An <i>H<sup>1</sup>
                  </i>-Kaczmarz reconstructor for atmospheric tomography

Eslitzbichler, Markus; Pechstein, Clemens; Ramlau, Ronny

doi:10.1515/jip-2013-0007

Cited by 8 publications

(5 citation statements)

References 16 publications

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“…Following the Kolmogorov model, considerations in [16] initiate to expect smoother functions for representing atmospheric turbulence than just L 2 with a high probability. This leads to the assumption of Φ ∈ H 11/6 R 2 as already suggested in [18].…”

Section: Pyramid and Roof Wavefront Sensor Forward Operatorsmentioning

confidence: 95%

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Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

2019

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We consider the mathematical background of the wavefront sensor type that is widely used in Adaptive Optics systems for astronomy, microscopy, and ophthalmology. The theoretical analysis of the pyramid sensor forward operators presented in this paper is aimed at a subsequent development of fast and stable algorithms for wavefront reconstruction from data of this sensor type. In our analysis we allow the sensor to be utilized in both the modulated and non-modulated fashion. We derive detailed mathematical models for the pyramid sensor and the physically simpler roof wavefront sensor as well as their various approximations. Additionally, we calculate adjoint operators which build preliminaries for the application of several iterative mathematical approaches for solving inverse problems such as gradient based algorithms, Landweber iteration or Kaczmarz methods.

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Section: Pyramid and Roof Wavefront Sensor Forward Operatorsmentioning

confidence: 95%

“…We choose any Φ, Ψ ∈ L 2 R 2 with support on the telescope pupil Ω. Due to the linearity of the inner product and with representation (18), it holds that…”

Section: Adjoint Operatorsmentioning

confidence: 99%

Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

2019

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“…where s = [s x , s y ] denotes pyramid wavefront sensor measurements, P : D (P ) → L 2 R 2 the non-linear pyramid sensor operator with D (P ) ⊆ H 11/6 R 2 and Φ ∈ H 11/6 R 2 the unknown incoming wavefront [12,13,29,50]. Due to the finite size of both the telescope pupil and the wavefront sensor detector, in the following denoted by Ω = Ω y × Ω x for simplicity, the involved wavefronts Φ and sensor measurements s have compact support on Ω.…”

Section: Non-linear Problem Of Wavefront Reconstruction Using Pyramidmentioning

confidence: 99%

Nonlinear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber–Kaczmarz iterations

Hutterer

Ramlau

2018

Appl. Opt.

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Accurate and robust wavefront reconstruction methods for pyramid wavefront sensors are in high demand as these sensors are planned to be part of many instruments currently under development for ground based telescopes. The pyramid sensor relates the incoming wavefront and its measurements in a non-linear way. Nevertheless, almost all existing reconstruction algorithms are based on a linearization of the model. The assumption of a linear pyramid sensor response is justifiable in closed loop AO when the measured phase information is small but may not be reasonable in reality due to unpreventable errors depending on the system such as non common path aberrations. In order to solve the non-linear inverse problem of wavefront reconstruction from pyramid sensor data we introduce two new methods based on the non-linear Landweber and Landweber-Kaczmarz iteration. Using these algorithms we experience high-quality wavefront estimation especially for the non-modulated sensor by still keeping the numerical effort feasible for large-scale AO systems.the Subaru Telescope (SCexAO), the Magellan Telescope (MagAO), the Mont Megantic Telescope (INO Demonstrator), and the Calar Alto Telescope (PYRAMIR).

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“…Figure . Note that it is also possible to consider the tomography operator acting between the spaces

\otimes_{l = 1}^{L} H^{1} ({normalΩ}_{l})

to H 1 (Ω D ); . Let us explain our notation: we write functions related to turbulence/wavefront distortions (and only those) at some layer l with an upper bracketed index, Φ ( l ) .…”

Section: Problem Settingmentioning

confidence: 99%

Towards analytical model optimization in atmospheric tomography

Helin

Kindermann²,

Saxenhuber³

2016

Math Methods in App Sciences

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Modern ground‐based telescopes rely on a technology called adaptive optics in order to compensate for the loss of angular resolution caused by atmospheric turbulence. Next‐generation adaptive optics systems designed for a wide field of view require a stable and high‐resolution reconstruction of the turbulent atmosphere. By introducing a novel Bayesian method, we address the problem via reconstructing the atmospheric turbulence strength profile and the turbulent layers simultaneously, where we only use wavefront measurements of incoming light from guide stars. Most importantly, we demonstrate how this method can be used for model optimization as well. We propose two different algorithms for solving the maximum a posteriori estimate: the first approach is based on alternating minimization and has the advantage of integrability into existing atmospheric tomography methods. In the second approach, we formulate a convex non‐differentiable optimization problem, which is solved by an iterative thresholding method. This approach clearly illustrates the underlying sparsity‐enforcing mechanism for the strength profile. By introducing a tuning/regularization parameter, an automated model reduction of the layer structure of the atmosphere is achieved. Using numerical simulations, we demonstrate the performance of our method in practice. Copyright © 2016 John Wiley & Sons, Ltd.

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An H¹ -Kaczmarz reconstructor for atmospheric tomography

Cited by 8 publications

References 16 publications

Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

Nonlinear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber–Kaczmarz iterations

Towards analytical model optimization in atmospheric tomography

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An H1 -Kaczmarz reconstructor for atmospheric tomography

Cited by 8 publications

References 16 publications

Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

Real-time adaptive optics with pyramid wavefront sensors: part I. A theoretical analysis of the pyramid sensor model

Nonlinear wavefront reconstruction methods for pyramid sensors using Landweber and Landweber–Kaczmarz iterations

Towards analytical model optimization in atmospheric tomography

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Product

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An H¹ -Kaczmarz reconstructor for atmospheric tomography