For large-scale system with tens of thousands of states and outputs the computation in the conventional Kalman filter becomes time-consuming such that Kalman filtering in large-scale real-time application is practically infeasible. A possible mathematical framework to lift the curse of dimensionality is to lift the problem in higher dimensions with the use of tensors and then decompose it. The tensor-train decomposition is chosen due to its computational advantages for systems with low tensor-train rank. Within this thesis two main limitations of the existing tensor Kalman filter are solved. First, a method is developed based on tensor-train rank truncation of the covariances to increase the computational speed for more general systems. Second, a MIMO tensor Kalman filter is developed for a specific class of systems. The power of the developed methods is shown on the example of adaptive optics which fits into the framework. A comparison with state-of-the-art large-scale estimation algorithms shows the computational advantage of the tensor Kalman filter at the cost of approximation errors.

Accurate and safe proton beam delivery is one of the most crucial tasks during Proton Therapy (PT) of ocular melanoma. The eye movement and gaze angle tracking system should be able to monitor in real time the eye position and orientation (6 degree of freedom) to exactly localize the tumor location inside the eye with respect to the proton beam. The system should also immediately switch the beam off if the tumor goes out of the irradiated area to protect vital organs and keep the non affected cells healthy. The non-invasive eye-tracking system will replace the painful surgical procedure of implantation of radio-opaque tantalum clips on the eye. In order to estimate accurately enough the eye gaze angle and torsion, a stereo imaging system consisting of two high-resolution imaging cameras and two infra-red beacons can be used. The six coordinates of the eye are extracted by image analysis of the acquired stereo camera images using the beacons reflections (glints) and location of the eye pupil. The accuracy of the method can be affected by motion artifacts and difficulty of pupil segmentation in some eyes. Further, the method is inclusive to cyclo-torsion (rotation of the eye about its optical axis). The eye-tracking system can thus benefit from motion artifacts suppression and implementation of analysis of the eye surface features, e.g. iris pattern. Another addition is to use steerable mounts and mirrors to improve measurements and, thus, eye tracking accuracy. The goal of this thesis is to build a prototype stereo eye-tracking system, write a tracking code and investigate the viability of this method for further improving eye tracking accuracy.