Enhancing Industrial X-ray Tomography by Data-Centric Statistical Methods

Abstract

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 lowcontrast 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. Impact StatementIndustrial X-ray tomography reconstruction accuracy depends on various factors, like the equipment, measurement geometry and constraints of the target. For example dynamical systems are harder targets than static ones. The harder and noisier the setting becomes, the more emphasis goes on mathematical modelling of the targets. Bayesian statistical inversion is a common choice for difficult measurement settings, and its limitations mainly come from the choice of the a priori models. Gaussian models are widely studied, but they provide smooth reconstructions. Total variation priors are not invariant under mesh changes, so doing systematic uncertainty quantification, like data-centric sensor optimisation, cannot be done with them. Besov and Cauchy priors however provide systematic non-Gaussian random field models, which can be used for contrast-boosting tomography. The drawback is higher computational cost. Hence, the techniques developed here are useful for non-time-critical applications with difficult measurement settings. In these cases, the methods developed may provide significantly better reconstructions than the traditional methods, like filtered back-projection.

Non-Stationary Multi-layered Gaussian Priors for Bayesian Inversion

“…The total number of parameter for each layer is 1985, which makes the total number of parameters for all layers 3970. The total number of parameters in this example is greatly reduced compared to [53] where each pixel in the target image count as a parameter, that is, for our example it translates to 261121 parameters.…”