Image Processing Reference
In-Depth Information
Figure 2.4 Examples of filter windows. Filter windows usually have an odd number of pixels
so that the pixel to be estimated may be at the center.
i
j
MAE (
ψ opt
)
MAE (
ψ opt
)
for i>j
(2.4)
It would appear that the best strategy is therefore to use the largest possible
window to give the minimum MAE and hence the best restoration. In theory this is
true. However, in practice the optimal filter may be difficult to determine for a large
window, which can mean that on balance it is better to use a smaller mask. This will
be discussed in more detail in Chapter 4.
2.6 Filter Design
The task of filter design is to determine the optimum filter within a sliding window,
e.g., for the problem described in Fig. 2.2. In this illustrative example, the simple
three-point horizontal window shown in Fig. 2.5 will be used.
The pixel values in the window may be considered as an input vector of binary
values x =( X 0 , X 1 , X 2 ). The filter output is an estimate of the value at the center of
the window (location corresponding to X 1 ). The filter output value may therefore be
represented as a binary function of the three input pixel values.
There are eight (= 2 3 ) possible combinations of the input vector x . The design
process of the filter
consists of allocating a value of either 0 or 1 for each possible
combination of x . There are 256 (= 2 2 3
ψ
) different ways of doing this, and therefore
256 different filters.
The optimum filter
ψ opt i is the one yielding the lowest value of MAE of all of
these possible filters. It would be possible to filter the noisy image with every one
of the 256 filters and to compare them to the ideal image and calculate the MAE.
Figure 2.5
Three-point horizontal filter window.
 
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