Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-24T04:29:01.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Poisson random balls: self-similarity and X-ray images

Part of: Integral transforms, operational calculus General convexity Stochastic analysis Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  08 September 2016

Hermine Biermé  and
Anne Estrade
Hermine Biermé*
Affiliation:
Université René Descartes
Anne Estrade*
Affiliation:
Université René Descartes
*
∗ Postal address: MAP5-UMR 8145, Université René Descartes, 45, rue des Saints-Pères, F 75270 Paris cedex 06, France.
∗ Postal address: MAP5-UMR 8145, Université René Descartes, 45, rue des Saints-Pères, F 75270 Paris cedex 06, France.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a random field obtained by counting the number of balls containing a given point when overlapping balls are thrown at random according to a Poisson random measure. We describe a microscopic process which exhibits multifractional behavior. We are particularly interested in the local asymptotic self-similarity (LASS) properties of the field, as well as in its X-ray transform. We obtain two different LASS properties when considering the asymptotics either in law or in the sense of second-order moments, and prove a relationship between the LASS behavior of the field and the LASS behavior of its X-ray transform. These results can be used to model and analyze porous media, images, or connection networks.

Keywords

Random fieldrandom setshot noiseoverlapping spheresPoisson point processX-ray transformasymptotic self-similarityfractional Brownian motion

MSC classification

Primary: 60G60: Random fields 60D05: Geometric probability and stochastic geometry 52A22: Random convex sets and integral geometry 44A12: Radon transform
Secondary: 60G12: General second-order processes 60G55: Point processes 60G57: Random measures 60G18: Self-similar processes 60H05: Stochastic integrals
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 853 - 872
Copyright
Copyright © Applied Probability Trust 2006 

References

Benassi, A., Cohen, S. and Istas, J. (2003). Local self-similarity and the Hausdorff dimension. C. R. Acad. Sci. Paris Ser. I 336, 267–272.Google Scholar
Benassi, A., Cohen, S. and Istas, J. (2004). On roughness indices for fractional fields. Bernoulli 10, 357–376.Google Scholar
Benassi, A., Jaffard, S. and Roux, D. (1997). Elliptic Gaussian random processes. Rev. Mat. Iberoamericana 13, 19–89.Google Scholar
Biermé, H. (2005). Champs aléatoires: autosimilarité, anisotropie et étude directionnelle. , University of Orléans, France. Available at http://www.math-info.univ-paris5.fr/∼bierme/recherche/Thesehb.pdf.Google Scholar
Biermé, H., Estrade, A. and Kaj, I. (2006). About scaling behavior of random balls models. In Proc. 6th Internat. Conf. Stereology, Spatial Statist. Stoch. Geometry, Prague, Union of Czech Mathematicians and Physicists, pp. 63–68.Google Scholar
Bonami, A. and Estrade, A. (2003). Anisotropic analysis of some Gaussian models. J. Fourier Anal. Appl. 9, 215–236.Google Scholar
Cioczek-Georges, R. and Mandelbrot, B. B. (1995). A class of micropulses and antipersistent fractional Brownian motion. Stoch. Process. Appl. 60, 1–18.Google Scholar
Cohen, S. and Taqqu, M. (2004). Small and large scale behavior of the Poissonized telecom process. Methodology Comput. Appl. Prob. 6, 363–379.Google Scholar
Falconer, K. J. (2003). The local structure of random processes. J. London Math. Soc. 67, 657–672.Google Scholar
Harba, R. et al. (1994). Determination of fractal scales on trabecular bone X-ray images. Fractals 2, 451–456.Google Scholar
Heinrich, L. and Schmidt, V. (1985). Normal convergence of multidimensional shot-noise and rates of this convergence. Adv. Appl. Prob. 17, 709–730.CrossRefGoogle Scholar
Kaj, I., Leskelä, L., Norros, I. and Schmidt, V. (2004). Scaling limits for random fields with long-range dependence. To appear in Ann. Prob. Google Scholar
Lacaux, C. (2004). Real harmonizable multifractional Lévy motions. Ann. Inst. H. Poincaré Prob. Statist. 40, 259–277.Google Scholar
Lacaux, C. (2005). Fields with exceptional tangent fields. J. Theoret. Prob. 18, 481–497.Google Scholar
Lévy-Véhel, J. and Peltier, R. F. (1995). Multifractional Brownian motion: definition and preliminary results. Tech. Rep. 2645, INRIA. Available at http://www.inria.fr/rrrt/rr-2645.html.Google Scholar
Ramm, A. G. and Katsevich, A. I. (1996). The Radon Transform and Local Tomography. CRC Press, Boca Raton, FL.Google Scholar
Serra, J. (1982). Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
Wicksell, S. D. (1925). The corpuscle problem: a mathematical study for a biometrical problem. Biometrika 17, 84–99.Google Scholar