In specifying prior clique potentials for continuous surfaces, only pair-site clique potentials are normally used for piecewise continuous surfaces. In the simplest case, that of flat surface, they can be defined by
where 
is a function penalizing the violation of
smoothness caused by the difference 
. For the purpose of
restoration, the function g is generally even, i.e.
and nondecreasing, i.e.
on 
. The derivative of g can be expressed as the
following form
One may choose appropriate g functions to impose different smoothness assumptions: complete smoothness or piecewise smoothness.
When the underlying 
does not contain discontinuities, i.e.
continuous everywhere, 
is usually a quadratic function
Under the quadratic g, the penalty to the violation of smoothness is
proportional to 
. The quadratic pair-site potential
function is not suitable for prior surface models in which the
underlying 
is only piecewise continuous. At
discontinuities, 
tends to be very large and the
quadratic function brings about a large smoothing force when the energy
is being minimized, giving an oversmoothed result.
To encode piecewise smoothness, g has to satisfy a necessary
condition
where 
is a constant. The above means that g should
saturate at its asymptotic upper-bound when 
to allow
discontinuities. A possible choice is the truncated quadratic
used in the line process model [Geman and Geman 1984 ; Marroquin 1985 ; Blake and Zisserman 1987] in
which
The above satisfies condition (2.16). How to properly choose the g (or h) function discontinuity-adaptive restoration is the subject to be studied in Chapter 3.
A significant difference between (2.11)
and (2.9) is due to the nature of label
set 
. For piecewise constant restoration where labels are
considered un-ordered, the difference between any two labels 
and
is symbolic, e.g.
in the set of 
. In the continuous
case where labels are ordered by the relation, e.g.
``smaller than'', the
difference takes a continuous quantitative value. We may define a
``softer'' version of (2.9) to provide a
connection between continuous and discrete restoration. For piecewise
constant restoration of a two-level image, for example, we may define
which in the limit is
.
The above instances of prior models, the clique potential is a function
of the first order difference 
which is an
approximation of 
. They have a tendency of producing surfaces of
constant or piecewise constant height, though in the continuous
restoration, continuous variation in height is allowed. They are not
very appropriate models for situations where the underlying surface is
not flat or piecewise flat.
According to the discussions in Section 1.3.3, we
may design potential function 
, 
and 
for
surfaces of constant grey level (horizontally flat), constant gradient
(planar but maybe slant) and constant curvature, respectively. It is
demonstrated, e.g.
in [Geman and Reynolds 1992], that potentials involving 
give better results for surfaces of constant gradient. Higher order
models are also necessary for reconstruction from sparse data
[Grimson 1981 ; Terzopoulos 1983a ; Poggio et al. 1985 ; Blake and Zisserman 1987].
