14-17 May 2018
New Orleans
US/Central timezone

Microstructural characterization via Minkowski-functional-based global descriptors

15 May 2018, 18:00
New Orleans

New Orleans

Poster MS 2.06: New Trends in Image Processing: From Discrete Tomography over Machine Learning to in-situ Contact Angle Measurement Poster 2


Willi Pabst (University of Chemistry and Technology, Prague)


The densities of Minkowski functionals (volume, surface, mean curvature and total curvature) represent a complete set of independent global microstructural descriptors [1-3], as a consequence of Hadwiger’s characterization theorem [4]. Similar to correlation functions [5, 6] they offer a systematic and principally automatizable approach for the quantitative description of microstructures, but unlike the latter they form a complete set of simple parameters that are readily determined on real microstructures and, if appropriately implemented into microstructure-property relations, could provide more accurate predictions of effective properties than micromechanical bounds or model predictions based on volume fractions alone. In this contribution we present examples of the application of Minkowski-functional-based global descriptors for the quantitative description of porous ceramics. We show that, apart from the porosity (pore volume fraction) and mean chord length (based on the phase-specific surface density) also a generalized Jeffries size (based on the mean curvature integral density) can be determined from planar sections [1,7,8]. The correlation between these two independent size measures is analyzed and the average pore size thus determined is compared to the characteristic values (quantiles and mean values) extracted from pore size distributions (number- and volume-weighted) determined via microscopic image analysis, after correcting for the random section problem (Wicksell’s problem [9]) via appropriate transformation matrices [10]. Moreover, it is shown how the 3D Euler characteristic can be determined on (appropriately binarized) serial sections of spatial images (obtained by X-ray computed tomography).


  1. Pabst W., Gregorová E., Uhlířová T.: Microstructure characterization via stereological relations – a shortcut for beginners, Mater. Charact. 105 (1), 1-12 (2015).
  2. Ohser J., Mücklich F.: Statistical Analysis of Microstructures in Materials Science. John Wiley & Sons, Chichester (2000).
  3. Pabst W., Uhlířová T., Gregorová E.: Microstructure characterization of porous ceramics via Minkowski functionals, 12 pp. in: Ceramic Transactions. Wiley, Hoboken (in press).
  4. Hadwiger H.: Vorlesungen über Inhalt, Oberfäche und Isoperimetrie. Springer, Berlin 1957.
  5. Torquato S.: Random Heterogeneous Materials – Microstructure and Macroscopic Properties. Springer, New York 2002.
  6. Milton G. W.: The Theory of Composites. Cambridge University Press, Cambridge 2002.
  7. Uhlířová T., Gregorová E., Pabst W., Nečina V.: Preparation of cellular alumina ceramics via biological foaming with yeast and its microstructural characterization via stereological relations, J. Eur. Ceram. Soc. 35 (1), 187-196 (2015).
  8. Gregorová E., Pabst W., Uhlířová T., Nečina V., Veselý M., Sedlářová I.: Processing, microstructure and elastic properties of mullite-based ceramic foams prepared by direct foaming with wheat flour, J. Eur. Ceram. Soc. 36 (1), 109-120 (2016).
  9. Wicksell S. D.: The corpuscle problem – A mathematical study of a biometric problem, Biometrika 17, 84-99 (1925).
  10. Pabst W., Uhlířová E.: A generalized class of transformation matrices for the reconstruction of sphere size distributions from section circle size distributions, Ceram. Silik. 61 (2), 147-157 (2017).
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Primary authors

Willi Pabst (University of Chemistry and Technology, Prague) Ms Tereza Uhlířová (UCT Prague) Dr Eva Gregorová (University of Chemistry and Technology, Prague)

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