Cumulative wavefront reconstructor for the Shack-Hartmann sensor

Previous Article
IPI Home
This Issue
Next Article
Computing the fibre orientation from Radon data using local Radon transform

November 2011, 5(4): 893-913. doi: 10.3934/ipi.2011.5.893

Cumulative wavefront reconstructor for the Shack-Hartmann sensor

Mariya Zhariy ^1, , Andreas Neubauer ^2, , Matthias Rosensteiner ^3, and Ronny Ramlau ^1,

1.	Industrial Mathematics Institute, Johannes Kepler University Linz, A-4040 Linz, Austria, Austria
2.	Industrial Mathematics Institute, Johannes Kepler University Linz A-4040 Linz
3.	MathConsult GmbH, Altenbergerstrae 69, A-4040 Linz, Austria

Received November 2010 Revised September 2011 Published November 2011

Abstract
Related Papers

We present a new direct algorithm aiming at the reconstruction of the optical wavefront from the Shack-Hartmann sensor measurements in Single Conjugate Adaptive Optics (SCAO) systems. The objective of an adaptive optics system designed for a large telescope can be only achieved if the wavefront reconstruction is sufficiently fast. Our scheme does not contain any explicit regularization for the reconstruction process but is still able to provide a good quality of reconstruction. The analysis of quality is given for three varying parameters: the diameter of the telescope, the number of subapertures and the level of photon noise. It has been shown both analytically and numerically that the quality of the reconstruciton, measured by the Strehl ratio, is reasonable for the small photon noise level and increases with the increasing number of subapertures for the same telescope size. The impact of the photon noise on the reconstruction gets higher with the increasing telescope diameter. The computational complexity of the method is linear in the number of unkowns. Counting all summation and multiplication steps the scaling factor is $14$. Moreover, due to its simple structure, the cumulative reconstructor is pipelinable and parallelizable, which makes the effective computation even faster.

Keywords: cumulative wavefront reconstructor., Adaptive optics, inverse problems.

Mathematics Subject Classification: 78A10, 78M2.

Citation: Mariya Zhariy, Andreas Neubauer, Matthias Rosensteiner, Ronny Ramlau. Cumulative wavefront reconstructor for the Shack-Hartmann sensor. Inverse Problems and Imaging, 2011, 5 (4) : 893-913. doi: 10.3934/ipi.2011.5.893

References:

[1]	J. M. Beckers, Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics, in "Proc. European Southern Observatory Conf. and Workshop on Very Large Telescopes and Their Instrumentation'' (ed. M. H. Ulrich), Vol. 30, (1988), 693-703.
[2]	M. A. Davison, The ill-conditioned nature of the limited angle tomography problem, SIAM J. Appl. Math., 43 (1983), 428-448. doi: 10.1137/0143028.
[3]	B. L. 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.
[4]	B. L. Ellerbroek and C. R. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes, in "Proc. SPIE vol 5169-23," Adaptive Optics System Technologies II, (2003), 206-217. doi: 10.1117/12.506580.
[5]	B. L. Ellerbroek and C. R. Vogel, Inverse problems in astronomical adaptive optics, Inverse Problems, 25 (2009), 063001, 37 pp.
[6]	H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
[7]	D. L. Fried, Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements, J. Opt. Soc. Am., 67 (1977), 370-375. doi: 10.1364/JOSA.67.000370.
[8]	L. Gilles, Order $N$ sparse minimum-variance open-loop reconstructor for extreme adaptive optics, Opt. Lett., 28 (2003), 1927-1929. doi: 10.1364/OL.28.001927.
[9]	L. Gilles, Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multi-conjugate adaptive optics, Appl. Opt., 44 (2004), 993-1002. doi: 10.1364/AO.44.000993.
[10]	L. Gilles, C. R. Vogel and B. L. 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.
[11]	J. Herrmann, Least-squares wave front errors of minimum norm, J. Opt. Soc. Am., 70 (1980), 28-35. doi: 10.1364/JOSA.70.000028.
[12]	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.
[13]	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. A, 19 (2002), 2100-2111. doi: 10.1364/JOSAA.19.002100.
[14]	L. A. Poyneer and J.-P. Véran, Optimal modal Fourier transform wave-front control, J. Opt. Soc. Am. A, 22 (2005), 1515-1526. doi: 10.1364/JOSAA.22.001515.
[15]	F. Roddier, "Adaptive Optics in Astronomy,'' Cambridge University Press, Cambridge, New York, 1999. doi: 10.1017/CBO9780511525179.
[16]	M. C. Roggemann and B. M. Welsh, "Imaging Through Turbulence,'' CRC Press, Boca Raton, Florida, 1996.
[17]	E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes, J. Opt. Soc. Am. A, 27 (2010), 1046-1059. doi: 10.1364/JOSAA.27.001046.
[18]	C. R. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction, Applied Optics, 45 (2006), 705-715. doi: 10.1364/AO.45.000705.
[19]	Q. Yang, C. R. Vogel and B. L. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography, Applied Optics, 45 (2006), 5281-5293. doi: 10.1364/AO.45.005281.

