Nonlinear heuristic regularization for astronomical imaging

Many inverse problems in applications are ill-posed, i.e., it is hardly possible to extract the wanted information from data by standard methods since, because of instabilities, small data error are amplified and can corrupt the result. Examples can be found in imaging, parameter identification, or learning theory. Such problems are often handled by regularization methods, where the data are filtered appropriately to get useful results. An essential component therein are the regularization parameters, which control the intensity of filtering and which have to be tuned, usually using additional information and by user interaction, in a problem-adapted way to get useful result. The focus of this project lies, however, on fully-automated data-driven (heuristic) regularization methods for nonlinear inverse problems, which do not require any additional input or information. The fact that these methods work in the more simple linear case is by no means obvious and has been proven only recently. Advanced regularization of inverse problems is nowadays often based on nonlinear filters, but a corresponding automatic parameter selection for it has not been established. One goal of this project is to develop, analyze, and generalize such automatic regularization scheme for nonlinear inverse problems and to apply them to problems in astronomical imaging for large-scale telescopes. In order to design well-performing methods, an accompanying convergence analysis has to be established based on nonlinear noise-conditions. Essentially this last issue requires a clear definition and a deep mathematical understanding what kind of noise and errors can occur in a specific problem. One of the problem, where we want to apply these heuristic schemes, involves the detection and correction of disturbances of the starlight due to turbulence when travelling through the earth`s atmosphere. Current telescope technology measure these disturbances and correct them by deformable mirrors (by adaptive optics systems). The problem of calculating the atmospheric turbulence from these measurements and finding the necessary corrections can be handled by nonlinear regularization methods, and it is intended to improve this technology by using fully automatic heuristic schemes that requires no user-interaction. A second benchmark problem is to estimate image and, e.g., telescope defects simultaneously by a regularization procedure, where advanced nonlinear heuristic methods are an essential ingredient. The new methods to be developed in this project will allow for fully-automatic algorithms with minimal user interaction that work more robust and more precise than traditional methods.

Many inverse problems are ill-posed, i.e., it is hardly possible to extract the wanted information from data by standard methods since, because of instabilities, small data errors are amplified and might corrupt the result. Examples can be found in imaging, parameter identification, or learning theory. Such problems are often handled by regularization methods, where the data are filtered appropriately to get useful results. An essential component therein is the regularization parameter, which controls the intensity of filtering and which has to be tuned, usually by means of additional information, by user interaction. The focus of this project lied, however, on data-driven so-called "heuristic" regularization methods, which do not require any additional input or information and which can detect the correct amount of filtering fully automatic. The fact that these methods work and how they do has been proven only recently and is within the scope of current research. In this project, such fully-automated regularization schemes were investigated in detail, improved, and applied to various inverse problems. Amongst other things, the existing theory for such methods could be extended, for instance, with regard to the inclusion of white noise as data error or for the case of operator perturbations which arise in case of model errors or by discretization. Moreover, a new data-driven method could be developed, which, conceived as a simplification of the popular L-curve method, is, however, superior to the latter with respect to reliability and efficiency. A further outstanding contribution in this project was the development and analysis of data-driven methods for the application to nonlinear filter methods that are current state-of-the-art in modern image processing technologies. Beside applications in image processing, these new methods could be applied to various typical inverse problems, for instance, to the inversion of the Radon transform, which is the basis for computerized tomography technology, or also for some problems in astronomy such as the control of the adaptive optics for large-scale telescopes. The results in this project can serve as foundation for the development for novel, fully-automatic regularization schemes with applications in current hot-topic research, for example, in machine learning or algorithms for artificial intelligence.