Cumulative wavefront reconstructor for the Shack-Hartmann sensor

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.