Boolean model

Summary

The Boolean model is a random set in $\R^d$. It is the union of independent random shapes, the grains, placed at the points of a Poisson point process, the germs. It is the basic germ-grain model of stochastic geometry. Many of its mean values have closed formulas. For example, a given point is covered with probability $1 - e^{-\lambda E[V_d(K)]}$, where $\lambda$ is the intensity and $E[V_d(K)]$ the mean grain volume. It is used to model porous media, coverage of wireless networks and continuum percolation.