Fitting Laguerre tessellation approximations to tomographic image data

The analysis of polycrystalline materials benefits greatly from accurate quantitative descriptions of their grain structures. Laguerre tessellations approximate such grain structures very well. However, it is a quite challenging problem to fit a Laguerre tessellation to tomographic data, as a high-dimensional optimization problem with many local minima must be solved. In this paper, we formulate a version of this optimization problem that can be solved quickly using the cross-entropy method, a robust stochastic optimization technique that can avoid becoming trapped in local minima. We demonstrate the effectiveness of our approach by applying it to both artificially generated and experimentally produced tomographic data.

[1]  Somnath Ghosh,et al.  A framework for automated analysis and simulation of 3D polycrystalline microstructures. Part 2: Synthetic structure generation , 2008 .

[2]  Cheng Li,et al.  Inverting Dirichlet Tessellations , 2003, Comput. J..

[3]  Thomas M. Liebling,et al.  The Laguerre model of grain growth in two dimensions I. Cellular structures viewed as dynamical Laguerre tessellations , 1996 .

[4]  Dirk P. Kroese,et al.  The cross-entropy method for estimation , 2013 .

[5]  Hendrik Schmidt,et al.  Fitting of random tessellation models to keratin filament networks. , 2006, Journal of theoretical biology.

[6]  André Liebscher,et al.  Laguerre approximation of random foams , 2015 .

[7]  Franz Aurenhammer,et al.  A criterion for the affine equivalence of cell complexes inRd and convex polyhedra inRd+1 , 1987, Discret. Comput. Geom..

[8]  Hiroshi Imai,et al.  Voronoi Diagram in the Laguerre Geometry and its Applications , 1985, SIAM J. Comput..

[9]  P. Gritzmann,et al.  Generalized balanced power diagrams for 3D representations of polycrystals , 2014, 1411.4535.

[10]  Volker Schmidt,et al.  Parametric Representation of 3D Grain Ensembles in Polycrystalline Microstructures , 2014 .

[11]  Dirk P. Kroese,et al.  The Cross-Entropy Method for Continuous Multi-Extremal Optimization , 2006 .

[12]  Volker Schmidt,et al.  Stochastic modeling and predictive simulations for the microstructure of organic semiconductor films processed with different spin coating velocities , 2015 .

[13]  A. Mocellin,et al.  The Laguerre model of grain growth in two dimensions II. Examples of coarsening simulations , 1996 .

[14]  D. Stoyan,et al.  Statistical Analysis and Modelling of Spatial Point Patterns , 2008 .

[15]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[16]  T. Sych,et al.  3D Characterization, Modeling and Effective Thermal Conductivity of Open Aluminium Foams , 2008 .

[17]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[19]  Luc Salvo,et al.  In situ investigation by X-ray tomography of the overall and local microstructural changes occurring during partial remelting of an Al–15.8 wt.% Cu alloy , 2007 .

[20]  D. Rowenhorst,et al.  Particle coarsening in high volume fraction solid-liquid mixtures , 2006 .

[21]  Dirk P. Kroese,et al.  Chapter 3 – The Cross-Entropy Method for Optimization , 2013 .

[22]  Tetyana Sych,et al.  3D IMAGE ANALYSIS OF OPEN FOAMS USING RANDOM TESSELLATIONS , 2011 .

[23]  Ethan D. Bolker,et al.  Recognizing Dirichlet tessellations , 1985 .

[24]  Dominique Jeulin,et al.  3D RECONSTRUCTION OF A MULTISCALE MICROSTRUCTURE BY ANISOTROPIC TESSELLATION MODELS , 2014 .

[25]  Atsuo Suzuki,et al.  APPROXIMATION OF A TESSELLATION OF THE PLANE BY A VORONOI DIAGRAM , 1986 .

[26]  Franziska Hoffmann,et al.  Spatial Tessellations Concepts And Applications Of Voronoi Diagrams , 2016 .

[27]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[28]  Dirk P. Kroese,et al.  The Cross Entropy Method: A Unified Approach To Combinatorial Optimization, Monte-carlo Simulation (Information Science and Statistics) , 2004 .

[29]  Claudia Redenbach,et al.  Random Tessellations and their Application to the Modelling of Cellular Materials , 2015 .

[30]  C. Lautensack,et al.  Fitting three-dimensional Laguerre tessellations to foam structures , 2008 .

[31]  Henning Friis Poulsen,et al.  On the Use of Laguerre Tessellations for Representations of 3D Grain Structures , 2011 .

[32]  Oliver Brunke,et al.  Quantitative methods for the analysis of synchrotron-µ CT datasets of metallic foams , 2005 .

[33]  P. D. Groen.  An Introduction to Total Least Squares , 1998, math/9805076.

[34]  Volker Schmidt,et al.  Stochastic 3D Models for the Micro-structure of Advanced Functional Materials , 2015 .

[35]  Kokichi Sugihara.  Three-dimensional convex hull as a fruitful source of diagrams , 2000, Theor. Comput. Sci..

[36]  Sergei Zuyev,et al.  Random Laguerre tessellations , 2008, Advances in Applied Probability.

[37]  P. Cloetens,et al.  New opportunities for 3D materials science of polycrystalline materials at the micrometre lengthscale by combined use of X-ray diffraction and X-ray imaging , 2009 .

[38]  Volker Schmidt,et al.  Inverting Laguerre Tessellations , 2014, Comput. J..

[39]  Uwe Wolfram,et al.  Particle tracking during Ostwald ripening using time-resolved laboratory X-ray microtomography , 2014 .

[40]  Franz Aurenhammer,et al.  Recognising Polytopical Cell Complexes and Constructing Projection Polyhedra , 1987, J. Symb. Comput..

[41]  C. D. Gelatt,et al.  Optimization by Simulated Annealing , 1983, Science.

[42]  Jos B. T. M. Roerdink,et al.  The Watershed Transform: Definitions, Algorithms and Parallelization Strategies , 2000, Fundam. Informaticae.

[43]  Dirk P. Kroese,et al.  Handbook of Monte Carlo Methods , 2011 .

[44]  David Hartvigsen,et al.  Recognizing Voronoi Diagrams with Linear Programming , 1992, INFORMS J. Comput..

[45]  D. G. Evans,et al.  Detecting Voronoi (area-of-influence) polygons , 1987 .

[46]  Thomas M. Liebling,et al.  Simulating and Modelling Grain Growth as the Motion of a Weighted Voronoi Diagram , 1992 .

[47]  Xuanhe Zhao,et al.  Simulation of polycrystalline structure with Voronoi diagram in Laguerre geometry based on random closed packing of spheres , 2004 .

[48]  Somnath Ghosh,et al.  A framework for automated analysis and simulation of 3D polycrystalline microstructures. , 2008 .

[49]  D Jeulin,et al.  Stereological reconstruction of polycrystalline materials , 2015, Journal of microscopy.

[50]  C. Rau,et al.  Quantitative characterization of the three-dimensional microstructure of polycrystalline Al-Sn using X-ray microtomography , 2004 .

[51]  Volker Schmidt,et al.  Stochastic 3D modeling of Ostwald ripening at ultra-high volume fractions of the coarsening phase , 2015 .

[52]  Lih-Yuan Deng,et al.  The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation, and Machine Learning , 2006, Technometrics.

[53]  Xinjian Xue,et al.  The Laguerre model for grain growth in three dimensions , 1997 .

[54]  P. D. de Groen.  An Introduction to Total Least Squares , 1996 .