A rank-constrained reformulation of the blind deconvolution problem on images taken with coherent illumination is proposed. Since in the reformulation the rank constraint is imposed on a matrix that is affine in the decision variables, we propose a novel convex heuristic for the blind deconvolution problem. The proposed heuristic allows for easy incorporation of prior information on the decision variables and the use of the phase diversity concept. The convex optimization problem can be iteratively re-parameterized to obtain better estimates. The proposed methods are demonstrated on numerically illustrative examples.

To optimally compensate for time-varying phase aberrations with adaptive optics, a model of the dynamics of the aberrations is required to predict the phase aberration at the next time step. We model the time-varying behavior of a phase aberration, expressed in Zernike modes, by assuming that the temporal dynamics of the Zernike coefficients can be described by a vector-valued autoregressive (VAR) model. We propose an iterative method based on a convex heuristic for a rank-constrained optimization problem, to jointly estimate the parameters of the VAR model and the Zernike coefficients from a time series of measurements of the point-spread function (PSF) of the optical system. By assuming the phase aberration is small, the relation between aberration and PSF measurements can be approximated by a quadratic function. As such, our method is a blind identification method for linear dynamics in a stochastic Wiener system with a quadratic nonlinearity at the output and a phase retrieval method that uses a time-evolution-model constraint and a single image at every time step.

Aberrations in optical systems, such as telescopes and microscopes, degrade the quality of the images that can be produced by these systems. For example, an object that is positioned out of focus produces a blurred image on a camera sensor and the turbulent air in the earth’s atmosphere reduces the imaging performance of telescopes. In this thesis we only consider wavefront aberrations. AO can be used to compensate for these wavefront aberrations. The working principle of AO is to quantify by measuring or estimation the wavefront aberration and to dynamically adjust wavefront modulating devices, such as Deformable Mirrors (DMs), to counteract the aberration and thereby improving the optical performance. The estimation of the wavefront aberration based on images of a point source is called phase retrieval, which is a highly nonlinear estimation problem. The success of the estimation usually depends on the (type of) algorithm, the available information on the aberration that is incorporated in the estimate, and the degree to which the model of the optical system corresponds to reality. In this thesis we propose a convex optimization-based method for phase retrieval. The method allows for easy inclusion of many types of prior information on the aberration. Furthermore, we develop an efficient implementation of the optimization. The robustness of the approach against measurement noise is investigated and compared with several other state of the art algorithms. Experimental validation shows the algorithmis well able to estimate aberrations in real-life circumstances. A new type of prior information is introduced to estimate dynamic wavefront aberrations. In literature and in practice, the optical path is split between either a wavefront sensor and a camera, or between multiple cameras in order to reliable estimate an aberration. The inherent problem is that between the sensor and cameras the aberration can differ (Non-Common Path (NCP) errors), and a wrong estimate is used in the compensation by the AO system. We propose a method to estimate the aberration from measurements by a single camera, by assuming that the aberration evolves according to (non-specific) model, i.e. the dynamics are contained in a model-set. At the same time that we estimate the aberration, we also identify the dynamics according to which the aberration evolves over time. The estimation of the wavefront aberration based on images of an unknown object is called blind deconvolution if both the aberration and object are estimated. Like phase retrieval, this too is a highly nonlinear estimation problem. We propose the first convexoptimization based estimation method for blind deconvolution problems that estimate aberration and object when the images are acquired using coherent illumination. The method allows for the inclusion of many existing types of prior information on the object and/or aberration. Finally, we analyze controllers for segmented mirrors in large ground-based telescopes. These mirrors consist of many interconnected hexagonal segments. This distributed nature of the system warrants the investigation into whether the controller that keeps the segments aligned can be designed in such a way that it can be distributed over the segments as well, essentially resulting in a distributed controller where local controllers communicate with each other. What complicates the analysis is that the dynamics across segments are not necessarily decoupled: the wind load can be correlated and the flexibility in the supporting structure of the segments can cause dynamic coupling. We investigate the design of a distributed controller that incorporates these global dynamics. Furthermore, we investigate the performance of the distributed controller and howit relates to the communication and interconnection pattern of the local controllers.

We present a convex relaxation-based algorithm for large-scale general phase retrieval problems. General phase retrieval problems include, e.g., the estimation of the phase of the optical field in the pupil plane based on intensity measurements of a point source recorded in the image (focal) plane. The non-convex problem of finding the complex field that generates the correct intensity is reformulated into a rank constraint problem. The nuclear norm is used to obtain the convex relaxation of the phase retrieval problem. A new iterative method referred to as convex optimization-based phase retrieval (COPR) is presented, with each iteration consisting of solving a convex problem. In the noise-free case and for a class of phase retrieval problems, the solutions of the minimization problems converge linearly or faster towards a correct solution. Since the solutions to nuclear norm minimization problems can be computed using semidefinite programming, and this tends to be an expensive optimization in terms of scalability, we provide a fast algorithm called alternating direction method of multipliers (ADMM) that exploits the problem structure. The performance of the COPR algorithm is demonstrated in a realistic numerical simulation study, demonstrating its improvements in reliability and speed with respect to state-of-the-art methods.

A systematic distributed optimal control design procedure is proposed for the rejection of wind load-induced disturbances on a truss-supported segmented mirror. The distributed nature of the controller is achieved by weighing of the interaction matrices between local (per-segment) controllers in a global H2 optimization. The procedure allows a tradeoff analysis between the controller implementation complexity versus the improved performance the extra communication brings. The procedure is demonstrated on a finite element model of a segmented mirror on a flexible supporting truss to which we apply the combined closed-loop performance and local controller interconnection structure optimization. The resulting set of controllers is compared to a set of baseline controllers including linear-quadratic-Gaussian control, singular value decomposition control, and a distributed controller where local controllers of neighboring segments communicate. The tradeoff analysis for the segmented mirror demonstrates that the communication between the local controllers can be greatly reduced without significantly compromising the rejection of wind-induced wavefront errors.

We analyse the very general class of uncertain systems that have Linear Fractional Representations (LFRs), and uncertainty blocks in a convex set with a finite number of vertices. For these systems we design static output feedback controllers. In the general case, computing a robust static output feedback controller with optimal performance gives rise to a bilinear matrix inequality (BMI). In this article we show how this BMI problem can be efficiently rewritten to fit in the framework of sequential convex relaxation, a method that searches simultaneously for a feasible controller and one with good performance. As such, our approach does not rely on being supplied with a feasible initial solution to the BMI. This sets it apart from methods that depend on a good initial, feasible starting point to progress from there using an alternating optimization scheme. In addition to using the proposed method, the controller matrices can be of a predetermined fixed structure. Alternatively, an L1 constraint can be easily added to the optimization problem as a convex variant of a cardinality constraint, in order to induce sparsity on the controller matrices.

We consider the use of the nuclear norm operator, and its tendency to produce low rank results, to provide a convex relaxation of Bilinear Matrix Inequalities (BMIs). The BMI is first written as a Linear Matrix Inequality (LMI) subject to a bi-affine equality constraint and subsequently rewritten into an LMI subject to a rank constraint on a matrix affine in the decision variables. The convex nuclear norm operator is used to relax this rank constraint. We provide an algorithm that iteratively improves on the sum of the objective function and the norm of the equality constraint violation. The algorithm is demonstrated on a controller synthesis example.