Stochastic modeling of multidimensionalparticle characteristics with parametric

copulas for the analysis of microstructuraleffects during separation processes of

particle systems

SPP 2045 – Zentralprojekt Z2

Contact Orkun Furat orkun.furat@uni-ulm.deVolker Schmidt volker.schmidt@uni-ulm.de

Objective

The goal of the SPP 2045 is to develop new separation processes for systems of particleswith sizes below 10 µm. For that purpose the central project, i.e. Zentralprojekt Z2 willanalyze tomographic image data before and after separation processes to statisticallysummarize distributions of multidimensional particle shape and size characteristics withparametric copulas. This reduces the complexity of large data sets to a few model pa-rameters which in return enables the analysis of microstructural effects on the quality ofseparation processes. Such an analysis can be supported with stochastic geometry modelswhich can generate, based on observed data, a large variety of virtual – but realistic –particle systems. For example, it is possible, through extensive simulations of such virtual3D particle systems, to derive stereological formulas which allow the estimation of sizeand shape distributions of a 3D particle system when only planar information is available.The working packages (AP) are structured as follows:

AP1 Segmentation and parametric representation of 3D particles

AP2 Stochastic modeling of particle size and shape

AP3 Determination of stereological formulas

AP4 Stochastic modeling of local particle density

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AP5 Stochastic modeling of grain boundaries

AP6 Copula models for multidimensional particle shape and size characteristics

AP7 Copula models of geometric and physical particle properties

AP8 Analysis of structure-property relationships

AP9 Experimental validation of forecast models

Particle system

Since the developed methods are scale-invariant they can be applied on tomographic 3Dimage data of a wide variety of particle systems. There is also a wide range of stochasticgeometry models for the generation of virtual particles. By utilizing the spherical har-monics representation of particle surfaces stochastic geometry models can be defined thatcan generate star-shaped (a generalization of convexity) particles, see Figure 1a. For thegeneration of faceted particles, see Figure 1b which display sharp edges typical cells ofrandom mosaics (tessellations) are viable stochastic geometry models. For the preliminarywork 3D images of systems of magnetite and quartz particles were analyzed.

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Figure 1: Possible realizations of two different stochastic geometry models. a) Particlegenerated through a stochastic model based on SHE. b) Particle generated as a typicalcell from a mosaic model.

Methods

For the analysis of a system of particles from tomographic image data a particle-wise seg-mentation, usually with a marker-based watershed algorithm, is necessary, such that it ispossible to extract each particle for further analysis. This approach allows the computa-tion of a multidimensional vector of characteristic quantities that summarize the size andshape of a particle. By doing so for the entirety of the particle system one receives a largesample of such vectors which in return makes it possible to determine a multidimensionalprobability distribution of characteristics for the system of particles.

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Characterization

Segmentation and parametric representation of particles The marker-based wa-tershed algorithm on distance maps of binarized images or gradient images produces aparticle-wise voxel-based segmentation. The complexity of the voxel-based representa-tion of a particle can be reduced by approximating the particle surface with so calledspherical-harmonics-expansions (SHE). In this representation just a few parameters suf-fice to describe a particle and by determining the multidimensional distribution of theSHE parameters for a system of particles this representation can be the basis for a stochas-tic geometry model that can generate virtual particles. A system of particles in SHE rep-resentation can be seen in Figure 2, where the particles were obtained by segmentationof tomographic image data.

Figure 2: Segmented system of particles in SHE representation.

Size-shape characteristics and copulas Due to the segmentation it is possible todetermine for each particle a vector of characteristics like volume, surface area, sphericity,convexity and elongation which describe its size and shape. This allows the determinationof the multidimensional distribution of such vectors of characteristics. Parametric copuladistributions are a viable option for modeling such distributions, since they can incor-porate correlations and dependencies between characteristics. In Figure 3 a copula withfive parameters was used two describe the multidimensional distribution of such vectorsof characteristics.

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(a) Empirical distribution.

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(b) Parametric copula distribution.

Figure 3: Joint two-dimensional distributions of particle sphericity and elongation. a)Empirical distribution visualized with kernel density estimators. b) Distribution of afitted copula which is described by only five parameters.

Analysis of microstructural effects on separation processes By comparing cop-ula distributions of size and shape characteristics of particle systems before and afterseparation processes it is possible to quantify the separation-efficiency. Furthermore, thisapproach makes it possible to systematically analyze the effect the microstructure of theparticle system and the parameters of the separation process have on the success of theseparation. Thus the analysis of multidimensional copula distributions does not only allowan evaluation of separation processes developed in the SPP, but it also provides insightinto the mechanisms during separation processes.

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