Singular Value and Frame Decomposition-based Reconstruction for Atmospheric Tomography

TL;DR

This work addresses reconstructing atmospheric turbulence profiles from wavefront sensor data by developing singular-value-type decompositions in Sobolev spaces and an explicit frame decomposition for atmospheric tomography. The SVTD extension yields a structured, efficient inversion in Sobolev settings, while the FD provides an explicit, nearly SVD-like reconstruction suitable for general aperture shapes and mixed NGS/LGS data. Numerical experiments in an ELT/MORFEO-like environment using the MOST simulator demonstrate competitive performance of SVTD and show how iterative FD can substantially improve reconstructions, with FEWHA remaining strong in mixed guide-star scenarios. The methods offer promising avenues for real-time or near-real-time AO tomography, with future work aimed at non-MATLAB implementations and optimization for real-time deployment.

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.

Figures (11)

• Figure 1.1: Schematic depiction of wavefront correction via deformable mirror (left, image from Auzinger_2017) and working principle of a SCAO system (right, image from Egner_2006).
• Figure 1.2: Schematic depiction of different types of AO systems. Magenta spirals represent astronomical objects of interest, while red and greens stars correspond to locations of NGS and LGS, respectively. The darker shaded areas correspond to the directions corrected for by the corresponding AO systems. Image taken from Auzinger_2017.
• Figure 1.3: Illustration of the atmospheric tomography problem with three turbulence layers, NGSs and corresponding WFSs (left). Light stemming from a single LGS is influenced by the cone effect (right). Images taken from Yudytskiy_2014.
• Figure 2.1: A schematic drawing of the domains $\Omega_A$, $\Omega_\ell,$ and $\Omega_T$.
• Figure 5.1: Graphical illustration of 3 DMs conjugated to 3 reconstructed layers.

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Corollary 3.4
• proof
• Definition 4.1
• Definition 4.2