2.3.1
Figure 2.4: A lattice of sites (in dots) and its dual sites (in bars) in
the 4-neighborhood system.
Rewrite the posterior energy (2.23) for the piecewise continuous restoration below
To impose the piecewise smoothness prior, the g function must have
the property (2.16). The 
is modified for the purpose of explicitly marking edges.
In addition to the existing MRF for pixel values, we introduce another
coupled MRF called line process (LP)
[Geman and Geman 1984 ; Marroquin 1985] in which each label takes an
value on 
, to signal the occurrences of edges. The two coupled
MRFs are defined on two spatially interwoven sets as
illustrated in Fig.2.4 (assuming the 4-neighborhood
system). One set is the existing lattice for the intensity field and the
other is the dual lattice for the introduced edge field. A possible
edge may exist between each pair of the neighboring pixel sites.
We use the following notations. Denote the lattice of the m pixel sites by
(corresponding to the dots in Fig.2.4) and the dual lattice (corresponding to the bars) by
where 
means i and 
are neighbors. Let 
,
, be an intensity label taking a value in a real interval
. Let 
, 
, be an edge label, also
called a line process variable [Geman and Geman 1984], taking a value in 
, with 0 and 1 representing
the absence and presence of an edge, respectively.
The interaction between the two coupled MRFs is determined by the joint
probability 
or the prior energy 
. Consider the
energy (2.23) with the potential function
defined in
(2.17). It was for piecewise continuous
restoration without explicit labeling of edge. We modify it into a
function of both intensity and edge variables:
where i and 
are neighbors. To minimize the above
alone, the intensity labels and the edge
labels are determined as follows: If 
,
then it is cheaper to pay the price 
and set
; otherwise it is more economical to set 
to insert an edge at the cost of 
.
The prior energy with the g function in (2.40) is
It can be expressed as
where 
and 
. In terms of
probability, the above corresponds to
Adding the prior energy and the likelihood energy yields the posterior energy
Minimization of the equation (2.44) has to
be performed over all 
and 
. Since the latter field assumes
discrete configurations, the minimization is a combination of real and
combinatorial problems. This can be converted to a simpler real
minimization with the following energy [Blake and Zisserman 1987]
where g is the truncated quadratic potential function
(2.40). The above is exactly the same
as (2.22). Its minimization can be
performed using algorithms working on real numbers, such as the GNC
[Blake and Zisserman 1987]. By minimizing it with respect to only 
, we
obtain the restored image 
in which the restored pixel values
near edges are properly preserved (not over-smoothed). The edge field
is then determined by thresholding 
:
where i and 
are neighbors.
Do we need the edge field 
explicitly? The current trend seems to
be using a g function without the explicit edge labels, i.e.
performing
minimization after eliminating the line process as illustrated previously. This is reflected by
e.g.
[Koch et al. ; Blake and Zisserman 1987 ; Geiger and Girosi 1989 ; Shulman and Herve 1989 ; Lange 1989 ; Li 1990b ; Rangarajan and Chellappa 1990 ; Geman and Reynolds 1992 ; Geman et al. 1992 ; Bouman and Sauer 1993 ; Stevenson et al. 1994]. It is less difficult to perform the real minimization than the combinatorial minimization. However, an explicit edge field may be useful for modeling contextual interactions of complex edge configurations, such as forbidden edge patterns which are naturally described by a discrete field.
