Abstract

We recently introduced the Kaczmarz algorithm for solving the atmospheric tomography problem in multiconjugate adaptive optics (MCAO). This iterative method solves the problem significantly faster than the standard matrix vector multiplication. We present the algorithm as well as an extension, which includes the effects of laser guide stars, such as the cone effect, tip/tilt indetermination, and spot elongation. We show that we can successfully cope with these effects and that the algorithm is suited for an MCAO system for the future generation of extremely large telescopes.

© 2013 Optical Society of America

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  1. F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).
  2. F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
    [CrossRef]
  3. B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
    [CrossRef]
  4. L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
    [CrossRef]
  5. B. Ellerbroek, L. Gilles, and C. Vogel, “Numerical simulations of multiconjugate adaptive optics wavefront reconstuction on giant telescopes,” Appl. Opt. 42, 4811–4818 (2003).
    [CrossRef]
  6. B. Ellerbroek and C. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
    [CrossRef]
  7. L. Gilles, B. Ellebroek, and C. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
    [CrossRef]
  8. L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007).
  9. L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. 25, 2427–2435 (2008).
    [CrossRef]
  10. C. Vogel and Q. Yang, “Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm,” Opt. Express 14, 7487–7498 (2006).
  11. Q. Yang, C. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
    [CrossRef]
  12. T. Fusco, J. Conan, G. Rousset, L. Mugnier, and V. Michau, “Optimal wave-front reconstruction strategies for multiconjugate adaptive optics,” J. Opt. Soc. Am. A 18, 2527–2538 (2001).
    [CrossRef]
  13. D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356–1373 (2004).
  14. R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
    [CrossRef]
  15. M. Rosensteiner, “Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition,” J. Opt. Soc. Am. A 29, 2328–2336 (2012).
    [CrossRef]
  16. S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A, Sciences Mathématiques 35, 355–357 (1937).
  17. N. Ageorges and C. Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy, NATO Asi Series. Series C, Mathematical and Physical Science (Springer, 2000).
  18. M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
    [CrossRef]
  19. M. Rosensteiner, “Cumulative reconstructor: fast wavefront reconstruction algorithm for extremely large telescopes,” J. Opt. Soc. Am. A 28, 2132–2138 (2011).
    [CrossRef]
  20. H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).
  21. F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).
  22. R. Kowar and O. Scherzer, “Convergence analysis of a Landweber–Kaczmarz method for solving nonlinear ill-posed equations,” in Ill-Posed and Inverse Problems (book series), Vol. 23, pp. 69–90 (2002).
  23. M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).
  24. J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).
  25. A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
    [CrossRef]
  26. M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press Laser and Optical Science and Technology Series (CRC Press, 1996).
  27. R. M. Clare, M. L. Louarn, and C. Béchet, “Optimal noise-weighted reconstruction with elongated Shack–Hartmann wavefront sensor images for laser tomography adaptive optics,” Appl. Opt. 49, G27–G36 (2010).
    [CrossRef]
  28. M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
    [CrossRef]

2012

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

M. Rosensteiner, “Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition,” J. Opt. Soc. Am. A 29, 2328–2336 (2012).
[CrossRef]

2011

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

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

2010

J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).

R. M. Clare, M. L. Louarn, and C. Béchet, “Optimal noise-weighted reconstruction with elongated Shack–Hartmann wavefront sensor images for laser tomography adaptive optics,” Appl. Opt. 49, G27–G36 (2010).
[CrossRef]

2008

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. 25, 2427–2435 (2008).
[CrossRef]

2007

M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).

2006

2004

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356–1373 (2004).

2003

2002

B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
[CrossRef]

2001

2000

F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

1937

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A, Sciences Mathématiques 35, 355–357 (1937).

Baumeister, J.

J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).

Béchet, C.

Clare, R. M.

Conan, J.

De Cezaro, A.

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

Ellebroek, B.

Ellerbroek, B.

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. 25, 2427–2435 (2008).
[CrossRef]

Q. Yang, C. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

B. Ellerbroek and C. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “Numerical simulations of multiconjugate adaptive optics wavefront reconstuction on giant telescopes,” Appl. Opt. 42, 4811–4818 (2003).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
[CrossRef]

F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007).

Engl, H.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Flicker, R.

F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

Fusco, T.

Gavel, D.

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356–1373 (2004).

Gilles, L.

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. 25, 2427–2435 (2008).
[CrossRef]

L. Gilles, B. Ellebroek, and C. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “Numerical simulations of multiconjugate adaptive optics wavefront reconstuction on giant telescopes,” Appl. Opt. 42, 4811–4818 (2003).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007).

Haltmeier, M.

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).

Hanke, M.

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Hubin, N.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Kaczmarz, S.

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A, Sciences Mathématiques 35, 355–357 (1937).

