Image Processing Reference
In-Depth Information
ing at the output of the filter. Hence it has strong noise rejection properties. The
standard median filter is formed by setting
T
=(1+
W
i
)/2 and
W
i
= 1 for all
i
. It has
the property of being self-dual. This means that it treats black and white pixels
equally. If the image were to be inverted, then median-filtered and inverted again, it
would give the same result as median-filtering the original image.
Two obvious ways of varying the median filter is to change the weights
W
i
or
the threshold value
T
.
Changing the weights gives more importance to certain pixels—usually those
closest to the center of the window. This is important especially for larger windows.
A WOS filter that has different values of weights but retains
T
=(1+
∑
W
i
)/2 is
known as a
weighted median
filter. If the weights are symmetrical about the middle,
i.e.,
W
i
=W
n
1
i
, it will also be self-dual.
Changing the threshold parameter
T
means that a different rank other than the
median is chosen. For values of
T
other than the center value, the filter is not
self-dual. If
T
is allowed to vary, but all the weights are set to 1(
W
i
= 1), then the fil-
ter becomes a rank-order filter. Trivial examples are for
T=
1 resulting in the mini-
mum and
T=n
giving the maximum.
In designing WOS filters, the critical question is: What combination of values
of
W
i
and
T
result in the optimum filter for any given task? It is of course possible to
search all values, but this is very time consuming. It is also possible to employ itera-
tive techniques to adjust the filter parameters until the error criterion is minimized.
This is necessary in more complicated examples, but in many cases the techniques
described in previous chapters may be adapted to determine the optimum parame-
ters. Other work in this field includes Shmulevich
3,4
and Arce
5
, which includes fil-
ters with negative weights.
All filters that can be put into the context of generalized WOS filters are in-
creasing filters. This means that they have two special properties:
∑
•
They may be implemented in terms of mathematical morphology.
•
They may be extended to grayscale via threshold decomposition.
The first property may or may not be a useful one. Where the filter results in a sim-
ple set of morphological structuring elements, it may be implemented in hardware
in terms of comparators, resulting in a fast simple circuit. However, some filters can
result in large sets of structuring elements, and more arithmetic-based implementa-
tions may give greater efficiency.
The mention of grayscale processing will come as a relief to many readers who
feared that they were reading a topic limited to binary image processing. Far from
it—the techniques will be extended to grayscale in coming chapters. The advantage
of several of these techniques is that the optimum grayscale filter may be determined
at a binary level through threshold decomposition and then extended to grayscale
without loss of optimality. This is a strong property because it drastically simplifies
the training.
