Reduced wavefront reconstruction mean square error using optimal priors: algebraic analysis and simulations

摘要

The turbulent wavefront reconstruction step in an adaptive optics system is an inverse problem. The Mean-Square Error (MSE) assessing the reconstruction quality is made of two terms, often called bias and variance. The latter is also commonly referred as the noise propagation. The aim of this paper is to investigate the evolution of these two error contributions when the number of parameters to be estimated becomes of the order of 10~4. Such dimensions are expected for the adaptive optics systems on the Extremely Large Telescopes. We provide an algebraic formalism to compare the MSE of Maximum Likelihood and Maximum A Posteriori linear reconstructors. A Generalized Singular Value Decomposition applied on the reconstructors theoretically enhances the differences between zonal and modal approaches, and demonstrates the gain in using Maximum A Posteriori method. Thanks to numerical simulations, we quantitatively study the evolution of the MSE contributions with respect to the pupil shape, to the outer scale of the turbulence, to the number of actuators and to the signal-to-noise ratio. Simulations results are consistent with previous noise propagation studies and with our algebraic analysis. Finally, using the Fractal Iterative Method as a Maximum A Posteriori reconstruction algorithm in our simulations, we demonstrate a possible reduction of the MSE of a factor 2 in large adaptive optics systems, for low signal-to-noise ratio.