A Kaczmarz type iterative reconstructor for Multi Conjugate Adaptive Optics
R. Ramlau and M. Rosensteiner
Industrial Mathematics Institute, Kepler University Linz and Mathconsult GmbH, Linz
Most of the large earthbound astronomical telescopes use Adaptive Optics technology (AO) in order to enhance the image quality. The degradation of the measured images is caused by atmospheric turbulences. Multi Conjugate Adaptive Optics (MCAO) is a technique that aims at a high imaging quality over a large field of view, which is achieved by using several (laser) guide stars, each assigned to a wavefront sensor, and several deformable mirrors that correct for turbulences in different atmospheric layers. Mathematically, the computation of a the optimal mirror deformations from wavefront measurements forms an inverse and ill posed problem. Therefore, regularization methods have to be used for a stable reconstruction. The solution to the MCAO problem involves three steps: reconstruction of the incoming wavefront, reconstruction of the turbulent layers (atmospheric tomography) and computation of the best mirror correction (fitting step). The standard approach collects these three operations into one matrix, leading to a large and ill conditioned matrix vector system. In our talk we present a method that solves the three subproblems subsequently. First, a fast wavefront reconstructor is used to reconstruct the incoming wavefronts from measurements of a Shack - Hartmann sensor. The atmospheric tomography problem is solved by a Kaczmarz type iterative reconstructor, which converges geometrically to a solution, meaning that only few iterations are needed for a sufficiently good reconstruction of the turbulent layers. The fitting step is finally achieved by a projection of the reconstructed turbulent layers to the mirrors. Due to the fast reconstructors for the subproblems the whole algorithm guarantees a fast reconstruction, which will be demonstrated with our numerical implementations. We wish to remark that the method can easily be adapted to Multi Object Adaptive Optics (MOAO), as only a different fitting step is needed.