August  2016, 10(3): 781-805. doi: 10.3934/ipi.2016021

A gradient-based method for atmospheric tomography

1. 

Altenbergerstrasse 69, Linz, 4040, Austria, Austria

Received  February 2015 Revised  January 2016 Published  August 2016

In Adaptive Optics (AO) systems for ground-based telescopes, one aims at mechanically correcting for atmospheric aberrations by means of quickly moving deformable mirrors. In complex AO systems, which are using several light sources and aim at a good reconstruction in a large field of view, the derivation of optimal mirror commands from measured light typically includes the problem of atmospheric tomography. As the computational effort for such a limited-angle tomography problem is strongly increasing for growing telescope sizes, fast algorithms are needed. We present a novel algorithm for atmospheric tomography that takes real-life effects such as tip/tilt indetermination, cone effect and spot elongation into account. Furthermore, we discuss two models for the tip and tilt components of an incoming wavefront and incorporate them into the reconstruction. We find a fast step size choice for our Gradient-based iteration and compare it with different existing step size choices. Numerical results are demonstrated for two different AO systems on a 42 m telescope, using the European Southern Observatory's end-to-end simulation tool, OCTOPUS.
Citation: Daniela Saxenhuber, Ronny Ramlau. A gradient-based method for atmospheric tomography. Inverse Problems and Imaging, 2016, 10 (3) : 781-805. doi: 10.3934/ipi.2016021
References:
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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, Monthly Notices of the Royal Astronomical Society, 376 (2007), 287-312.

[2]

J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

[3]

C. Béchet, M. L. Louarn, R. Clare, M. Tallon, I. Tallon-Bosc and E. Thiébaut, Closed-loop ground layer adaptive optics simulations with elongated spots: Impact of modeling noise correlations, in 1st AO4ELT conference - Adaptive Optics for Extremely Large Telescopes, 2010, 03004-03009, URL http://dx.doi.org/10.1051/ao4elt/201003004.

[4]

C. Béchet, M. Tallon, I. Tallon-Bosc, Éric Thiébaut, M. L. Louarn and R. M. Clare, Optimal reconstruction for closed-loop ground-layer adaptive optics with elongated spots, J. Opt. Soc. Am. A, 27 (2010), A1-A8, URL http://josaa.osa.org/abstract.cfm?URI=josaa-27-11-A1.

[5]

E. Brunner, C. BÉchet and M. Tallon, Optimal projection of reconstructed layers onto deformable mirrors with fractal iterative method for ao tomography, in SPIE Astronomical Telescopes+ Instrumentation, International Society for Optics and Photonics, 847 (2012), 84475I-84475I. doi: 10.1117/12.926809.

[6]

B. Ellerbroek and C. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes, Proc. SPIE, 5169 (2003), 206-217. doi: 10.1117/12.506580.

[7]

B. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques, J. Opt. Soc. Am., 19 (2002), 1803-1816. doi: 10.1364/JOSAA.19.001803.

[8]

B. Ellerbroek and C. Vogel, Inverse problems in astronomical adaptive optics, Inverse Problems, 25 (2009), 063001, 37pp. doi: 10.1088/0266-5611/25/6/063001.

[9]

M. Eslitzbichler, C. Pechstein and R. Ramlau, An h1-kaczmarz reconstructor for atmospheric tomography, Journal of Inverse and Ill-Posed Problems, 21 (2013), 431-450. doi: 10.1515/jip-2013-0007.

[10]

T. Fusco, J.-M. Conan, G. Rousset, L. Mugnier and V. Michau, Optimal wave-front reconstruction strategies for multi conjugate adaptive optics, J. Opt. Soc. Am. A, 18 (2001), 2527-2538.

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D. Gavel, Tomography for multiconjugate adaptive optics systems using laser guide stars, SPIE Astronomical Telescopes and Instrumentation, 5490 (2004), 1356-1373. doi: 10.1117/12.552402.

