Germ-Grain Model

In continuum percolation theory, the so-called germ-grain model is an obvious generalization of both the Boolean and Boolean-Poisson models which is driven by an arbitrary stationary point process  and which assigns to the points  arbitrary compact sets  in  rather than the standard closed balls.

In this scenario, the points  are known as the germs while the sets  are known as grains. It is not uncommon to consider the union of all grains in a germ-grain model, a collection which is sometimes referred to as the grain cover (Kuronen and Leskelä 2012). The grain cover is sometimes referred to as the basis of the model in question.

In older literature, it is not uncommon to define a germ-grain model by assuming any of several other conditions, e.g., that the grains fail to be independent, by allowing the subsets  to be random closed sets (not necessarily compact) in , and by considering as a basis all unions of the form

(1)

where  is a non-Poisson marked point process (MPP) on  with mark space ; this is in contrast to, e.g., the Boolean-Poisson model which has a basis of the form

(2)

where  is a Poisson point process in  and where  is the compact set in  assigned to each  (Hanisch 1981). Though dated, this notion of a germ-grain model can be further elaborated upon using more rigorous mathematical formality by explicitly writing the MPP  as

(3)

assuming it satisfies either

(4)

or

(5)

and by defining as a model basis the almost surely (with respect to ) closed set

(6)

The resulting model is said to be driven by  or to have been derived from  (Heinrich 1992).

REFERENCES

Hanisch, K. H. "On Classes of Random Sets and Point Process Models." Serdica Bulgariacae Mathematicae Publicationes 7, 160-166, 1981.

Heinrich, L. "On Existence and Mixing Problems of Germ-Grain Models." Statistics 23, 271-286, 1992.

Kuronen, M. and Leskelä, L. "Hard-Core Thinnings of Germ-Grain Models with Power-Law Grain Sizes." 5 Apr 2012. http://arxiv.org/abs/1204.1208.

Meester, R. and Roy, R. Continuum Percolation. New York: Cambridge University Press, 2008.