A germ-grain model in continuum percolation theory is a 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 marked point process (MPP) on 
with mark space 
consisting of closed sets in
. This construction includes, 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). The general construction can be made explicit
by writing the MPP 
as
![Phi=sum_(i in N)delta_([X_i(Phi),Z_i(Phi)]),](/images/equations/Germ-GrainModel/NumberedEquation3.svg) |
(3)
|
where 
is a germ and 
is its closed set grain. As a sufficient condition,
assume that, almost surely, the translated grains
form a locally finite family. In 
, this means that every compact
set 
meets only finitely many translated grains 
. Under this assumption, the union
 |
(4)
|
is almost surely a closed set. The resulting model is said to be driven by 
or to have been derived from 
(Heinrich 1992).
