This paper presents a computationally efficient framework in which a single focal-plane image is used to obtain a high-resolution reconstruction of dynamic aberrations. Assuming small-phase aberrations, a non-linear Kalman filter implementation is developed whose computational complexity scales close to linearly with the number of pixels of the focal-plane camera. The performance of themethod is tested in a simulation of an adaptive optics system, where the small-phase assumption is enforced by considering a closed-loop system that uses a low-resolution wavefront sensor to control a deformable mirror. The results confirmthe computational efficiency of the algorithm and showa large robustness against noise and model uncertainties.

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Phase aberrations in optical systems,which occur in various applications such as astronomy, microscopy and ophthalmology, degrade the quality of obtained images. The exact cause and nature of the aberrations depends on the application. In astronomy, turbulence within the Earth’s atmosphere creates fluctuations of the refractive index, leading to phase aberrations. In order to compensate for the distorting effect of aberrations, adaptive optics (AO) systems are used to correct for phase aberrations in real-time. A deformable mirror (DM) is often used to apply the necessary corrections to improve the image quality. Due to the temporally dynamic nature of atmospheric turbulence and the corresponding phase aberrations, estimation errors caused by delays within the AO control loop are a significant part of the total estimation error. Accurate prediction of the phase aberrations is therefore an important aspect when aiming to improve the performance of AO systems. Reconstructing the phase aberrations from focal plane images only is known as focal plane wavefront sensing. Many focal plane sensing methods are based on solving the phase retrieval problem, which is the problem of reconstructing the phase aberrations from the point spread function (PSF). Due to the non-linear optimization problem that underlies phase retrieval, developing real-time solvers is very challenging and a wavefront sensor (WFS) is often included to avoid the phase retrieval problem. Due to the linear relation between the phase aberration (i.e. wavefront) and WFS signal, WFSbased AO is often preferred over the wavefront sensorless (WFSless) AO systems that use focal plane sensing. There are, however, also a number of disadvantages. First, the addition of extra hardware components, including the WFS and a beam splitter, makes the system more complex and expensive than WFSless systems. Second, splitting the light between the focal plane camera and WFS results in non-common path aberrations (NCPAs), which can be a limiting factor in high-resolution imaging systems. @en

This paper presents a computationally efficient wavefront aberration prediction framework for data-driven control in large-scale adaptive optics systems. Our novel prediction algorithm splits prediction into two stages: A highresolution and a low-resolution stage. For the former, we exploit sparsity structures in the system matrices in a data-driven Kalman filtering algorithm and constrain the identified gain to be likewise sparse; for the latter, we identify a denseKalman gain and performcorrections to the suboptimal predictions of the former on a smaller grid. This novel prediction framework is able to retain the robustness to measurement noise of the standardKalman filter in a much more computationally efficient manner, in both its offline and online aspects, while minimally sacrificing performance; its data-driven nature further compensates for modeling errors. As an intermediate result, we present a sparsity-exploiting data-drivenKalman filtering algorithm able to quickly estimate an approximateKalman gain without solving the Riccati equation.

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This paper discusses various practical problems arising in the design and simulation of predictive control methods for adaptive optics. Although there has been increased attention towards optimal prediction and control methods for AO systems, they are often tested in simplified simulation environments. The use of advanced AO simulators however, is a valuable alternative to the use of real data or laboratory experiments, as they provide both a flexible environment which is ideal for testing a new algorithm and are more accessible to non-experts. Topics that are often not explicitly discussed, such as the identification of a turbulence dynamics model from data, the use of matrix structures in AO systems to decrease the computational complexity and the implementation of Kalman filters to optimally deal with realistic noise conditions are examined. All topics discussed are illustrated by an accompanying Matlab code, which is based on the existing Matlab AO toolbox OOMAO.

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A new wavefront sensorless adaptive optics method is presented that can accurately correct for time-varying aberrations using a single focal plane image at each sample instance. The linear relation between the mean square of the aberration gradient and the change in second moment of the image forms the basis of the presented method. The new algorithm results in significant improvements when an accurate model of the aberration’s temporal dynamics is known, by applying a Kalman filter and optimal control. Moreover, where existing wavefront sensorless adaptive optics methods update all modes sequentially, the information of the Kalman filter is used to select and update the modes that are expected to give the greatest improvement in performance. The performance is analyzed in a simulation of an adaptive optics system for atmospheric turbulence. The results show that the new method is able to correct for the aberration more accurately for higher wind speeds and higher noise levels than existing algorithms.

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An extension of the Tensor Network (TN) Kalman filter [2], [3] for large scale LTI systems is presented in this paper. The TN Kalman filter can handle exponentially large state vectors without constructing them explicitly. In order to have efficient algebraic operations, a low TN rank is required. We exploit the possibility to approximate the covariance matrix as a TN with a low TN rank. This reduces the computational complexity for general SISO and MIMO LTI systems with TN rank greater than one significantly while obtaining an accurate estimation. Improvements of this method in terms of computational complexity compared to the conventional Kalman filter are demonstrated in numerical simulations for large scale systems.

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In this paper, we propose the use of Gaussian radial basis functions (GRBFs) to model the generalized pupil function for phase retrieval. The selection of the GRBF hyper-parameters is analyzed to achieve an increased accuracy of approximation. The performance of the GRBF-based method is compared in a simulation study with another modal-based approach considering extended Nijboer–Zernike (ENZ) polynomials. The almost local character of the GRBFs makes them a much more flexible basis with respect to the pupil geometry. It has been shown that for aberrations containing higher spatial frequencies, the GRBFs outperform ENZ polynomials significantly, even on a circular pupil. Moreover, the flexibility has been demonstrated by considering the phase retrieval problem on an annular pupil.

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