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Processes on manifolds form a more general concept; here, stochastic geometry is linked with the theory of random sets see [1]. The following is a characteristic result in this direction see [2]. A process is called non-singular if its Palm distribution is absolutely continuous relative to the unconditional distribution of the process.

Stochastic and Integral Geometry by Rolf Schneider (ebook)

All non-singular processes of straight lines are doubly-stochastic Poisson processes i. Poisson processes controlled by a random measure. A number of no less unexpected properties of other geometric processes which are invariant relative to groups have been discovered using the tool of combinatorial integral geometry see [3]. The following result, among others, has been obtained by the method of averaging combinatorial decompositions over the realization space of a process see [3]. The concept of a "typical" element of a given geometric process is of considerable importance in stochastic geometry [7].

The solution of such problems has been obtained for Poisson processes.

Stochastic and Integral Geometry

Similar problems arise, for example, in astrophysics. Problems of so-called stereology also relate to stochastic geometry if they are applied to processes of geometric figures [7]. In stereology a multi-dimensional image has to be described through its intersections with straight lines or planes of a smaller number of dimensions.

Results have been obtained here on stereology of the first and second moment measures. Only the most typical problems have been mentioned above, since the boundaries of stochastic geometry are hard to define accurately.

Stochastic Geometry

The following areas adjoin stochastic geometry: geometric statistics [4] , the theory of random sets of fractional dimensions [5] , mathematical morphology and image analysis [6] , random shape theory [7]. The more recent development of stochastic geometry, with a special view to various applications, is described in [a3]. The influence from integral geometry and its use in certain parts of stochastic geometry can be seen in [a4] and [a2].

A Stochastic Geometry Approach to Analyzing Cellular Networks with Semi-static Clustering

Its main theme, once the foundations have been laid, is the quantitative investigation of the basic models. This comprises the introduction of suitable parameters, in the form of functional densities, relations between them, and approaches to their estimation.

Much additional information on stochastic geometry is collected in the section notes. The interplay between both disciplines is demonstrated by various fundamental results. A chapter on selected problems about geometric probabilities and an outlook to non-stationary models are included, and much additional information is given in the section notes.

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