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Families of Poisson processes defined on general state spaces and with the intensity measure scaled by a positive parameter are investigated. In particular, mean value relations with respect to the scale parameter are established and used to derive various Gamma-type results for certain geometrical characteristics determined by finite subprocesses. In particular, we deduce Miles' complementary theorem. Applications of the results within stochastic geometry and particularly for random tesselations are discussed.
A new class of Gibbsian models with potentials associated with the connected components or homogeneous parts of images is introduced. The relationship with Markov random fields and marked point processes is explored and spatial Markov properties are established. Further, extensions to infinite lattices are studied. Statistical inference problems including geostatistical applications and statistical image analysis are also discussed. Finally, simulation studies are presented which show that the models may be appropriate for a variety of interesting patterns.
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