Stochastic Modeling of Multidimensional Particle Properties Using Parametric Copulas

Abstract

In this paper, prediction models are proposed which allow the mineralogical characterization of particle systems observed by X-ray micro tomography (XMT). The models are calibrated using 2D image data obtained by a combination of scanning electron microscopy and energy dispersive X-ray spectroscopy in a planar cross-section of the XMT data. To reliably distinguish between different minerals the models are based on multidimensional distributions of certain particle characteristics describing, for example, their size, shape, and texture. These multidimensional distributions are modeled using parametric Archimedean copulas which are able to describe the correlation structure of complex multidimensional distributions with only a few parameters. Furthermore, dimension reduction of the multidimensional vectors of particle characteristics is utilized to make non-parametric approaches such as the computation of distributions via kernel density estimation viable. With the help of such distributions the proposed prediction models are able to distinguish between different types of particles among the entire XMT image.

PARROT: A Pilot Study on the Open Access Provision of Particle-Discrete Tomographic Datasets

“…To begin with, particle systems can be efficiently characterized by modeling the distribution of individual particle characteristics using univariate parametric probability distributions (Johnson et al, 1994(Johnson et al, , 1995. Moreover, the particle-discrete vectors of characteristics allow for the modeling of multivariate distributions which capture the correlation structure of considered characteristics (Ditscherlein et al, 2020a;Furat et al, 2019a), see Appendix A-2 of the Supplementary Material. Besides this, the segmented tomographic image data can be used to calibrate stochastic geometry models.…”

“…More precisely, univariate distributions do not provide any information on whether and how particle characteristics are correlated with each other (for example, if larger particles are more spherically shaped, etc.). Therefore, we deploy multivariate parametric distributions for modeling the joint distribution of characteristicsthus, capturing the correlation between characteristics (Furat et al, 2019a(Furat et al, , 2021aDitscherlein et al, 2020a). Then, the considered particle system can be described by just a few model parameters.…”

Multidimensional Characterisation of Time-dependent Image Data: A Case Study for the Peripheral Nervous System in Ageing Mice

“…Therefore, we use a copula approach (Durante and Sempi, 2010;Joe, 1997) for modelling the joint distribution of axon area and myelin sphericity. Note that the copula approach has demonstrated its benefits in various other applications in order to fit parametric models to multivariate probability distributions, see, e.g., Furat et al (2019a); Neumann et al (2021);von Loeper et al (2020). A short introduction to this topic is given in the following section.…”

Evaluation of Recyclability of a WEEE Slag by Means of Integrative X-Ray Computer Tomography and SEM-Based Image Analysis

“…Different approaches exist to face this problem. These are direct three-dimensional analysis of the chemical composition using X-ray fluorescence tomography [21,22] or the combination of two-dimensional analysis of the chemical composition with XCT [19,[23][24][25].…”