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By virtue of the MU technique, in order to achieve some critical value
for data fitting, not only the number of outer iterations
but also the number of inner iterations in each outer iteration is reduced.
Consequently, the total number of iterations is greatly reduced.
The computational time is reduced accordingly.
Using synthetic images (Gaussian peaks on a flat background),
ordinary images (portraits), and astronomical images (point sources,
extended objects), experiments show that compared with
(without model updating), the CPU time is typically
reduced for 
(the most frequent model updating)
by factors of 3 to 10.
Let us look into the iteration procedure in some more detail.
For simplicity we assume that some ``critical value'' 
is achieved after two outer iterations. This value is common to
the two cases: without and with model updating. They share the same
first outer iteration (see Fig. 1),
after which the ME solution 
, determined from
in Eq. (2) with 
, satisfies
where 
and 
are 
and 
used
in the iteration, and the residual
The second outer iteration starts off with the initial value 
, and
ends up with the solution 
satisfying
Note that we still have .
The ME solution from the first outer iteration is used as the model,
i.e., 
in Eq. (2) is equal to 
.
The initial value in the second outer iteration is still 
.
Now the solution 
satisfies
Note that 
is fixed in the iteration.
The 
values are equal for both cases:
The equality of 
and 
means that the two solutions
and
are located in the same
hyper-ellipsoid of 
. They are two distinct points when 
.
Assuming that exact data fitting could be achieved, 
, then
the hyper-ellipsoid would shrink to a single point representing both
and 
. Even in this case, 
and 
would be two solutions that are not
equal because the ``solution space'' (a set formed by all the possible
solutions) without model updating is different from that with model updating.
This can be understood by comparing Eq. (4) with Eq. (5)
and noticing the term 
in the latter's
exponent. The situation here is similar to that in which we have two different
coordinate systems in a space; the same point in the space
has different coordinate values in different systems.
Now we revert to the case where 
.
Based on the above discussion, it is easy to see that
and 
are different.
As a digression we would like to point out that the consistency of MEM solution
ensures that the final solution depends only on the given data but
not on the order in which these data are used (Wu 1991).
However, the argument given there does not apply to the current case
concerning image restoration.
In the case where more than two outer iterations are performed,
it is clear from the above discussion that the final solutions
for a certain value, say 
, of 
are different
with and without model updating. But we are not able to
quantify theoretically the difference between them.
We have observed in experiments that the solutions with and without
model updating are similar in morphology but noticeably different
in photometry. The former is less biased or has reduced nonlinearity
in photometry. Why is that?
It is generally accepted that better models will result
in solutions better in photometry. The flat model used throughout
the iteration is by no means the best one. In the iteration,
as is increasing and data fitting becomes tighter and tighter,
the large-scale structure of image is built up first and the small-scale
structure comes later. The model is improving by updating it
in a reasonable way. This explains the improvement of the solution
in photometry. Model updating does make difference to the solution.
Fortunately, it becomes better, not worse.
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Up: Model Updating in the
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