EIKONAL-BASED MODELS OF RANDOM TESSELLATIONS

Bruno Figliuzzi

doi:10.5566/ias.2061

Authors

Bruno Figliuzzi Centre for Mathematical Morphology, Mines ParisTech - PSL Research University, Fontainebleau, 77300

DOI:

https://doi.org/10.5566/ias.2061

Keywords:

Eikonal equation, Fast marching algorithm, Stochastic geometry, Voronoi tessellations,

Abstract

In this article, we propose a novel, efficient method for computing a random tessellation from its vectorial representation at each voxel of a discretized domain. This method is based upon the resolution of the Eikonal equation and has a complexity in O(N log N), N being the number of voxels used to discretize the domain. By contrast, evaluating the implicit functions of the vectorial representation at each voxel location has a complexity of O(N²) in the general case. The method also enables us to consider the generation of tessellations with rough interfaces between cells by simulating the growth of the germs on a domain where the velocity varies locally. This aspect constitutes the main contribution of the article. A final contribution is the development of an algorithm for estimating the multi-scale tortuosity of the boundaries of the tessellation cells. The algorithm computes the tortuosity of the boundary at several scales by iteratively deforming the boundary until it becomes a straight line. Using this algorithm, we demonstrate that depending on the local velocity model, it is possible to control the roughness amplitude of the cells boundaries.

References

Belhadj J, Romary T, Gesret A, Noble M and Figliuzzi B (2018). New parameterizations for Bayesian seismic tomography. Inverse Probl 34(6):065007

Bortolussi V, Figliuzzi B, Willot F, Faessel M, Jeandin M (2018). Morphological modeling of cold spray coatings. Image Anal Stereol 37(2):145-58

Chiu SN, Stoyan D, Kendall WS and Mecke J (2013). Stochastic geometry and its applications. John Wiley & Sons

Figliuzzi B, Jeulin D, Faessel M, Willot F, Koishi M and Kowatari N (2016). Modelling the microstructure and the viscoelastic behaviour of carbon black filled rubber materials from 3D simulations. Tech Mechanik 32(1-2):22-46

Gasnier J-B, Willot F, Trumel H, Figliuzzi B, Jeulin D and Biessy M (2015). A Fourier-based numerical homogenization tool for an explosive material. Matériaux & Techniquess 103(3):308

Jeulin D (1991). Modèles morphologiques de structures aléatoires et de changement d'échelle. Doctoral dissertation, Caen

Jeulin D (2012). Morphology and effective properties of multi-scale random sets: A review. Comptes Rendus Mécanique 340(4-5):219-29

Jeulin D (2015). Boolean random functions. In: Stochastic Geometry, Spatial Statistics and Random Fields. Springer: 143-69

Jeulin D (2017). Morphological models. In: Encyclopedia of Continuum Mechanics. Springer, 1-12

Johnson W and Mehl R (1939). Reaction kinetics in processes of nucleation and growth. Trans. Aime 135(8):396-415

Koishi M, Kowatari N, Figliuzzi B, Faessel M, Willot F and Jeulin D (2017). Computational material design of filled rubbers using multi-objective design exploration. 10th European Conference on Constitutive Models for Rubbers (ECCMR)

Lee T and Cowan R (1994). A stochastic tessellation of digital space. In: Mathematical morphology and its applications to image processing. Springer, 217-24

Lee T (1999). A stochastic tessellation for modelling and simulating colour aluminium grain images. J Microsc-Oxford 193(2):109-26

Malladi R, Sethian J and Vemuri B (1995). Shape modeling with front propagation: A level set approach. IEEE trans pattern anal 17(2):158-75

Möller J (1989). Random tessellations in $mathbb{R}^d$. Adv Appl Probab 21(1):37-73

Möller J (1992). Random Johnson-Mehl tessellations. Adv Appl Probab 24(4):814-44

Möller J (1994). Lectures on random Voronoi tessellations. Springer

John Wiley & Sons

Moreaud M, Jeulin D, Morard V and Revel R (2012). TEM image analysis and modelling: application to boehmite nanoparticles. J Microsc-Oxford 245(2):186-99

Ohser J and Schladitz K (2009). 3D images of materials structures: processing and analysis. John Wiley & Sons

Parikh N and Boyd S (2014). Proximal algorithms. Founds Trends Optimization 1(3):127--239

Sethian J (1996). A fast marching level set method for monotonically advancing fronts. P Natl Acad Sci USA 93(4):1591-95

Sethian J (1999). Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. Cambridge university press

Sethian J (1999). Fast marching methods. SIAM rev 41(2):199-235

Torquato S (2013). Random heterogeneous materials: microstructure and macroscopic properties.

Springer Science & Business Media

Wang H, Pietrasanta A, Jeulin D, Willot F, Faessel M, Sorbier L and Moreaud M (2015). Modelling mesoporous alumina microstructure with 3D random models of platelets. J Microsc-Oxford 260(3):287-301