Kaltenbacher, B.

J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).

Korkiakoski, V.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Kowar, R.

R. Kowar and O. Scherzer, “Convergence analysis of a Landweber–Kaczmarz method for solving nonlinear ill-posed equations,” in Ill-Posed and Inverse Problems (book series), Vol. 23, pp. 69–90 (2002).

Le Louarn, M.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Leitao, A.

M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).

Leitäo, A.

J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).

Leito, A.

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

Louarn, M. L.

Marchetti, E.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Michau, V.

Mugnier, L.

Natterer, F.

F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).

Neubauer, A.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

Ramlau, R.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

Rigaut, F.

F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

Roddier, F.

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

Roggemann, M. C.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press Laser and Optical Science and Technology Series (CRC Press, 1996).

Rosensteiner, M.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

M. Rosensteiner, “Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition,” J. Opt. Soc. Am. A 29, 2328–2336 (2012).
[CrossRef]

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

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

Rousset, G.

Scherzer, O.

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).

R. Kowar and O. Scherzer, “Convergence analysis of a Landweber–Kaczmarz method for solving nonlinear ill-posed equations,” in Ill-Posed and Inverse Problems (book series), Vol. 23, pp. 69–90 (2002).

Verinaud, C.

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

Vogel, C.

C. Vogel and Q. Yang, “Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm,” Opt. Express 14, 7487–7498 (2006).

Q. Yang, C. Vogel, and B. Ellerbroek, “Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography,” Appl. Opt. 45, 5281–5293 (2006).
[CrossRef]

B. Ellerbroek and C. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

L. Gilles, B. Ellebroek, and C. Vogel, “Preconditioned conjugate gradient wave-front reconstructors for multiconjugate adaptive optics,” Appl. Opt. 42, 5233–5250 (2003).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “Numerical simulations of multiconjugate adaptive optics wavefront reconstuction on giant telescopes,” Appl. Opt. 42, 4811–4818 (2003).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007).

Welsh, B.

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press Laser and Optical Science and Technology Series (CRC Press, 1996).

Yang, Q.

Zhariy, M.

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

Appl. Math. Comput

A. De Cezaro, M. Haltmeier, A. Leito, and O. Scherzer, “On steepest-descent-Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Appl. Math. Comput 202, 596–607 (2008).
[CrossRef]

Appl. Opt.

Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A, Sciences Mathématiques

S. Kaczmarz, “Angenäherte auflösung von systemen linearer gleichungen,” Bulletin International de l’Académie Polonaise des Sciences et des Lettres. Classe des Sciences Mathématiques et Naturelles. Série A, Sciences Mathématiques 35, 355–357 (1937).

Inverse Probl.

R. Ramlau and M. Rosensteiner, “An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration,” Inverse Probl. 28, 095004 (2012).
[CrossRef]

Inverse Problems and Imaging

M. Haltmeier, A. Leitao, and O. Scherzer, “Kaczmarz methods for regularizing nonlinear ill-posed equations i: Convergence analysis,” Inverse Problems and Imaging 1, 289–298 (2007).

J. Baumeister, B. Kaltenbacher, and A. Leitäo, “On Levenberg-Marquardt Kaczmarz methods for regularizing systems of nonlinear ill-posed equations,” Inverse Problems and Imaging 4, 335–350 (2010).

M. Zhariy, A. Neubauer, M. Rosensteiner, and R. Ramlau, “Cumulative wavefront reconstructor for the Shack–Hartman sensor,” Inverse Problems and Imaging 5, 893–913 (2011).
[CrossRef]

J. Opt. Soc. Am.

L. Gilles and B. Ellerbroek, “Split atmospheric tomography using laser and natural guide stars,” J. Opt. Soc. Am. 25, 2427–2435 (2008).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Proc. SPIE

M. Le Louarn, C. Verinaud, V. Korkiakoski, N. Hubin, and E. Marchetti, “Adaptive optics simulations for the European Extremely Large Telescope,” Proc. SPIE 6272, 627234 (2006).
[CrossRef]

D. Gavel, “Tomography for multiconjugate adaptive optics systems using laser guide stars,” Proc. SPIE 5490, 1356–1373 (2004).

F. Rigaut, B. Ellerbroek, and R. Flicker, “Priciples, limitations and performance of multiconjugate adaptive optics,” Proc. SPIE 4007, 1022–1031 (2000).
[CrossRef]

B. Ellerbroek, L. Gilles, and C. Vogel, “A computationally efficient wavefront reconstructor for simulation or multi-conjugate adaptive optics on giant telescopes,” Proc. SPIE 4839, 989–1000 (2002).
[CrossRef]

L. Gilles, B. Ellerbroek, and C. Vogel, “Layer-oriented multigrid wavefront reconstruction algorithms for multi-conjugate adaptive optics,” Proc. SPIE 4839, 1011–1022 (2002).
[CrossRef]

B. Ellerbroek and C. Vogel, “Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes,” Proc. SPIE 5169, 206–217 (2003).
[CrossRef]

Other

L. Gilles, B. Ellerbroek, and C. Vogel, “A comparison of multigrid V-cycle versus fourier domain preconditioning for laser guide star atmospheric tomography,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007).