show all references

References:

[1]	J. M. Beckers, Increasing the size of the isoplanatic patch with multi-conjugate adaptive optics, in "Proc. European Southern Observatory Conf. and Workshop on Very Large Telescopes and Their Instrumentation'' (ed. M. H. Ulrich), Vol. 30, (1988), 693-703.
[2]	M. A. Davison, The ill-conditioned nature of the limited angle tomography problem, SIAM J. Appl. Math., 43 (1983), 428-448. doi: 10.1137/0143028.
[3]	B. L. 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.
[4]	B. L. Ellerbroek and C. R. Vogel, Simulations of closed-loop wavefront reconstruction for multiconjugate adaptive optics on giant telescopes, in "Proc. SPIE vol 5169-23," Adaptive Optics System Technologies II, (2003), 206-217. doi: 10.1117/12.506580.
[5]	B. L. Ellerbroek and C. R. Vogel, Inverse problems in astronomical adaptive optics, Inverse Problems, 25 (2009), 063001, 37 pp.
[6]	H. Engl, M. Hanke and A. Neubauer, "Regularization of Inverse Problems,'' Mathematics and its Applications, 375, Kluwer Academic Publishers Group, Dordrecht, 1996.
[7]	D. L. Fried, Least-square fitting a wave-front distortion estimate to an array of phase-difference measurements, J. Opt. Soc. Am., 67 (1977), 370-375. doi: 10.1364/JOSA.67.000370.
[8]	L. Gilles, Order $N$ sparse minimum-variance open-loop reconstructor for extreme adaptive optics, Opt. Lett., 28 (2003), 1927-1929. doi: 10.1364/OL.28.001927.
[9]	L. Gilles, Closed-loop stability and performance analysis of least-squares and minimum-variance control algorithms for multi-conjugate adaptive optics, Appl. Opt., 44 (2004), 993-1002. doi: 10.1364/AO.44.000993.
[10]	L. Gilles, C. R. Vogel and B. L. 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.
[11]	J. Herrmann, Least-squares wave front errors of minimum norm, J. Opt. Soc. Am., 70 (1980), 28-35. doi: 10.1364/JOSA.70.000028.
[12]	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.
[13]	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. A, 19 (2002), 2100-2111. doi: 10.1364/JOSAA.19.002100.
[14]	L. A. Poyneer and J.-P. Véran, Optimal modal Fourier transform wave-front control, J. Opt. Soc. Am. A, 22 (2005), 1515-1526. doi: 10.1364/JOSAA.22.001515.
[15]	F. Roddier, "Adaptive Optics in Astronomy,'' Cambridge University Press, Cambridge, New York, 1999. doi: 10.1017/CBO9780511525179.
[16]	M. C. Roggemann and B. M. Welsh, "Imaging Through Turbulence,'' CRC Press, Boca Raton, Florida, 1996.
[17]	E. Thiébaut and M. Tallon, Fast minimum variance wavefront reconstruction for extremely large telescopes, J. Opt. Soc. Am. A, 27 (2010), 1046-1059. doi: 10.1364/JOSAA.27.001046.
[18]	C. R. Vogel and Q. Yang, Multigrid algorithm for least-squares wavefront reconstruction, Applied Optics, 45 (2006), 705-715. doi: 10.1364/AO.45.000705.
[19]	Q. Yang, C. R. Vogel and B. L. Ellerbroek, Fourier domain preconditioned conjugate gradient algorithm for atmospheric tomography, Applied Optics, 45 (2006), 5281-5293. doi: 10.1364/AO.45.005281.

