Image Processing Reference
In-Depth Information
There is no guarantee that observations collected from any given test set will
possess the weight-monotonic property. The model, however, is not unreasonable
for ideal images in which the microgeometry is somewhat random and the noise is
white and symmetric. Simulations show that these assumptions hold for restora-
tion-type problems where the noisy and ideal images have similar pixel values, but
they do not hold for inverted or edge-detected images.
In the cases where the weight-monotonic property does not hold, it would sug-
gest that rank-order filters are not applicable for these problems. The weight-
monotonic property may be used as a test to check if increasing filters in general
and rank-order filters in particular are suitable for a given problem.
6.7 Design of Weighted Median Filters
The previous example showed the design of a filter constrained to be a rank-order
filter. This approach may be extended to weighted median filters.
Weighted median filters are self-dual. This means that they treat black and
white pixels equally. Therefore, as well as constraining the filter to depend on a
weighted ordering of its inputs, it must also be constrained to be self-dual. These
constraints may be enforced by placing the design problem in the context of a dif-
ferencing filter, D. The design of the optimum center-weighted median filter within
a window B reduces to the problem of determining the pixel weighting W for which
the MAE is a minimum.
As a result of the constraint of self duality , it is easier to analyze the weighted
median filter by considering the conditions under which the center pixel, X c
switches state, either from 0 to 1 or vice versa. This is done by defining W med in
terms of a differencing filter D ( x ):
W med =X c
D ( x ),
(6.11)
where
is the set difference (XOR) operator.
Rather than specifying an absolute value of 0 or 1, the differencing filtering
D ( x ) indicates whether the value at the center of the window X c should be changed.
Examples of the four cases of D ( x ), X c , and W med are given in Table 6.2.
Table 6.2
Operation of the differencing filter D (
x
).
Pixel at Center
of window, X c
Differencing filter
value, D ( x )
Output of Weighted
Median Filter, W med
0
0
0
0
1
1
1
0
0
1
1
1
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