Image Processing Reference
In-Depth Information
Table 6.3 Switching strength of weighted median filter for various center weightings.
Center weight, W
Number of neighbors of opposite state required to
cause center pixel to switch state
1
5
3
6
5
7
7
8
> 7
Not possible
neighboring pixel is required to trigger a switch. It can easily be shown that there
are therefore only four valid center weights for the filter defined in a 3
3 window
and that these are 1, 3, 5, and 7. When the center weighting is W= 1, the filter is
identical to the standard median. For center weights greater than seven, it becomes
impossible to switch the center value even if all other eight pixels have the opposite
value. In this case, the filter becomes an identity filter and it is neither extensive nor
antiextensive. This relationship is shown in Table 6.3.
For simplicity, let d=D ( x ) and P ( d= 1 | | x '|) be the probability that X c will
switch value when a total corresponding to | x '| of its neighbors have the opposite
value. Similarly, P ( d= 0 | | x '|) is the probability that X c will remain unchanged under
the same conditions. The prior probability of | x '| is given by P (| x '|) and P ( d=
1 | | x '|) = 1 -P ( d= 0 | | x '|).
Assuming that the weight-monotonic property holds, then the probability that a
pixel will switch state P ( d =1 | | x '|) increases monotonically with the number of
neighbors it has of the opposite value | x '|. It is expected that this property would be
reflected in the training set data for an imaging problem capable of being corrected
by an increasing filter such as the weighted median.
By the same argument as in the general case and the WOS filter, the optimum
differencing filter D opt ( I ) is determined by | x '| opt , the minimum value of | x '| for
which P ( d= 1 || x ' | )
×
0.5. The total MAE may be calculated in a similar way as will
be seen.
The probability that the center value will switch P ( y X c | | x '|) is used to design
the differencing filter and is estimated from the training set. A variation on the fa-
miliar table of observations is formed. The training images are scanned with the fil-
ter window and at each location a count is kept as to whether the ideal image value y
differs from the noisy value X c for each value of | x '| in the window. The switching
probability P ( y
X c | | x '|), is then determined as
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yX
Py X
x
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(6.12)
c
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c
(
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+
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yX
yX
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