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, as well as chemical, biomedical, 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 the 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 favor smoothness, to non-Gaussian total variation (TV), Besov, and Cauchy priors which promote sharp edges and high- 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. We study the applicability of a no-U-turn variant of Hamiltonian Monte Carlo (HMC) methods and of a more classical adaptive Metropolis-within-Gibbs (MwG) algorithm to enable full uncertainty quantification of the reconstructions. We use maximum a posteriori (MAP) estimates with limited-memory BFGS (Broyden–Fletcher–Goldfarb–Shanno) optimization 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. In that case, we compare the priors without uncertainty quantification. We show that Cauchy priors produce smaller number of artefacts than other choices, especially with sparse high-noise measurements, and choosing HMC enables systematic uncertainty quantification, provided that the posterior is not pathologically multimodal or heavy-tailed.

Structural Gaussian Priors for Bayesian CT reconstruction of Subsea Pipes

“…The prior's effect on the uncertainty estimates have also been seen in other studies. For instance, edge-preserving priors give rise to high uncertainty on discontinuities in CT reconstructions [3,4]. In future work we aim to further improve uncertainty estimates of the pipe and defect structure, for example to help distinguish artifacts from real features in the reconstruction.…”

“…In Bayesian CT reconstruction [2,3,4], it is important to choose an appropriate prior. Typical choices in this setting are Markov Random Field (MRF) priors given by multivariate Gaussian and Laplacian distributions.…”

Geometry Parameter Estimation for Sparse X-ray Log Imaging

“…As the third image reconstruction method, we employ the maximum a posteriori (MAP) estimate of the posterior distribution with the first-order isotropic Cauchy difference prior [24,28,29] and the following data likelihood term…”

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 [67] where each pixel in the target image count as a parameter, that is, for our example it translates to 261 121 parameters.…”