Streaming video completion is the practice that aims to fill in missing or corrupted pixels in a video stream by using past uncorrupted data. A method to tackle this problem is recently introduced called a Tensor Networked Kalman Filter (TNKF). It shows promising results in terms of performance compared to state-of-the-art methods for high percentages of missing pixels (≥ 95%). The main drawback of using a TNKF is the computational speed, which needs to be improved to compete with other existing methods and to be carried out in real-time by a regular computer. This work discusses three methods that reduce the computational load of the algorithm, which speeds up computations. The first method is replacing the existing algorithm with a Block Update TNKF. Secondly, the use of randomized rounding instead of deterministic rounding is investigated. The last method is the simplification of the TNKF update. Results that are presented in this report show that significant speedups of up to +132% can be achieved. In most situations, the considered speedup methods compromise the reconstruction’s accuracy. This thesis discusses the effects this has on the quality of the reconstruction.

Indoor positioning systems cannot rely on conventional localization methods, such as GPS, to locate devices because of interference with the structure of buildings. One solution is to use magnetic positioning, which is based on spatial variations in the patterns of the ambient magnetic field. To model magnetic fields, Gaussian process regression is used, providing predictions of the magnetic field at unvisited locations along with uncertainty quantification. These predictions and their uncertainties are valuable information for probabilistic localization algorithms used for magnetic positioning. Full Gaussian process regression has poor scalability, becoming computationally intractable from roughly 10,000 one-dimensional measurements due to its associated cubic computational complexity. In the existing literature, approximations for Gaussian process regression have been extensively studied to reduce this computational complexity. Of these approximations, only approximations involving basis functions and local experts have been used in the context of scalable magnetic field modeling. A favorable approximation framework from existing literature uses structured kernel interpolation (SKI), allowing for fast regression through efficient Krylov subspace methods. The SKI framework is favorable as it allows for fast regression in low dimensions without introducing boundary effects. In this thesis, the SKI framework is used to approximate two distinct magnetic field models: the shared model, which considers independence between the magnetic field components with shared hyperparameters, and the scalar potential model, which includes physical properties of the magnetic field (Maxwell’s equations) in the model. The scalability of the approach is shown using simulations and experiments with magnetic field measurements. Through the simulations, it is shown the SKI framework accurately and efficiently approximates the models. The applicability of the SKI framework for scalable magnetic field modeling is investigated using data collected using a motion capture suit, the Xsens MVN Link Suit. In the final experiment, a magnetic field map is constructed based on more than 40,000 three-dimensional measurements without splitting the data set, which took less than one minute on a standard laptop.

Online video completion aims to complete corrupted frames of a video in an online fashion. Consider a surveillance camera that suddenly outputs corrupted data, where up to 95% of the pixels per frame are corrupted. Real time video completion and correction is often desirable in such scenarios. Therefore, this thesis improves the Tensor-Networked Kalman Filter (TNKF) as presented in [12] by developing the Tensor-Networked Square-Root Kalman Filter (TNSRKF). The TNSRKF is a Square-Root Kalman Filter (SRKF) realized in a Tensor Networks (TN) structure. The square-root Kalman filter is an inherently stable algorithm, and indirectly allows for more information retention throughout the algorithm. Thereby, the filter aims to improve the performance of the TNKF. Furthermore, implementing the filter in TN results in an intuitive implementation of the filter while allowing for larger video dimensions than the widely used matrix format. This thesis concludes that the TNSRKF is too computationally burdensome to complete a video in an online fashion due to the implementation of the Modified Gram-Schmidt (MGS) algorithm in Tensor-Train (TT)-format. In addition, results demonstrate that a low-rank orthogonality matrix is not realizable, making it impractical to update the state covariance matrix via any method that uses an orthonormal matrix.