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Modern Stochastics: Theory and Applications


Description of the symmetric convex random closed sets as zonotopes from their Feret diameters
Pub. online: 3 January 2017      Type: Research Article     
Received
24 October 2016
Revised
7 December 2016
Accepted
14 December 2016
Published
3 January 2017

Abstract

In this paper, the 2-D random closed sets (RACS) are studied by means of the Feret diameter, also known as the caliper diameter. More specifically, it is shown that a 2-D symmetric convex RACS can be approximated as precisely as we want by some random zonotopes (polytopes formed by the Minkowski sum of line segments) in terms of the Hausdorff distance. Such an approximation is fully defined from the Feret diameter of the 2-D convex RACS. Particularly, the moments of the random vector representing the face lengths of the zonotope approximation are related to the moments of the Feret diameter random process of the RACS.

References

[1] 
Ahmad, O.S., Debayle, J., Pinoli, J.-C.: A geometric-based method for recognizing overlapping polygonal-shaped and semi-transparent particles in gray tone images. Pattern Recognit. Lett. 32(15), 2068–2079 (2011)
[2] 
Ballani, F.: The surface pair correlation function for stationary boolean models. Adv. Appl. Probab. 39(1), 1–15 (2007). doi:10.1239/aap/1175266466.
[3] 
Bárány, I., Reitzner, M.: On the variance of random polytopes. Adv. Math. 225(4), 1986–2001 (2010). doi:10.1016/j.aim.2010.04.012.
[4] 
Bronstein, E.M.: Approximation of convex sets by polytopes. J. Math. Sci. 153(6), 727–762 (2008). doi:10.1007/s10958-008-9144-x.
[5] 
Buwa, V.V., Ranade, V.V.: Dynamics of gas–liquid flow in a rectangular bubble column: Experiments and single/multi-group CFD simulations. Chem. Eng. Sci. 57(22), 4715–4736 (2002)
[6] 
Calderon De Anda, J., Wang, X.Z., Roberts, K.J.: Multi-scale segmentation image analysis for the in-process monitoring of particle shape with batch crystallisers. Chem. Eng. Sci. 60(4), 1053–1065 (2005)
[7] 
Campi, S., Haas, D., Weil, W.: Approximation of zonoids by zonotopes in fixed directions. Discrete Comput. Geom. 11(4), 419–431 (1994). doi:10.1007/BF02574016.
[8] 
Chiu, S.N., Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications. John Wiley & Sons (2013). doi:10.1002/9781118658222.
[9] 
Cox, D.R., Miller, H.D.: The Theory of Stochastic Processes, vol. 134. CRC Press (1977)
[10] 
Dafnis, N., Giannopoulos, A., Tsolomitis, A.: Asymptotic shape of a random polytope in a convex body. J. Funct. Anal. 257(9), 2820–2839 (2009). doi:10.1016/j.jfa.2009.06.027.
[11] 
Eppstein, D.: Zonohedra and zonotopes. Math. Educ. Res. 5, 15–21 (1996)
[12] 
Galerne, B.: Modèles d’image aléatoires et synthèse de texture. PhD thesis, Ecole normale supérieure de Cachan – ENS Cachan (2010)
[13] 
Gardner, R.J.: Geometric Tomography, vol. 6. Cambridge University Press, Cambridge (1995).
[14] 
Glasauer, S., Schneider, R.: Asymptotic approximation of smooth convex bodies by polytopes. In: Forum Mathematicum, vol. 8, pp. 363–378 (1996). doi:10.1515/form.1996.8.363.
[15] 
Gray, R.M.: Toeplitz and Circulant Matrices: II. Information Systems Laboratory, Stanford Electronics Laboratories, Stanford University (1977)
[16] 
Heinrich, L., Molchanov, I.S.: Central limit theorem for a class of random measures associated with germ–grain models. Advances in Applied Probability, 283–314 (1999). doi:10.1239/aap/1029955136.
[17] 
Hoffmann, L.M.: On weak stationarity and weak isotropy of processes of convex bodies and cylinders. Advances in Applied Probability, 864–882 (2007).
[18] 
McClure, D.E., Vitale, R.A.: Polygonal approximation of plane convex bodies. J. Math. Anal. Appl. 51(2), 326–358 (1975).
[19] 
Michielsen, K., De Raedt, H.: Integral-geometry morphological image analysis. Phys. Rep. 347(6), 461–538 (2001). doi:10.1016/S0370-1573(00)00106-X.
[20] 
Miles, R.E.: Random polygons determined by random lines in a plane. Proc. Natl. Acad. Sci. 52(4), 901–907 (1964).
[21] 
Molchanov, I.S.: Statistics of the boolean model: From the estimation of means to the estimation of distributions. Adv. Appl. Probab. 27(1), 63–86 (1995). doi:10.2307/1428096.
[22] 
Molchanov, I.S.: Statistics of the Boolean Model for Practitioners and Mathematicians. Wiley, Chichester (1997)
[23] 
Molchanov, I.S., Stoyan, D.: Asymptotic properties of estimators for parameters of the boolean model. Advances in Applied Probability, 301–323 (1994). doi:10.2307/1427437.
[24] 
Peyrega, C.: Prediction des proprietes acoustiques de materiaux fibreux heterogenes a partir de leur microstructure 3d. PhD thesis, École Nationale Supérieure des Mines de Paris (2010)
[25] 
Rahmani, S., Pinoli, J.-C., Debayle, J.: Characterization and estimation of the variations of a random convex set by its mean n-variogram: Application to the boolean model. In: International Conference on Geometric Science of Information, pp. 296–308, Springer (2015). doi:10.1007/978-3-319-25040-3_33.
[26] 
Rahmani, S., Pinoli, J.-C., Debayle, J.: Geometrical stochastic modeling and characterization of 2-d crystal population. In: 14th International Congress for Stereology and Image Analysis, 14th ICSIA FA06, Acta Stereologica (2015).
[27] 
Schneider, R.: Convex Bodies: The Brunn–Minkowski Theory, vol. 151. Cambridge University Press (2013). doi:10.1017/CBO9780511526282.
[28] 
Sundararajan, D.: The Discrete Fourier Transform: Theory, Algorithms and Applications. World Scientific (2001). doi:10.1142/9789812810298.
[29] 
Torquato, S.: Random Heterogeneous Materials: Microstructure and Macroscopic Properties, vol. 16. Springer (2002). doi:10.1007/978-1-4757-6355-3.
[30] 
Zafari, S., Eerola, T., Sampo, J., Kälviäinen, H., Haario, H.: Segmentation of overlapping elliptical objects in silhouette images. IEEE Trans. Image Process. 24(12), 5942–5952 (2015). doi:10.1109/TIP.2015.2492828.
[31] 
Zhang, D., Qi, L., Ma, J., Cheng, H.: Morphological control of calcium oxalate dihydrate by a double-hydrophilic block copolymer. Chem. Mater. 14(6), 2450–2457 (2002)
[32] 
Zhang, W.-H., Jiang, X., Liu, Y.-M.: A method for recognizing overlapping elliptical bubbles in bubble image. Pattern Recognit. Lett. 33(12), 1543–1548 (2012)

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Keywords
Zonotopes random closed set the Feret diameter polygonal approximation

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