Random Sets and Integral Geometry.

Abstract

16. Random Sets and Integral Geometry. By G. Matheron. New York and London, Wiley, 1975. xxiii, 261 p. 914″. £9.10. (Wiley Series in Probability and Mathematical Statistics.)

Variance asymptotics for the area of planar cylinder processes generated by Brillinger-mixing point processes

“…The distribution of a random closed set Ξ is determined by its Choquet functional (see, e.g., [15] or [16])…”

“…. , d − 1, are basic random set models in stochastic geometry; see, for example, [15,19,21] and [16]. Meanwhile CPs also receive much attention due to numerous applications (for d = 2, 3) in modern technologies as models for materials consisting of long fibers or to model dynamic telecommunication networks.…”

Three-Dimensional Generalized Discrete Fuzzy Number and Applications in Color Mathematical Morphology

“…It is a simple and efficient nonlinear processing technology, which consists of basic dilation operator and erosion operator. Mathematical morphology is first introduced in binary images processing by Matheron and Serra [13]- [15]. Due to its strict mathematical foundation and ability to deal with the spatial relationship of image pixels, it is extended to gray-scale images processing [16].…”

Random sets to model uncertainty in remotely sensed objects

“…A random set is a generalization of a random variable, taking sets as its elements. Random set theory has become an inherent part of probability theory since Matheron [104]. This theory studies probability measures on the space of closed subsets with the hit-or-miss topology.…”

“…Random set theory was developed as part of stochastic geometry and was used to simulate and model patterns in natural or technical process [7,81,104]. For example, it was used to model ore-sintering structure by Serra [149] and incidence of heather by Diggle [31] and simulate natural textures in binary images [50].…”