Abstract
Open cell foams are a class of modern materials which is interesting for a wide variety of applications and which is not accessible to classical materialography based on 2d images. 3d imaging by micro computed tomography is a practicable alternative. Analysis of the resulting volume images is either based on a simple binarisation of the image or on so-called cell reconstruction by image processing. The first approach allows to estimate mean characteristics like the mean cell volume using the typical cell of a random spatial tessellation as model for the cell shape. The cell reconstruction allows estimation of empirical distributions of cell characteristics. This paper summarises the theoretical background for the first method, in particular estimation of the intrinsic volumes and their densities from discretized data and models for random spatial tessellations. The accuracy of the estimation method is assessed using the dilated edge systems of simulated random spatial tessellations.
Article PDF
Similar content being viewed by others
References
Aurenhammer F (1987) A criterion for the affine equivalence of cell complexes in \({\mathbb R}^d\) and convex polyhedra in \({\mathbb R}^{d+1}\). Discrete Comput Geom 2:49–64
Bezrukov A, Bargieł M, Stoyan D (2002) Statistical analysis of simulated random packings of spheres. Part Part Syst Charact 19:111–118
Fan Z, Wu Y, Zhao X, Lu Y (2004) Simulation of polycrystalline structure with Voronoi diagram in Laguerre geometry based on random closed packing of spheres. Comput Mater Sci 29:301–308
Fraunhofer ITWM, Department of Image Processing (2005) MAVI—modular algorithms for volume images. http://www.itwm.fhg.de/mab/projects/MAVI/
Godehardt M, Schladitz K (2006) Geometric characterisation of light weight composites using computer tomographic images. In: Proceedings of the 9th European NDT conference. Berlin
Lautensack C (2007) Random Laguerre Tessellations. PhD thesis, Universität Karlsruhe, Verlag Lautensack, Weiler bei Bingen
Lautensack C (2008) Fitting three-dimensional Laguerre tessellations to foam structures. J Appl Stat 35(9):985–995
Lautensack C, Giertzsch M, Godehardt M, Schladitz K (2008) Modelling a ceramic foam using locally adaptable morphology. J Microsc 230(3):396–404
Lautensack C, Sych T (2006) 3d image analysis of open foams using random tessellations. Image Anal Stereol 25:87–93
Lautensack C, Sych T (2008) A random Weaire–Phelan foam. In: 8th international conference on stereology and image analysis in materials science STERMAT 2008. Zakopane, Poland
Lautensack C, Zuyev S (2008) Random Laguerre tessellations. Adv Appl Probab 40:630–650
Lorz U, Hahn U (1993) Geometric characteristics of spatial Voronoi tessellations and planar sections. Tech. Rep. 93-05, Fachbereich Mathematik, TU Bergakademie Freiberg. http://www.math.uni-augsburg.de/stochastik/hahn/papers/Lorz_Hahn_1993.pdf
Mecke J (1980) Palm methods for stationary random mosaics. In: Combinatorial principles in stochastic geometry. Work Collect., Erevan, pp 124–132
Ohser J, Mücklich F (2000) Statistical analysis of microstructures in materials science. John Wiley & Sons, Chichester, New York
Ohser J, Schladitz K (2009) 3D images of materials and structures—processing and analysis. Wiley-VCH, Weinheim
Ohser J, Nagel W, Schladitz K (2009) Miles formulae for Boolean models observed on lattices. Image Anal Stereol 28(2):77–92
Redenbach C (2009) Microstructure models for cellular materials. Comput Mater Sci 44:1397–1407
Schneider R (1993) Convex bodies. The Brunn–Minkowski theory. Cambridge University Press, Cambridge
Schneider R, Weil W (2008) Stochastic and integral geometry. Springer, Berlin
Serra J (1982) Image analysis and mathematical morphology, vol 1. Academic Press, London
Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd edn. Wiley, Chichester
Sych T (2004) Estimation of geometric characteristics of foam structures. Master’s thesis, Universität Kaiserslautern/Fraunhofer ITWM, Kaiserslautern
Weaire D (ed) (1996) The Kelvin problem: foam structures of minimal surface area. Taylor & Francis, London
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially funded by the project “Virtual material design” within project 01 SF 0708 (Fraunhofer–Carnot Cooperation) of the German Federal Ministry of Education and Research.
Rights and permissions
About this article
Cite this article
Schladitz, K., Redenbach, C., Sych, T. et al. Model Based Estimation of Geometric Characteristics of Open Foams. Methodol Comput Appl Probab 14, 1011–1032 (2012). https://doi.org/10.1007/s11009-010-9208-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-010-9208-5
Keywords
- Image analysis
- 3D images
- Porous media
- Solid foams
- Intrinsic volumes
- Spatial tessellation
- Voronoi tessellation
- Laguerre tessellation