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

Rodolphe Conan

doi:10.1117/12.2054472

21 July 2014 Fast iterative optimal estimation of turbulence wavefronts with recursive block Toeplitz covariance matrix

Rodolphe Conan

Proceedings Volume 9148, Adaptive Optics Systems IV; 91480R (2014) https://doi.org/10.1117/12.2054472
Event: SPIE Astronomical Telescopes + Instrumentation, 2014, Montréal, Quebec, Canada

Abstract

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.

Citation Download Citation

Rodolphe Conan "Fast iterative optimal estimation of turbulence wavefronts with recursive block Toeplitz covariance matrix", Proc. SPIE 9148, Adaptive Optics Systems IV, 91480R (21 July 2014); https://doi.org/10.1117/12.2054472

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