F. Roddier, Adaptive Optics in Astronomy (Cambridge University, 1999).

H. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer, 1996).

F. Natterer, The Mathematics of Computerized Tomography (Teubner, 1986).

R. Kowar and O. Scherzer, “Convergence analysis of a Landweber–Kaczmarz method for solving nonlinear ill-posed equations,” in Ill-Posed and Inverse Problems (book series), Vol. 23, pp. 69–90 (2002).

N. Ageorges and C. Dainty, eds., Laser Guide Star Adaptive Optics for Astronomy, NATO Asi Series. Series C, Mathematical and Physical Science (Springer, 2000).

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press Laser and Optical Science and Technology Series (CRC Press, 1996).

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Figures (7)

Fig. 1.
Fig. 1.

Atmospheric tomography: setting.

Fig. 2.
Fig. 2.

LGS cone effect.

Fig. 3.
Fig. 3.

Control scheme of the Kaczmarz pseudo open-loop control.

Fig. 4.
Fig. 4.

Comparison of the Kaczmarz algorithm with an MVM for high flux, cone effect, and tip/tilt indetermination considered (42 m telescope).

Fig. 5.
Fig. 5.

Comparison of the Kaczmarz algorithm with an MVM for high flux, cone effect, and tip/tilt indetermination considered (8 m telescope).

Fig. 6.
Fig. 6.

Comparison of the Kaczmarz algorithm with an MVM for low flux, cone effect, and tip/tilt indetermination considered.

Fig. 7.
Fig. 7.

Comparison of the Kaczmarz algorithm with an MVM for low-flux-taking cone effect, tip/tilt indetermination, and spot elongation into account.

Tables (2)

Tables Icon

Algorithm 1. Kaczmarz Iteration

Tables Icon

Algorithm 2. Kaczmarz Iteration with Integrated Tip/Tilt Reconstruction

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

s x [ i , j ] = ( Γ x φ ) [ i , j ] 1 | Ω i j | Ω i j φ x ( x , y ) d ( x , y ) ,
Γ φ = ( s x , s y ) ,
Ω D = { r = ( x , y ) R 2 : r D } .
Ω l = g = 1 G Ω D ( h l α g ) ,
Ω D ( h l α g ) { r R 2 : r h l α g Ω D } .
Φ = ( Φ ( 1 ) , , Φ ( L ) ) T ( L 2 ( Ω l ) ) L
Φ , Ψ = l = 1 L 1 γ l Φ ( l ) , Ψ ( l ) L 2 ( Ω l ) .
l = 1 L γ l = 1 .
( A α g Φ ) ( r ) := φ α g ( r ) = l = 1 L Φ ( l ) ( r + h l α g ) , r Ω D .
A α g Φ = φ α g , g = 1 ,… , G .
φ α g ( r ) = l = 1 L Φ ( l ) ( H g h l H g r + h l α g ) , r Ω D ,
( S H α h ϕ ) ( r ) ϕ ( H h H r + h α ) , r Ω D , H R , h [ 0 , H ] , α R 2 ,
L α g Φ l = 1 L S H l α g h l ϕ ( l ) .
L α g * Ψ = ( γ 1 ( S H 1 α g h 1 ) * Ψ γ 2 ( S H 2 α g h 2 ) * Ψ γ L ( S H L α g h L ) * Ψ ) ,
( S H g α g h l ) * Ψ = Ψ ( H g H g h l ( r α g h l ) ) χ Ω D ( α g h l ) ( H g H g h l r ) .
φ α g L ( r ) = Π φ α g ( r ) = φ α g ( r ) x 1 | Ω D | Ω D x φ α g ( r ) d r y 1 | Ω D | Ω D y φ α g ( r ) d r , r Ω D ,
N β n Φ = t β n = ( t β n x t β n y ) = ( l = 1 L 1 | Ω D | Ω D x Φ ( l ) ( r + h l β n ) d r l = 1 L 1 | Ω D | Ω D y Φ ( l ) ( r + h l β n ) d r ) ,
u ( t β n ) = t β n x · x + t β n y · y in Ω D .
s α spot = s + C α g , η w ,
A ϕ s δ C η 1 2 + ϕ C ϕ 1 2 min ,
s α corr = arg min { s s α spot C α g , η 1 2 + γ s 2 } , g = 1 , , G .

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