Poisson random balls: self-similarity and X-ray images. (English) Zbl 1118.60044
The authors 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. A microscopic model which yields a macroscopic self-similarity property is investigated. In order to obtain this model some power law behavior in the radius distribution and special intensities of Poisson random measures are introduced. A parameter that is supposed to contain tangible information on the structure, the local asymptotic self-similar (LASS) index, is explored. The action of an \(X\)-ray transform on the field is investigated. A relationship between the LASS behavior of the field and the LASS behavior of its \(X\)-ray transform is obtained.
Reviewer: Andrew Olenko (Victoria)
MSC:
| 60G60 | Random fields |
| 60D05 | Geometric probability and stochastic geometry |
| 52A22 | Random convex sets and integral geometry (aspects of convex geometry) |
| 44A12 | Radon transform |
| 60G12 | General second-order stochastic processes |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
Keywords:
random fields; random set; shot noise; Poisson point process; X-ray transform; fractional Brownian motion; self-similaritySoftware:
LASSReferences:
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