2.2.1
The underlying surface from which f is sampled is a graph surface
defined on a continuous domain where 
is the height
of the surface at 
. The first factor affecting the specification
of the MRF prior distributions is whether 
takes a continuous or
discrete value. We are interested in 
which is either
piecewise continuous or piecewise constant, i.e.
being piecewise because
of involved discontinuities. In the image representation, 
is sampled at an image lattice, giving sampled surface heights
. In the subsequent presentation, the double subscript
(i,j) is replaced by the single subscript i and the MRF samples is
denoted as 
unless there is a need for the
elaboration.
The set of the sampled heights 
is assumed to be
a realization of an MRF, that is 
at a particular location i
depends on those in the neighborhood. According to MRF-Gibbs
equivalence, specifying the prior distribution of an MRF amounts to
specifying the clique potential functions 
in the corresponding
Gibbs prior distribution (1.24). The assumption of
being continuous, piecewise continuous or piecewise
constant leads to different Gibbs distributions for the MRF.