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L. Gilles and B. Ellerbroek, Split atmospheric tomography using laser and natural guide stars, J. Opt. Soc. Am., 25 (2008), 2427-2435. doi: 10.1364/JOSAA.25.002427.

[13]

L. Gilles, C. Vogel and B. Ellebroek, Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction, J. Opt. Soc. Am. A, 19 (2002), 1817-1822. doi: 10.1364/JOSAA.19.001817.

[14]

T. Helin and M. Yudytskiy, Wavelet methods in multi-conjugate adaptive optics, Inverse Problems, 29 (2013), 085003, 18pp, URL http://stacks.iop.org/0266-5611/29/i=8/a=085003. doi: 10.1088/0266-5611/29/8/085003.

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M. Le Louarn, C. Verinaud and V. Korkiakoski, Simulation of mcao on (extremely) large telescopes, Comptes Rendus Physique, 6 (2005), 1070-1080. doi: 10.1016/j.crhy.2005.10.004.

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M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin and E. Marchetti, Adaptive optics simulations for the european extremely large telescope, in SPIE Astronomical Telescopes+ Instrumentation, International Society for Optics and Photonics, 6272 (2006), 627234-627234. doi: 10.1117/12.670187.

[18]

A. Neubauer, On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction, J. Inv. Ill-Posed Problems, 18 (2010), 551-576. doi: 10.1515/JIIP.2010.025.

[19]

A. Neubauer, A new cumulative wavefront reconstructor for the Shack-Hartmann sensor, J. Inv. Ill-Posed Problems, 21 (2013), 451-476. doi: 10.1515/jip-2013-0003.

[20]

D. W. Phillion and K. Baker, Two-sided pyramid wavefront sensor in the direct phase mode, Proc. SPIE, 6272 (2006), 627228. doi: 10.1117/12.671961.

[21]

L. Poyneer, D. Gavel and J. Brase, Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform, J. Opt. Soc. Am. A, 19 (2002), 2100-2111. doi: 10.1364/JOSAA.19.002100.

[22]

F. Quirós-Pacheco, C. Correia and S. Esposito, Fourier transform-wavefront reconstruction for the pyramid wavefront sensor, in 1st AO4ELT conference - Adaptive Optics for Extremely Large Telescopes, 2010, 07005-07010.

[23]

R. Ramlau, A. Obereder, M. Rosensteiner and D. Saxenhuber, Efficient iterative tip/tilt reconstruction for atmospheric tomography, Inverse Problems in Science and Engineering, 22 (2014), 1345-1366. doi: 10.1080/17415977.2013.873534.

[24]

R. Ramlau and M. Rosensteiner, An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration, Inverse Problems, 28 (2012), 095004, 23pp. doi: 10.1088/0266-5611/28/9/095004.

[25]

R. Ramlau, D. Saxenhuber and M. Yudytskiy, Iterative reconstruction methods in atmospheric tomography: FEWHA, Kaczmarz and Gradient-based algorithm, in Proc. SPIE, 9148 (2014), 91480Q. doi: 10.1117/12.2057379.

[26]

F. Rigaut, B. Ellerbroek and R. Flicker, Principles, limitations and performance of multiconjugate adaptive optics, Proc. SPIE, 4007 (2000), 1022-1031.

[27]

F. Roddier, The effects of atmospheric turbulence in optical astronomy, Progress in Optics, 19 (1981), 281-376. doi: 10.1016/S0079-6638(08)70204-X.

[28]

F. Roddier, Adaptive Optics in Astronomy, Cambridge, U.K. ; New York : Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511525179.

[29]

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press laser and optical science and technology series, CRC Press, 1996.

[30]

M. Rosensteiner, Cumulative reconstructor: Fast wavefront reconstruction algorithm for Extremely Large Telescopes, J. Opt. Soc. Am. A, 28 (2011), 2132-2138. doi: 10.1364/JOSAA.28.002132.

[31]

M. Rosensteiner, Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition, J. Opt. Soc. Am. A, 29 (2012), 2328-2336. doi: 10.1364/JOSAA.29.002328.

