Fast minimum variance wavefront reconstruction for extremely large telescopes

TL;DR

The paper introduces FRiM, a fast wavefront reconstruction algorithm for extremely large telescopes that leverages fractal approximations and conjugate gradient methods to achieve O(N) complexity, significantly outperforming traditional methods.

Contribution

The paper presents a novel O(N) wavefront reconstruction algorithm using fractal basis approximation and efficient preconditioning, optimized for extremely large telescopes.

Findings

FRiM is over 100 times faster than classical methods for large sensors.

The algorithm achieves solution in 5-10 conjugate gradient iterations regardless of N.

It effectively enforces Kolmogorov statistics through fractal approximation.

Abstract

We present a new algorithm, FRiM (FRactal Iterative Method), aiming at the reconstruction of the optical wavefront from measurements provided by a wavefront sensor. As our application is adaptive optics on extremely large telescopes, our algorithm was designed with speed and best quality in mind. The latter is achieved thanks to a regularization which enforces prior statistics. To solve the regularized problem, we use the conjugate gradient method which takes advantage of the sparsity of the wavefront sensor model matrix and avoids the storage and inversion of a huge matrix. The prior covariance matrix is however non-sparse and we derive a fractal approximation to the Karhunen-Loeve basis thanks to which the regularization by Kolmogorov statistics can be computed in O(N) operations, N being the number of phase samples to estimate. Finally, we propose an effective preconditioning which also scales as O(N) and yields the solution in 5-10 conjugate gradient iterations for any N. The resulting algorithm is therefore O(N). As an example, for a 128 x 128 Shack-Hartmann wavefront sensor, FRiM appears to be more than 100 times faster than the classical vector-matrix multiplication method.