Abstract
AbstractX-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.
Highlights
X-ray tomography has applications in various industrial fields such as sawmill industry, where it can be used for detecting knots, rotten parts, and foreign objects in logs (Shustrov et al, 2019; Zolotarev et al, 2019)
X-ray microtomography can be used to measure the internal structure of substances at the micrometer level (Ou et al, 2017)
X-ray microtomography can be used to measure soil properties in laboratories, while a closely related travel-time tomography can be used to measure the structure of soil and rocks from cross-borehole measurements (Ernst et al, 2007)
Summary
X-ray tomography has applications in various industrial fields such as sawmill industry, where it can be used for detecting knots, rotten parts, and foreign objects in logs (Shustrov et al, 2019; Zolotarev et al, 2019). We study so-called Bayesian methods (Kaipio and Somersalo, 2005), where we introduce a statistical prior model for the possible internal structures in form of a probability distribution. This prior model encodes the information on what kind of structures are more likely and which are less likely than others. X-ray tomography as a Bayesian statistical inverse problem Reducing the number of measurements tends to add artefacts to the tomographic reconstruction This means that we need to carefully evaluate the accuracy of the reconstruction for different levels of sparsity. Where πðXÞ is the a priori density, that is, the probabilistic description of the unknown we know before any measurements are taken, πðyjXÞ is the likelihood density, and πðyÞ is a normalization constant which we can be omitted in computations
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