[32]

M. Rosensteiner and R. Ramlau, Efficient iterative atmospheric tomography reconstruction from LGS and additional tip/tilt measurements, in SPIE 8447, Adaptive Optics Systems III, 2012, 84475S. doi: 10.1117/12.945876.

[33]

M. Rosensteiner and R. Ramlau, The Kaczmarz algorithm for multi-conjugate adaptive optics with laser guide stars, J. Opt. Soc. Am. A, 30 (2013), 1680-1686.

[34]

I. Shatokhina, A. Obereder, M. Rosensteiner and R. Ramlau, Preprocessed cumulative reconstructor with domain decomposition: A fast wavefront reconstruction method for pyramid wavefront sensor, {Applied Optics, 52 (2013), 2640-2652. doi: 10.1364/AO.52.002640.

[35]

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin and E. Thiébaut, Fractal iterative method for fast atmospheric tomography on extremely large telescopes, in Proc. SPIE 7736, Adaptive Optics Systems II, 2010, 77360X. doi: 10.1117/12.858042.

[36]

M. Tallon, I. Tallon-Bosc, C. Béchet and E. Thiébaut, Shack-hartmann wavefront reconstruction with elongated sodium laser guide stars: Improvements with priors and noise correlations, Proc. SPIE, Adaptive Optics Systems, 7015 (2008), 70151N, URL http://link.aip.org/link/?PSI/7015/70151N/1. doi: 10.1117/12.788902.

[37]

M. Tallon, E. Thiébaut and C. Béchet, A fractal iterative method for fast wavefront reconstruction for extremely large telescopes, in Adaptive Optics: Methods, Analysis and Applications, Optical Society of America, 2007, paper PMA2 doi: 10.1364/AOPT.2007.PMA2.

[38]

E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes, J. Opt. Soc. Am. A, 27 (2010), 1046-1059.

[39]

A. Tokovinin, M. L. Louarn and M. Sarazin, Isoplanatism in a multi-conjugate adaptive optics system, J. Opt. Soc. Am., 17 (2000), 1819-1827.

[40]

A. Tokovinin and E. Viard, Limiting precision of tomographic phase estimation, J. Opt. Soc. Am., 18 (2001), 873-882. doi: 10.1364/JOSAA.18.000873.

[41]

G. Tyler, Bandwidth considerations for tracking trough turbulence, J. Opt. Soc. Am. A, 11 (1994), 358-367.

[42]

F. Vidal, E. Gendron and G. Rousset, Tomography approach for multi-object adaptive optics, JOSA A, 27 (2010), A253-A264. doi: 10.1364/JOSAA.27.00A253.

[43]

C. Vogel and Q. Yang, Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm, Optics Express, 14 (2006), 7487-7498. doi: 10.1364/OE.14.007487.

[44]

C. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction, Applied Optics, 45 (2006), 705-715. doi: 10.1364/AO.45.000705.

[45]

Q. Yang, C. Vogel and B. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography, Applied Optics, 45 (2006), 5281-5293.

[46]

M. Yudytskiy, Wavelet Methods in Adaptive Optics, PhD thesis, Johannes Kepler University Linz, 2014.

[47]

M. Yudytskiy, T. Helin and R. Ramlau, A frequency dependent preconditioned wavelet method for atmospheric tomography, in Third AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes, 2013.

[48]

M. Yudytskiy, T. Helin and R. Ramlau, Finite element-wavelet hybrid algorithm for atmospheric tomography, J. Opt. Soc. Am. A, 31 (2014), 550-560, URL http://josaa.osa.org/abstract.cfm?URI=josaa-31-3-550. doi: 10.1364/JOSAA.31.000550.

[49]

M. Zhariy, A. Neubauer, M. Rosensteiner and R. Ramlau, Cumulative wavefront reconstructor for the Shack-Hartman sensor, Inverse Problems and Imaging, 5 (2011), 893-913. doi: 10.3934/ipi.2011.5.893.

show all references

References:
[1]

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, Monthly Notices of the Royal Astronomical Society, 376 (2007), 287-312.

