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The entropy expression generally accepted in image restoration is (in 1-D notation)
where represents the image to be restored, and
is the model or prior estimate of the image.
The model in the entropy expression plays a special role in
MEM image restoration. In most cases, e.g., using the
data constraint,
the MEM solution is unique, depending on the model.
More specifically, once the model is given, the solution is unique.
However, any change of the model will result in the change of the solution
even if the data constraints remain the same. This will be shown specifically
in the next section.
Let us consider the following specific formulation in the MEM algorithm.
The entropy of an image defined as in Eq. (1)
is maximized subject to the data fitting constraint in the image domain
and the total power constraint
where
represents the data, i.e., degraded image,
the noise variance, and
(
) the PSF (point spread function) combined with the
ICF (intrinsic correlation function);
the subscript c stands for critical value,
and ob for observational value.
The method is implemented by forming the objective function
and using the Newton method to find ME solutions, i.e.,
to maximize
for particular
values of the Lagrange multipliers
and
(inner iteration). The desired
values of
and
are achieved by choosing appropriate
and
(outer iteration).
In what follows we will concentrate on
and
pay minimum attention to
since the total power constraint
is of little interest in the problem.
The iterates are updated according to
where
is an optimal step determined by a one-dimensional search.
In order to calculate the change
,
we need to solve a set of linear equations of very large size:
where the gradient and matrix elements are calculated from
,
i.e., the current
and given by, respectively,
and
This is a time consuming operation.
In the underlying algorithm for the program,
the non-diagonal elements of the
PSF matrix are simply ignored in calculating the Hessian
under the assumption
that the diagonal elements dominate, so that the matrix becomes
a diagonal one:
where
In this way the solution of the
equations becomes a simple operation, specifically,
The scheme described above is attractive because of its simplicity.
However, it has the following drawback:
At the beginning of iteration, is zero and
.
Normally the model
is a flat image. In order to reduce the
value of
,
gradually steps up in the
course of the iteration as depicted schematically by the lines located
in the upper part of Fig. 1.
As such, it is fairly easy to find the ME solutions
for the particular
's at the early stage of iteration
since the ignored non-diagonal elements in Eq. (3) are
relatively insignificant.
However, as the iteration advances,
it becomes more and more difficult to find the ME solutions
for the particular
's because the ignored non-diagonal
elements (proportional to
) become larger and larger
relative to the diagonal elements (depending partly on
).
(Of course, this kind of difficulty will be encountered
by any algorithm - generally speaking, convergence slows down
at the later stage of iteration.)
From the above analysis we see that if the values of can be
reduced, then the convergence will be faster. Can we do it?
To achieve the desired value of
, we need to increase
to a
certain value.
and
are in a one-to-one relationship.
At first glance, it seems to be impossible.
The key of the new technique is to update the model in the iteration.
At the beginning of the iteration, as usual, and
.
is increased to some value to start
the first outer iteration, and an ME solution is obtained at the end.
Then this ME solution is used as the model to start the second
outer iteration. Accordingly,
is increased to some value different
from zero but not from the value used in the first outer iteration.
The later outer iterations proceed in the same manner.
In this way
will jump up and down,
but not progressively step up in the course of the iterations, as depicted
by the thick lines located in the lower part of Fig. 1.
As a consequence, the values of
are kept small.
Better approximation in the solution of the equations is achieved, and
therefore the convergence is speeded up.
In the program cmem a parameter called m_update is used to control the frequency of model updating. Specifically, the model will be updated every m_update'th outer iteration. Setting m_update to 1 realizes the above described model updating procedure and achieves the most rapid convergence.