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
![Phi=sum_(i in N)delta_([X_i(Phi),Z_i(Phi)]),](/images/equations/Germ-GrainModel/NumberedEquation3.svg) |
(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).
