6.2.1
Let us consider a heuristic procedure for simultaneous image restoration
and parameter estimation [Besag 1986]. Our interest is to restore the
underlying MRF realization, f, from the observed data, d. During
this process, two sets of parameters are to be estimated: a set of the
MRF parameters 
and a set of the observation parameters
. The procedure is summarized below:
of the true MRF, and
guess for 
and 
.
by 
.
by 
.
by 
based on the current 
, 
and 
.
is
approximately converged.
Assume that the data 
is determined by the
additive noise model
where 
is the independent zero-mean Gaussian
noise corrupting the true label 
. Then, for step 2, we can maximize
where 
. Step 2, which is not subject to the
Markovianity of f, amounts to finding the ML estimate of
which is, according to (6.5),
Steps 3 and 4 have to deal with the Markovianity. Consider again the homogeneous and isotropic auto-logistic MRF, characterized by (6.7) and (6.8). For step 3, we can maximize
where 
. Heuristics may be used to alleviate
difficulties therein. For example, the likelihood function 
in step 3 may be replaced by the pseudo-likelihood
(6.14) or the mean field approximation. A method for computing
the maximum pseudo-likelihood estimate of 
has
been given in Section 6.1.2. The maximization in step 4 may be
performed by using a local method such as ICM to be introduced in
Section 8.2.1.
Fig.6.5 shows an example. The true MRF, f, (shown in
upper-left) is generated by using a multi-level logistic
model, a generalization to the bi-level
auto-logistic MRF, with a number of M=6 (given) level labels in 
and the parameters being 
(known throughout the estimation) and 
,
respectively. The MRF is corrupted by the addition of Gaussian noise
with variance 
, giving the observed data, d,
(upper-right). With the correct parameter values 
and
known, no parameters are to be estimated and only step 3
needs to be done, e.g.
using ICM. ICM, with 
fixed, converges
after six iterations and the misclassification rate of the restored
image is 2.1%. The rate reduces to 1.2% (lower-left) after eight
iterations when 
is gradually increased to 
over the first
six cycles. With 
known 
estimated, the procedure
consists of all the steps but step 2. The error rate is 1.1% after
eight iterations (lower-right), with 
. When 
is
also to be estimated, the error rate is 
with 
and
.
Figure 6.5: Image restoration with unknown parameters.
(Upper-left) True MRF. (Upper-right) Noisy observation.
(Lower-left) Restoration with the exact parameters.
(Lower-right) Restoration with 
estimated.
From (Besag 1986) with permission; © 1986 Royal Statistical Society.
