Image Processing Reference
In-Depth Information
Furthermore, the number of foreground pixels placed into the list is
n
2
+
2
n+W
,
(6.7)
where
W
is the weighting given to the center pixel. In order that the corner pixel is
preserved
n
2
+2
n
+
W
>3
n
2
+2
n
(6.8)
Since all quantities are integers, the critical point occurs when
n
2
+
2
n+W=
3
n
2
+
2
n+
1,
(6.9)
W=
2
n
2
+
1.
therefore
This is consistent with a weighting of 3 for a 3
×
3 window. Other weightings are
given in Table 6.1.
Weightings may be calculated for the preservation of other fine detail in a simi-
lar way. In general, for larger windows weightings are to be applied to other win-
dow locations in addition to the center. The weightings may be determined by
forming and solving a series of simultaneous equations. This may also be carried
out for more general weighted rank-order filters where the rank is also a parameter.
The method described in Chapter 2 may be extended for the design of optimum
WOS filters and weighted median filters. In Chapters 2 and 3, a table of observa-
tions was generated and the optimum filter was derived. The filters were uncon-
strained in their function and the output was set independently for every input
combination. WOS and median filters represent a constraint on the function that
may be implemented by restricting the output to depend on a weighted summation
Table 6.1
Weighting required for corner preservation for various sizes of center-weighted
median filters.
Window size
Weighting
2
n
2
n
+1
(2
n
+1)
(2
n
+1)
1
3
3
×
3
2
9
5
×
5
3
19
7
×
7
4
33
9
×
9
