1)]=0.
σ=B0Pu1,0c+n,0=Ac
A[Hh]R(EV+1)×Jd^
σx(x,y)=ax0c1,0,0+ax1c0,1,0+ax2c0,0,1+nx(x,y),σy(x,y)=ay0c1,0,0+ay1c0,1,0+ay2c0,0,1+ny(x,y)
σ=dBd1Pud,d1c+n,0=Ac
σ=2B1Pu2,1c+n.
σx(x,y)=2b0(ax0c2,0,0+ax1c1,1,0+ax2c1,0,1)+2b1(ax0c1,1,0+ax1c0,2,0+ax2c0,1,1)+2b2(ax0c1,0,1+ax1c0,1,1+ax2c0,0,2)+nx(x,y),σy(x,y)=2b0(ay0c2,0,0+ay1c1,1,0+ay2c1,0,1)+2b1(ay0c1,1,0+ay1c0,2,0+ay2c0,1,1)+2b2(ay0c1,0,1+ay1c0,1,1+ay2c0,0,2)+ny(x,y)
σu(x,y)=d!(d1)!Bd1(b(x,y))Pd,d1(au)·ct+n(x,y)
σ=Dc+n
D=dBd1Pud,d1NA
J(c)=(σDc)(σDc).
c^LS=NA(DD)1Dσ,=Qσ
ϕ^LS(x)=Bd(bx)c^LS.
c=[c1,0,0t1c0,1,0t1c0,0,1t1c1,0,0t2c0,1,0t2c0,0,1t2].
axt1=[011],ayt1=[101],axt2=[101],ayt2=[011].
P1,0=[011000101000000101000011].
H=[100100010010].
h=[100000].
NA=[001000010010000001].
D=P1,0NA=[110100001011],
ϵ=ϕ^ϕ2
ER=ϵϕ2.
Kν=ϵσν|ϕ=0.
Ikl=|FF{Akl·ei·ϕkl}|R256×256
I(k·Mp+m,l·Mp+n)=Ikl(m,n),m=1,2,,Mp,n=1,2,,Mp
xckl=m=0Mp1n=0Mp1m·I(k·Mp+m,l·Mp+n)m=0Mp1n=0Mp1I(k·Mp+m,l·Mp+n),yckl=m=0Mp1n=0Mp1n·I(k·Mp+m,l·Mp+n)m=0Mp1n=0Mp1I(k·Mp+m,l·Mp+n).
σxkl=(xcklx0kl)f+wνσykl=(yckly0kl)f+wν
ϕ^FD(xi,yj)F10(Ω(M,N)),1iM,1jN.
F10(Ω(M,N))S10(TII(Ω(M,N))).
s10(x,y)=B1(b(x,y))cωS10(T4(ω))
cω=[cvc*]R12
cv=[cv1cv1cv2cv2cv3cv3cv4cv4],c*=[cv*cv*cv*cv*].
s10(vm)=B1(b(vm))·cω=cvm,1m4
ϕ^FD(vm)=cvm,1m4
cϕ=[ϕ^FD(v1)ϕ^FD(v1)ϕ^FD(v4)ϕ^FD(v4)].
s10(VT4)=B1(b(VT4))·[cϕc*]=[ϕ^FD(Vω)v*]
F10(ω)S10(T4(ω)).
d(ϕ(x,y),Sdr)q={O(|T|0)ifd<3r+22,r>0O(|T|d)if3r+22d2r+1,r>0O(|T|d+1)ifd3r+2,r0
ϕ(x,y)sdrq<ϕ(x,y)fq
ϕ(x,y)s10qϕ(x,y)fq
d(ϕ(x,y),Sdr(T))q<d(ϕ(x,y),S10(T))q,

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