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

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

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.
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Abstract

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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 

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