  X-ray tomography has applications in various industrial fields such as
sawmill industry, oil and gas industry, chemical engineering, and geotechnical
engineering. In this article, we study Bayesian methods for the X-ray
tomography reconstruction. In Bayesian methods, the inverse problem of
tomographic reconstruction is solved with help of a statistical prior
distribution which encodes the possible internal structures by assigning
probabilities for smoothness and edge distribution of the object. We compare
Gaussian random field priors, that favour smoothness, to non-Gaussian total
variation, Besov, and Cauchy priors which promote sharp edges and high-contrast
and low-contrast areas in the object. We also present computational schemes for
solving the resulting high-dimensional Bayesian inverse problem with
100,000-1,000,000 unknowns. In particular, we study the applicability of a
no-U-turn variant of Hamiltonian Monte Carlo methods and of a more classical
adaptive Metropolis-within-Gibbs algorithm for this purpose. These methods also
enable full uncertainty quantification of the reconstructions. For faster
computations, we use maximum a posteriori estimates with limited-memory BFGS
optimisation algorithm. As the first industrial application, we consider
sawmill industry X-ray log tomography. The logs have knots, rotten parts, and
even possibly metallic pieces, making them good examples for non-Gaussian
priors. Secondly, we study drill-core rock sample tomography, an example from
oil and gas industry. We show that Cauchy priors produce smaller number of
artefacts than other choices, especially with sparse high-noise measurements,
and choosing Hamiltonian Monte Carlo enables systematic uncertainty
quantification.
