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Having defined the prior and observational models, let us move on
to estimate the MAP.
Following the R-L method, which, as we have said,
corresponds to maximum a posterior estimation with
a uniform image prior, we seek to find
Applying logarithms and differentiating with respect to f we
obtain
the following equation:
where we have assumed that 
. To solve
(6)
Lucy uses the iterative scheme
where 
denotes iteration and 
component of the vector. This
iterative scheme is justified as an iterative scheme derived
from EM principles.
However, it can also be obtained by multiplying both sides of (6)
by f.
Let us now assume that we want to impose smoothness constraints on the
solution by using a CAR prior model, we have
Differentiating 
with respect to 
we obtain
or
Multiplying both sides of (8) by 
we obtain
the following iterative scheme:
where denotes iteration and component of the vector. This equation
can be rewritten as
where 
, 
corresponds to the classical R-L restoration method.
Before examining an example, let us briefly comment on the problem
of estimating 
. Although it is possible to estimate
on a trial and error basis we are currently working on the use of the
hierarchical Bayesian approach to image restoration (Molina 1993). The idea
is to use the joint distribution defined as
integrate (10) on to obtain 
, and then
find 
which satisfies
For the problem we have at hand we can use improper
noninformative priors, 
, and also
gamma distributions. The relationship between this hierarchical model
and the method developed by Katsaggelos, Kang, and Banham (1994) are
under investigation.