Singular Value and Frame Decomposition-based Reconstruction for Atmospheric Tomography

TL;DR

This paper develops advanced singular value and frame decomposition techniques for atmospheric tomography, enabling more efficient and realistic numerical reconstructions crucial for adaptive optics in large telescopes.

Contribution

It extends singular value decompositions to Sobolev spaces with weighted inner products and derives explicit frame-based solution operators for atmospheric tomography.

Findings

Enhanced numerical reconstruction algorithms for atmospheric turbulence profiles.

Validated methods through simulations of the Extremely Large Telescope's adaptive optics system.

Demonstrated improved efficiency and realism in atmospheric profile reconstruction.

Abstract

Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems used for enhancing the image quality of ground-based telescopes. Singular-value and frame decompositions of the underlying atmospheric tomography operator can reveal useful analytical information on this inverse problem, as well as serve as the basis of efficient numerical reconstruction algorithms. In this paper, we extend existing singular value decompositions to more realistic Sobolev settings including weighted inner products, and derive an explicit representation of a frame-based (approximate) solution operator. These investigations form the basis of efficient numerical solution methods, which we analyze via numerical simulations for the challenging, real-world Adaptive Optics system of the Extremely Large Telescope using the entirely MATLAB-based simulation tool MOST.