[1]	Michael Herty, Giuseppe Visconti. Kinetic methods for inverse problems. Kinetic and Related Models, 2019, 12 (5) : 1109-1130. doi: 10.3934/krm.2019042
[2]	Guanghui Hu, Peijun Li, Xiaodong Liu, Yue Zhao. Inverse source problems in electrodynamics. Inverse Problems and Imaging, 2018, 12 (6) : 1411-1428. doi: 10.3934/ipi.2018059
[3]	Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems and Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1
[4]	Sergei Avdonin, Pavel Kurasov. Inverse problems for quantum trees. Inverse Problems and Imaging, 2008, 2 (1) : 1-21. doi: 10.3934/ipi.2008.2.1
[5]	Maciej Zworski. A remark on inverse problems for resonances. Inverse Problems and Imaging, 2007, 1 (1) : 225-227. doi: 10.3934/ipi.2007.1.225
[6]	Zhi-An Wang. Wavefront of an angiogenesis model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (8) : 2849-2860. doi: 10.3934/dcdsb.2012.17.2849
[7]	Janne M.J. Huttunen, J. P. Kaipio. Approximation errors in nonstationary inverse problems. Inverse Problems and Imaging, 2007, 1 (1) : 77-93. doi: 10.3934/ipi.2007.1.77
[8]	Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380
[9]	Masoumeh Dashti, Stephen Harris, Andrew Stuart. Besov priors for Bayesian inverse problems. Inverse Problems and Imaging, 2012, 6 (2) : 183-200. doi: 10.3934/ipi.2012.6.183
[10]	Xiaosheng Li, Gunther Uhlmann. Inverse problems with partial data in a slab. Inverse Problems and Imaging, 2010, 4 (3) : 449-462. doi: 10.3934/ipi.2010.4.449
[11]	Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems and Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002
[12]	Tony Liimatainen, Lauri Oksanen. Counterexamples to inverse problems for the wave equation. Inverse Problems and Imaging, 2022, 16 (2) : 467-479. doi: 10.3934/ipi.2021058
[13]	Manuel Gutiérrez. Lorentz geometry technique in nonimaging optics. Conference Publications, 2003, 2003 (Special) : 386-392. doi: 10.3934/proc.2003.2003.386
[14]	Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89
[15]	Zheng-Ru Zhang, Tao Tang. An adaptive mesh redistribution algorithm for convection-dominated problems. Communications on Pure and Applied Analysis, 2002, 1 (3) : 341-357. doi: 10.3934/cpaa.2002.1.341
[16]	Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553
[17]	François Monard, Guillaume Bal. Inverse diffusion problems with redundant internal information. Inverse Problems and Imaging, 2012, 6 (2) : 289-313. doi: 10.3934/ipi.2012.6.289
[18]	Johannes Elschner, Guanghui Hu. Uniqueness in inverse transmission scattering problems for multilayered obstacles. Inverse Problems and Imaging, 2011, 5 (4) : 793-813. doi: 10.3934/ipi.2011.5.793
[19]	Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems and Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511
[20]	Simopekka Vänskä. Stationary waves method for inverse scattering problems. Inverse Problems and Imaging, 2008, 2 (4) : 577-586. doi: 10.3934/ipi.2008.2.577

2020 Impact Factor: 1.639

Tools

Metrics

Other articles
by authors

on AIMS
Mariya Zhariy Andreas Neubauer Matthias Rosensteiner Ronny Ramlau
on Google Scholar
Mariya Zhariy Andreas Neubauer Matthias Rosensteiner Ronny Ramlau

[Back to Top]