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# Workshop - Spatial Statistics and Image Analysis in Biology

Thursday, May 26, 2016 - 11:35 to 12:00
Jakub Stanek
Charles Univ. Prague, Czech Republic
Stochastic 3D modelling of three-phase microstructures with fully connected phases and related characteristics
Abstract:

This contribution concerns modelling of a given physical material using methods of stochastic geometry and spatial statistics. Particularly, a parametric stochastic model for microstructures consisting of three phases (pores, Nickle and YSZ) is presented. Denote the phases as $\Xi_i,\ i = 1, 2, 3$. Each of the phases is considered to be completely connected. In order to obtain the connectivity in the model, connected graphs $G_i = (X_i , E_i),\ i = 1, 2, 3,$ are used as backbones of the phases. In the basic model, there is a strong correlation between expected volume fraction of phases $\Xi_i$ and expectation of weighted total edge length of the graphs $G_i$ for all $i = 1, 2, 3,$ so the model parameters and the volume fractions of the phases are quantitatively related. Usage of this relation for estimating the model parameters will be shown. Further, the model can be generalised in two different ways. The first generalisation allows to include anisotropy effects in the model. The second one leads to a model which is flexible enough to fit experimental image data with respect to volume fractions, tortuosities and constrictivities. In this generalised model, the Nelder-Mead method was used for estimation of all model parameters from experimental image data. These two generalisations are presented, too. Finally, two geometrical characteristics describing the transport properties through given phase are defined. The first one is mean geodesic tortuosity and it describes the length of transport paths with respect to the material thickness. The second one, called constrictivity, is a quantitative measure for bottleneck effect in a microstructure. Their mathematical definitions are introduced and the methods of their estimation are described.
Joint work with Matthias Neumann, Lorenz Holzer, Viktor Benes and Volker Schmidt

slides