Abstract
The Boolean model is the basic random set model for many applications. Its main advantage lies in the fact that it is determined by a single measure-valued parameter, the intensity measure. Whereas classically Boolean models were studied which are stationary and isotropic, some of the methods and results have been extended to the non-isotropic situation. More recent investigations consider inhomogeneous Boolean models, i.e. random sets without any invariance property. Density formulae for inhomogeneous Boolean models make use of local variants of the classical quermassintegrals, the surface area measures and curvature measures. Iterations of translative integral formulae for curvature measures lead to further measures of mixed type.
In this survey, we describe some of these local and mixed-type functionals from integral geometry and show how they can be used to extend density formulae for Boolean models from the stationary and isotropic case to the non-isotropic situation, and finally to inhomogeneous Boolean models.
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Weil, W. (2000). Mixed Measures and Inhomogeneous Boolean Models. In: Mecke, K.R., Stoyan, D. (eds) Statistical Physics and Spatial Statistics. Lecture Notes in Physics, vol 554. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45043-2_5
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DOI: https://doi.org/10.1007/3-540-45043-2_5
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