A probabilistic model reasons about physical quantities as random variables that can be estimated from measured data. The Gaussian process is a respected member of this family, being a flexible non-parametric method that has proven strong capabilities in modelling a wide range of nonlinear functions. This thesis focuses on advanced Gaussian process techniques; the contribution consist of practical methodologies primarily intended for inverse tomographic applications.

In our most theoretical formulation, we propose a constructive procedure for building a customised covariance function given any set of linear constraints. These are explicitly incorporated in the prior distribution and thereby guaranteed to be fulfilled by the prediction.

One such construction is employed for strain field reconstruction, to which end we successfully introduce the Gaussian process framework. A particularly well-suited spectral based approximation method is used to obtain a significant reduction of the computational load. The formulation has seen several subsequent extensions, represented in this thesis by a generalisation that includes boundary information and uses variational inference to overcome the challenge provided by a nonlinear measurement model.

We also consider X-ray computed tomography, a field of high importance primarily due to its central role in medical treatments. We use the Gaussian process to provide an alternative interpretation of traditional algorithms and demonstrate promising experimental results. Moreover, we turn our focus to deep kernel learning, a special construction in which the expressiveness of a standard covariance function is increased through a neural network input transformation. We develop a method that makes this approach computationally feasible for integral measurements, and the results indicate a high potential for computed tomography problems.