[2]

J. Barzilai and J. Borwein, Two-point step size gradient methods, IMA Journal of Numerical Analysis, 8 (1988), 141-148. doi: 10.1093/imanum/8.1.141.

[3]

C. Béchet, M. L. Louarn, R. Clare, M. Tallon, I. Tallon-Bosc and E. Thiébaut, Closed-loop ground layer adaptive optics simulations with elongated spots: Impact of modeling noise correlations, in 1st AO4ELT conference - Adaptive Optics for Extremely Large Telescopes, 2010, 03004-03009, URL http://dx.doi.org/10.1051/ao4elt/201003004.

[4]

C. Béchet, M. Tallon, I. Tallon-Bosc, Éric Thiébaut, M. L. Louarn and R. M. Clare, Optimal reconstruction for closed-loop ground-layer adaptive optics with elongated spots, J. Opt. Soc. Am. A, 27 (2010), A1-A8, URL http://josaa.osa.org/abstract.cfm?URI=josaa-27-11-A1.

[5]

E. Brunner, C. BÉchet and M. Tallon, Optimal projection of reconstructed layers onto deformable mirrors with fractal iterative method for ao tomography, in SPIE Astronomical Telescopes+ Instrumentation, International Society for Optics and Photonics, 847 (2012), 84475I-84475I. doi: 10.1117/12.926809.

[6]

B. Ellerbroek and C. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes, Proc. SPIE, 5169 (2003), 206-217. doi: 10.1117/12.506580.

[7]

B. Ellerbroek, Efficient computation of minimum-variance wave-front reconstructors with sparse matrix techniques, J. Opt. Soc. Am., 19 (2002), 1803-1816. doi: 10.1364/JOSAA.19.001803.

[8]

B. Ellerbroek and C. Vogel, Inverse problems in astronomical adaptive optics, Inverse Problems, 25 (2009), 063001, 37pp. doi: 10.1088/0266-5611/25/6/063001.

[9]

M. Eslitzbichler, C. Pechstein and R. Ramlau, An h1-kaczmarz reconstructor for atmospheric tomography, Journal of Inverse and Ill-Posed Problems, 21 (2013), 431-450. doi: 10.1515/jip-2013-0007.

[10]

T. Fusco, J.-M. Conan, G. Rousset, L. Mugnier and V. Michau, Optimal wave-front reconstruction strategies for multi conjugate adaptive optics, J. Opt. Soc. Am. A, 18 (2001), 2527-2538.

[11]

D. Gavel, Tomography for multiconjugate adaptive optics systems using laser guide stars, SPIE Astronomical Telescopes and Instrumentation, 5490 (2004), 1356-1373. doi: 10.1117/12.552402.

[12]

L. Gilles and B. Ellerbroek, Split atmospheric tomography using laser and natural guide stars, J. Opt. Soc. Am., 25 (2008), 2427-2435. doi: 10.1364/JOSAA.25.002427.

[13]

L. Gilles, C. Vogel and B. Ellebroek, Multigrid preconditioned conjugate-gradient method for large-scale wave-front reconstruction, J. Opt. Soc. Am. A, 19 (2002), 1817-1822. doi: 10.1364/JOSAA.19.001817.

[14]

T. Helin and M. Yudytskiy, Wavelet methods in multi-conjugate adaptive optics, Inverse Problems, 29 (2013), 085003, 18pp, URL http://stacks.iop.org/0266-5611/29/i=8/a=085003. doi: 10.1088/0266-5611/29/8/085003.

[15]

J. Kaipio and E. Somersalo, Statistical and Computational Inverse Problems, vol. 160 of Applied Mathematical Sciences, Springer Science+Business Media, Inc, 2005.

