We consider a class of random point and germ-grain processes, obtained using a rather natural weighting procedure. Given a Poisson point process, on each point one places a grain, a (possibly random) compact convex set. Let S be the union of all grains. One can now construct new processes whose density is derived from an exponential of a linear combination of quermass functionals of S. If only the area functional is used, then the area-interaction point process is recovered. New point processes arise if we include the perimeter length functional, or the Euler functional (number of components minus number of holes). The main question addressed by the paper is that of when the resulting point process is well-defined: geometric arguments are used to establish conditions for the point process to be stable in the sense of Ruelle. The analysis of digital images and spatial patterns calls for tractable stochastic models of random sets and point processes. In this paper, we investigate new point process and germ-grain models which are constructed by weighting a Poisson point process (or germgrain process) using exponentials of (sums of) quermass integrals (Minkowski functionals) of a Boolean model based on the reference random process. These functionals are obtained from local geometric measurements including set volume and integrals of curvature over the boundary, and include the Euler-Poincare characteristic.In the point process case the model under investigation generalises the Widom-Rowlinson penetrable spheres model [65] the area-interaction point process [4] and the morphological model in [34,37,38].In this paper our main focus will be on the conditions under which planar quermassinteraction processes are stable in the sense of Ruelle (inequality (9) in Section 2.1 below). This is important because stability is an accessible condition for the density to be proper (to
The area-interaction process and the continuum random-cluster model are characterized in terms of certain functional forms of their respective conditional intensities. In certain cases, these two point process models can be derived from a bivariate point process model which in many respects is simpler to analyse and simulate. Using this correspondence we devise a two-component Gibbs sampler, which can be used for fast and exact simulation by extending the recent ideas of Propp and Wilson. We further introduce a Swendsen -Wang type algorithm. The relevance of the results within spatial statistics as well as statistical physics is discussed.
We propose a new bandwidth selection method for kernel estimators of spatial point process intensity functions. The method is based on an optimality criterion motivated by the Campbell formula applied to the reciprocal intensity function. The new method is fully nonparametric, does not require knowledge of higher-order moments, and is not restricted to a specific class of point process. Our approach is computationally straightforward and does not require numerical approximation of integrals.
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