Image restoration and denoising is an essential preprocessing step for almost every subsequent task in computer vision. Markov Random Fields offer a wellfounded, sophisticated approach for this purpose, but unfortunately the associated computation procedures are not sufficiently fast, due to a high-dimensional optimization problem. While the increase of computing power could not solve this runtime issue appropriately, we address it in a mathematical way: we suggest an analytical solution for the optimum of the inference problem, which provides desirable mathematical properties. In practice, our new method accelerates the runtime via reducing the computational complexity of the image restoration task by orders of magnitude, independent from the smoothing intensity. As a result, Markov Random Fields can be considered for modern big data problems in computer vision, especially if numerous images with equal sizes are processed.

A novel Bayesian approach to the problem of variable selection using Gaussian process regression is proposed. The selection of the most relevant variables for a problem at hand often results in an increased interpretability and in many cases is an essential step in terms of model regularization. In detail, the proposed method relies on so-called nearest neighbor Gaussian processes, that can be considered as highly scalable approximations of classical Gaussian processes. To perform a variable selection the mean and the covariance function of the process are conditioned on a random set A. This set holds the indices of variables that contribute to the model. While the specification of a priori beliefs regarding A allows to control the number of selected variables, so-called reference priors are assigned to the remaining model parameters. The application of the reference priors ensures that the process covariance matrix is (numerically) robust. For the model inference a Metropolis within Gibbs algorithm is proposed. Based on simulated data, an approximation problem from computer experiments and two real-world datasets, the performance of the new approach is evaluated.

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