[16]

M. Le Louarn, C. Verinaud and V. Korkiakoski, Simulation of mcao on (extremely) large telescopes, Comptes Rendus Physique, 6 (2005), 1070-1080. doi: 10.1016/j.crhy.2005.10.004.

[17]

M. Le Louarn, C. Vérinaud, V. Korkiakoski, N. Hubin and E. Marchetti, Adaptive optics simulations for the european extremely large telescope, in SPIE Astronomical Telescopes+ Instrumentation, International Society for Optics and Photonics, 6272 (2006), 627234-627234. doi: 10.1117/12.670187.

[18]

A. Neubauer, On the ill-posedness and convergence of the Shack-Hartmann based wavefront reconstruction, J. Inv. Ill-Posed Problems, 18 (2010), 551-576. doi: 10.1515/JIIP.2010.025.

[19]

A. Neubauer, A new cumulative wavefront reconstructor for the Shack-Hartmann sensor, J. Inv. Ill-Posed Problems, 21 (2013), 451-476. doi: 10.1515/jip-2013-0003.

[20]

D. W. Phillion and K. Baker, Two-sided pyramid wavefront sensor in the direct phase mode, Proc. SPIE, 6272 (2006), 627228. doi: 10.1117/12.671961.

[21]

L. Poyneer, D. Gavel and J. Brase, Fast wave-front reconstruction in large adaptive optics systems with use of the Fourier transform, J. Opt. Soc. Am. A, 19 (2002), 2100-2111. doi: 10.1364/JOSAA.19.002100.

[22]

F. Quirós-Pacheco, C. Correia and S. Esposito, Fourier transform-wavefront reconstruction for the pyramid wavefront sensor, in 1st AO4ELT conference - Adaptive Optics for Extremely Large Telescopes, 2010, 07005-07010.

[23]

R. Ramlau, A. Obereder, M. Rosensteiner and D. Saxenhuber, Efficient iterative tip/tilt reconstruction for atmospheric tomography, Inverse Problems in Science and Engineering, 22 (2014), 1345-1366. doi: 10.1080/17415977.2013.873534.

[24]

R. Ramlau and M. Rosensteiner, An efficient solution to the atmospheric turbulence tomography problem using Kaczmarz iteration, Inverse Problems, 28 (2012), 095004, 23pp. doi: 10.1088/0266-5611/28/9/095004.

[25]

R. Ramlau, D. Saxenhuber and M. Yudytskiy, Iterative reconstruction methods in atmospheric tomography: FEWHA, Kaczmarz and Gradient-based algorithm, in Proc. SPIE, 9148 (2014), 91480Q. doi: 10.1117/12.2057379.

[26]

F. Rigaut, B. Ellerbroek and R. Flicker, Principles, limitations and performance of multiconjugate adaptive optics, Proc. SPIE, 4007 (2000), 1022-1031.

[27]

F. Roddier, The effects of atmospheric turbulence in optical astronomy, Progress in Optics, 19 (1981), 281-376. doi: 10.1016/S0079-6638(08)70204-X.

[28]

F. Roddier, Adaptive Optics in Astronomy, Cambridge, U.K. ; New York : Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511525179.

[29]

M. C. Roggemann and B. Welsh, Imaging Through Turbulence, CRC Press laser and optical science and technology series, CRC Press, 1996.

[30]

M. Rosensteiner, Cumulative reconstructor: Fast wavefront reconstruction algorithm for Extremely Large Telescopes, J. Opt. Soc. Am. A, 28 (2011), 2132-2138. doi: 10.1364/JOSAA.28.002132.

[31]

M. Rosensteiner, Wavefront reconstruction for extremely large telescopes via CuRe with domain decomposition, J. Opt. Soc. Am. A, 29 (2012), 2328-2336. doi: 10.1364/JOSAA.29.002328.

[32]

M. Rosensteiner and R. Ramlau, Efficient iterative atmospheric tomography reconstruction from LGS and additional tip/tilt measurements, in SPIE 8447, Adaptive Optics Systems III, 2012, 84475S. doi: 10.1117/12.945876.

