The first step is the same as that for piecewise constant restoration. The second step is to define the prior clique potential functions. Here, the single-site clique potentials are set to zero; only the pair-site cliques are considered with potential function (2.11) rewritten below
To impose the piecewise smoothness prior, the g function is required to satisfy condition (2.16).
The third step is to derive the likelihood energy. When the noise is
i.i.d. additive Gaussian, 
, the likelihood energy
is given (2.5) with continuously valued
.
In the fourth step, the prior energy 
and the likelihood energy
are added to obtain the posterior energy
or equivalently
where 
. When no noise is present, 
,
only the first term is effective and the MAP solution is exactly the
same as the data, 
, which is also also the maximum likelihood
solution. As 
becomes larger, the solution is more influenced
by the second, i.e.
the smoothness, term. Fig.2.1
shows the effect of 
on 
with the quadratic g function.
It can be seen that the larger is 
, the smoother is 
.
Figure 2.1: Solution 
obtained with 
(left), 20 (middle)
and 200 (right). The original signals are in thinner solid lines,
noisy data in dotted lines, and the minimal solutions in thicker solid
lines.
In the 1D case where 
can be an ordered set such
that the the nearest neighbor set for each interior site i is
, (2.22) can be written
as
(the terms involving the two boundary points are omitted.) When
is the sample of a continuous function 
,
, the above is an approximation to the regularization
energy
( cf.
Section 1.5.3.) When 
is as in
(2.15), the above is the energy for the standard
quadratic regularization
[Tikhonov and Arsenin 1977 ; Poggio et al. 1985]. When it is defined as
(2.17), the energy is called weak string
and its two-dimensional counterpart is called weak membrane
[Blake and Zisserman 1987].
Regularization of ill-posed problem will be discussed further in surface
reconstruction.
In the 2D case where 
is a lattice,
each site 
(except at the boundaries) have four nearest neighbors
. The corresponding
posterior energy is
This is a direct extension to (2.24).
