The next generation ground-based telescopes rely heavily on adaptive optics for overcoming the limitation of atmospheric turbulence. In the future adaptive optics modalities, like multi-conjugate adaptive optics (MCAO), atmospheric tomography is the major mathematical and computational challenge. In this severely ill-posed problem a fast and stable reconstruction algorithm is needed that can take into account many real-life phenomena of telescope imaging. We introduce a novel reconstruction method for the atmospheric tomography problem and demonstrate its performance and flexibility in the context of MCAO. Our method is based on using locality properties of compactly supported wavelets, both in the spatial and frequency domain. The reconstruction in the atmospheric tomography problem is obtained by solving the Bayesian MAP estimator with a conjugate gradient based algorithm. An accelerated algorithm with preconditioning is also introduced. Numerical performance is demonstrated on the official end-to-end simulation tool OCTOPUS of European Southern Observatory.
Reconstruction of the refractive index fluctuations in the atmosphere, or atmospheric tomography, is an underlying problem of many next generation adaptive optics (AO) systems, such as the multiconjugate adaptive optics or multiobject adaptive optics (MOAO). The dimension of the problem for the extremely large telescopes, such as the European Extremely Large Telescope (E-ELT), suggests the use of iterative schemes as an alternative to the matrix-vector multiply (MVM) methods. Recently, an algorithm based on the wavelet representation of the turbulence has been introduced in [Inverse Probl.29, 085003 (2013)] by the authors to solve the atmospheric tomography using the conjugate gradient iteration. The authors also developed an efficient frequency-dependent preconditioner for the wavelet method in a later work. In this paper we study the computational aspects of the wavelet algorithm. We introduce three new techniques, the dual domain discretization strategy, a scale-dependent preconditioner, and a ground layer multiscale method, to derive a method that is globally O(n), parallelizable, and compact with respect to memory. We present the computational cost estimates and compare the theoretical numerical performance of the resulting finite element-wavelet hybrid algorithm with the MVM. The quality of the method is evaluated in terms of an MOAO simulation for the E-ELT on the European Southern Observatory (ESO) end-to-end simulation system OCTOPUS. The method is compared to the ESO version of the Fractal Iterative Method [Proc. SPIE7736, 77360X (2010)] in terms of quality.
Multi conjugate adaptive optics (MCAO) is a system planned for all future extremely large telescopes to compensate in real-time for the optical distortions caused by atmospheric turbulence over a wide field of view. The principles of MCAO are based on two inverse problems: a stable tomographic reconstruction of the turbulence profile followed by the optimal alignment of multiple deformable mirrors (DMs), conjugated to different altitudes in the atmosphere. We present a novel method to treat the optimal mirror deformation problem for MCAO. Contrary to the standard approach where the problem is formulated over a discrete set of optimization directions we focus on the solution of the continuous optimization problem. In the paper we study the existence and uniqueness of the solution and present a Tikhonov based regularization method. This approach gives us the flexibility to apply quadrature rules for a more sophisticated discretization scheme. Using numerical simulations in the context of the European extremely large telescope we show that our method leads to a significant improvement in the reconstruction quality over the standard approach and allows to reduce the numerical burden on the computer performing the computations.
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