[33]

M. Rosensteiner and R. Ramlau, The Kaczmarz algorithm for multi-conjugate adaptive optics with laser guide stars, J. Opt. Soc. Am. A, 30 (2013), 1680-1686.

[34]

I. Shatokhina, A. Obereder, M. Rosensteiner and R. Ramlau, Preprocessed cumulative reconstructor with domain decomposition: A fast wavefront reconstruction method for pyramid wavefront sensor, {Applied Optics, 52 (2013), 2640-2652. doi: 10.1364/AO.52.002640.

[35]

M. Tallon, I. Tallon-Bosc, C. Béchet, F. Momey, M. Fradin and E. Thiébaut, Fractal iterative method for fast atmospheric tomography on extremely large telescopes, in Proc. SPIE 7736, Adaptive Optics Systems II, 2010, 77360X. doi: 10.1117/12.858042.

[36]

M. Tallon, I. Tallon-Bosc, C. Béchet and E. Thiébaut, Shack-hartmann wavefront reconstruction with elongated sodium laser guide stars: Improvements with priors and noise correlations, Proc. SPIE, Adaptive Optics Systems, 7015 (2008), 70151N, URL http://link.aip.org/link/?PSI/7015/70151N/1. doi: 10.1117/12.788902.

[37]

M. Tallon, E. Thiébaut and C. Béchet, A fractal iterative method for fast wavefront reconstruction for extremely large telescopes, in Adaptive Optics: Methods, Analysis and Applications, Optical Society of America, 2007, paper PMA2 doi: 10.1364/AOPT.2007.PMA2.

[38]

E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes, J. Opt. Soc. Am. A, 27 (2010), 1046-1059.

[39]

A. Tokovinin, M. L. Louarn and M. Sarazin, Isoplanatism in a multi-conjugate adaptive optics system, J. Opt. Soc. Am., 17 (2000), 1819-1827.

[40]

A. Tokovinin and E. Viard, Limiting precision of tomographic phase estimation, J. Opt. Soc. Am., 18 (2001), 873-882. doi: 10.1364/JOSAA.18.000873.

[41]

G. Tyler, Bandwidth considerations for tracking trough turbulence, J. Opt. Soc. Am. A, 11 (1994), 358-367.

[42]

F. Vidal, E. Gendron and G. Rousset, Tomography approach for multi-object adaptive optics, JOSA A, 27 (2010), A253-A264. doi: 10.1364/JOSAA.27.00A253.

[43]

C. Vogel and Q. Yang, Fast optimal wavefront reconstruction for multi-conjugate adaptive optics using the Fourier domain preconditioned conjugate gradient algorithm, Optics Express, 14 (2006), 7487-7498. doi: 10.1364/OE.14.007487.

[44]

C. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction, Applied Optics, 45 (2006), 705-715. doi: 10.1364/AO.45.000705.

[45]

Q. Yang, C. Vogel and B. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography, Applied Optics, 45 (2006), 5281-5293.

[46]

M. Yudytskiy, Wavelet Methods in Adaptive Optics, PhD thesis, Johannes Kepler University Linz, 2014.

[47]

M. Yudytskiy, T. Helin and R. Ramlau, A frequency dependent preconditioned wavelet method for atmospheric tomography, in Third AO4ELT Conference - Adaptive Optics for Extremely Large Telescopes, 2013.

[48]

M. Yudytskiy, T. Helin and R. Ramlau, Finite element-wavelet hybrid algorithm for atmospheric tomography, J. Opt. Soc. Am. A, 31 (2014), 550-560, URL http://josaa.osa.org/abstract.cfm?URI=josaa-31-3-550. doi: 10.1364/JOSAA.31.000550.

[49]

M. Zhariy, A. Neubauer, M. Rosensteiner and R. Ramlau, Cumulative wavefront reconstructor for the Shack-Hartman sensor, Inverse Problems and Imaging, 5 (2011), 893-913. doi: 10.3934/ipi.2011.5.893.

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