Fast iterative optimal estimation of turbulence wavefronts with recursive block Toeplitz covariance matrix

The estimation of a corrugated wavefront after propagation through the atmosphere is usually solved optimally with a Minimum-Mean-Square-Error algorithm. The derivation of the optimal wavefront can be a very computing intensive task especially for large Adaptive Optics (AO) systems that operates in real-time. For the largest AO systems, efficient optimal wavefront reconstructor have been proposed either using sparse matrix techniques or relying on the fractal properties of the atmospheric wavefront. We propose a new method that exploits the Toeplitz structure in the covariance matrix of the wavefront gradient. The algorithm is particularly well-suited to Shack-Hartmann wavefront sensor based AO systems. Thanks to the Toeplitz structure of the covariance, the matrices are compressed up to a thousand-fold and the matrix-to-vector product is reduced to a simple one-dimension convolution product. The optimal wavefront is estimated iteratively with the MINRES algorithm which exhibits better convergence properties for ill-conditioned matrices than the commonly used Conjugate Gradient algorithm. The paper describes, in a first part, the Toeplitz structure of the covariance matrices and shows how to compute the matrix-to-vector product using only the compressed version of the matrices. In a second part, we introduced the MINRES iterative solver and shows how it performs compared to the Conjugate Gradient algorithm for different AO systems.