Image Processing Reference
In-Depth Information
increasing it to 21 points or more, although requiring very large amounts of training
data did not have a significant effect on the error.
The message from these results is clear:
•
Large masks have low constraint error but high estimation error.
•
Small masks have high constraint error but lower estimation error.
Estimation error can be very severe, especially for large window sizes. In practice it
can far outweigh any constraint error. Therefore, it is often better to use a smaller
window.
Consider two filtering windows, one with
n
1
and
n
2
points respectively where
n
1
>
n
2
.
It is better to use the smaller window if the inequality below is true.
[
]
[
]
E
∆
ψψ
,
+
∆
ψψ
,
<
E
∆
ψψ
,
(4.3)
nN n
,
n n
nN n
,
2
2
2
1
1
1
This means that it is better to use a smaller window and accept a slight increase in
constraint error
2
, that is more than outweighed by the drastic reduction
in estimation error. Notice again that the constraint error is deterministic whereas
the estimation error is stochastic.
At this point, many researchers have simply given up on this type of approach.
The early promise and excellent results produced with simple problems and small
windows has disappeared as the combinatorial complexity exploded for larger win-
dows. While salt-and-pepper noise can be successfully removed with small opera-
tors, it is clear that many real-world image processing tasks require large windows.
Yet, these are difficult to design because of the large precision error.
It might seem, therefore, that the problem is just too complex to develop any
working solutions for practical problems. This is not the case; it requires further
constraints on the problem, and in particular on the nature of the function within the
filter.
This is the level at which heuristics and human intervention are valuable, in the
selection of the constraints. Human intervention is not appropriate in the selection
of filtering functions since this is too complex for most practical situations. How-
ever, intelligent selection of constraints is the key to obtaining excellent results to
real-world problems. This is the subject of the next chapter.
∆ψ
ψ
n
n
4.3 In Defense of Training Set Approaches
A criticism sometimes leveled at these filter design methods is that a representative
training set is required. This means that the “ideal” version of an image is required
in order to restore the noisy version. While this is a valid criticism, it is unreason-
able to dismiss such approaches simply because they make use of a test set.
