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Singular Value-based Atmospheric Tomography with Fourier Domain Regularization (SAFR)

Lukas Weissinger, Simon Hubmer, Bernadett Stadler, Ronny Ramlau

TL;DR

SAFR presents a singular-value-based, Fourier-domain regularization approach to atmospheric tomography within a three-step adaptive optics framework for MCAO on extremely large telescopes. By combining a Fourier basis, precomputed inverses, and an FFT-enabled workflow, SAFR achieves a matrix-free, memory-efficient solution with favorable parallelization properties. Numerical experiments in a MORFEO-like COMPASS setup show SAFR qualitatively matching or surpassing LS and approaching FEWHA performance while reducing computational cost and memory demands. The method offers practical advantages for real-time AO control, though a fully parallelized hard real-time implementation remains a subject for future work.

Abstract

Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems. It is used to enhance the image quality of ground-based telescopes, such as for the Multiconjugate Adaptive Optics Relay For ELT Observations (MORFEO) instrument on the Extremely Large Telescope (ELT). To solve this problem, a singular-value decomposition (SVD) based approach has been proposed before. In this paper, we focus on the numerical implementation of the SVD-based Atmospheric Tomography with Fourier Domain Regularization Algorithm (SAFR) and its performance for Multi-Conjugate Adaptive Optics (MCAO) systems. The key features of the SAFR algorithm are the utilization of the FFT and the pre-computation of computationally demanding parts. Together, this yields a fast algorithm with less memory requirements than commonly used Matrix Vector Multiplication (MVM) approaches. We evaluate the performance of SAFR regarding reconstruction quality and computational expense in numerical experiments using the simulation environment COMPASS, in which we use an MCAO setup resembling the physical parameters of the MORFEO instrument of the ELT.

Singular Value-based Atmospheric Tomography with Fourier Domain Regularization (SAFR)

TL;DR

SAFR presents a singular-value-based, Fourier-domain regularization approach to atmospheric tomography within a three-step adaptive optics framework for MCAO on extremely large telescopes. By combining a Fourier basis, precomputed inverses, and an FFT-enabled workflow, SAFR achieves a matrix-free, memory-efficient solution with favorable parallelization properties. Numerical experiments in a MORFEO-like COMPASS setup show SAFR qualitatively matching or surpassing LS and approaching FEWHA performance while reducing computational cost and memory demands. The method offers practical advantages for real-time AO control, though a fully parallelized hard real-time implementation remains a subject for future work.

Abstract

Atmospheric tomography, the problem of reconstructing atmospheric turbulence profiles from wavefront sensor measurements, is an integral part of many adaptive optics systems. It is used to enhance the image quality of ground-based telescopes, such as for the Multiconjugate Adaptive Optics Relay For ELT Observations (MORFEO) instrument on the Extremely Large Telescope (ELT). To solve this problem, a singular-value decomposition (SVD) based approach has been proposed before. In this paper, we focus on the numerical implementation of the SVD-based Atmospheric Tomography with Fourier Domain Regularization Algorithm (SAFR) and its performance for Multi-Conjugate Adaptive Optics (MCAO) systems. The key features of the SAFR algorithm are the utilization of the FFT and the pre-computation of computationally demanding parts. Together, this yields a fast algorithm with less memory requirements than commonly used Matrix Vector Multiplication (MVM) approaches. We evaluate the performance of SAFR regarding reconstruction quality and computational expense in numerical experiments using the simulation environment COMPASS, in which we use an MCAO setup resembling the physical parameters of the MORFEO instrument of the ELT.
Paper Structure (11 sections, 33 equations, 5 figures, 6 tables, 3 algorithms)

This paper contains 11 sections, 33 equations, 5 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Atmospheric Tomography in Three-step Adaptive Optics
  3. The Singular Value-based Atmospheric Tomography with Fourier Domain Regularization Algorithm
  4. Mathematical description
  5. Numerical Implementation
  6. The SAFR Algorithm in Three-step AO
  7. Computational Cost
  8. Numerical Experiments
  9. Test Configuration and Simulation Environment
  10. Numerical Results
  11. Conclusion

Figures (5)

  • Figure 1.1: 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 3.1: Structure of the matrix $B$, for $L=3$, $G=6$.
  • Figure 4.1: Different configurations of faint NGSs (orange) and LGSs (blue). The $1$ arcmin FoV is marked in gray.
  • Figure 4.2: Average SE Strehl ratio over time (left) and LE Strehl ratio vs. separation (right) after 10,000 time steps for different seeing conditions (different Fried parameter $r_0$) in the lowflux setting with 500 photons per subaperture per frame.
  • Figure 4.3: Average SE Strehl ratio over time (left) and LE Strehl ratio vs. separation (right) after 10,000 time steps for medium seeing conditions $r_0=0.157m$ in the highflux setting with 10,000 photons per subaperture per frame.