Piecewise constant surfaces, or homogeneous blob-like regions, can be properly characterized by the MLL (multi-level logistic), more specifically the homogeneous and isotropic MLL model described in Section 1.3.2. For cliques containing more than one site, the clique potentials are defined as (The definition in (2.7) is effectively the same as (1.52) for the restoration purpose.)
where 
is a constant dependent on c. That ``all sites in
c have the same label'', that is, all 
are the
same, means the entire smoothness of labels f on the clique c. Any
violation of the entire smoothness incurs a penalty of the positive
number 
. Because the more probable configurations are those
with higher 
, or lower 
, values, the MLL model
(2.7) favors smooth f.
For single site cliques, the clique potentials depend on the label assigned to the site
where 
is the penalty against that 
is labeled l; see
(1.53). The higher 
is, the less pixels will
be assigned the value l. This has an effect of controlling the
percentage of the sites labeled l.
A special case of (2.7) is such that 
is nonzero only
for the pair-site cliques and zero for all the other types. In this
case, 
for 
, and for 
it is
where 
is the Kronecker delta function and 
is
the penalty against non-equal labels on two-site cliques. The prior
energy is the sum of all the clique potentials, that is,
where ``
'' is equivalent to
``
''. This simple form has been used for the
restoration of piecewise constant images by
[Geman and Geman 1984 ; Elliott et al. 1984 ; Derin and Cole 1986 ; Derin and Elliott 1987 ; Leclerc 1989 ; Li 1990a].
