How Accurate Is the Logic-Based Filter? - Logic-based Nonlinear Image Processing - page 20

Figure 3.2 An error image. The figure shows the pixels not repaired following filtering with an
optimal filter.

For the simple example shown in Chapter 2, the error calculation is shown in
Fig. 3.1. The smaller value in the two columns, either N 0 or N 1 , is added to the total
and it can be seen that the overall number of pixels in error is 14. Figure 3.2 shows
the error image, the correctly restored pixels, and shows the errors marked on the
image. It can be seen that all isolated error pixels have been correctly restored.
However, where a number of adjacent pixels are in error, the filter has been unable
to correct them, which is not surprising as it only operates within a three-point win-
dow.
When a suboptimal filter function is used to filter the image, the MAE in-
creases relative to the optimal. The amount by which the error increases may be
computed from the table of observations. The error only increases for those inputs
where the suboptimal filter has a different output to the optimal. For these inputs, it
increases by the difference between N 0 and N 1 . For example, when the noisy image
from Fig. 2.2 was filtered with the function ψ =X 2 instead of the optimum filter

opt , the resulting error was as shown in Fig. 3.3.
The extreme right-hand column of the table in Fig. 3.3 corresponds to |N 0 - N 1 |
and represents the increase in error resulting from switching the output value for
that particular input. It is also known as the advantage . 1
For the two filters described, their outputs differ only for inputs (0, 1, 1) and (1,
0, 0) and the error therefore increases by nine and thirteen pixels respectively, giv-
ing a total increase of 22. The overall error would therefore be the error from the op-
timal filter plus the increase in error using the suboptimal filter (i.e., 14 + 22 = 36
pixels). It can be seen from the extreme right-hand column of the table in Fig. 3.3
that the consequences of getting the filter output wrong can be very different for
different inputs. Switching the filter output values for some inputs may have little
effect on the MAE because either those inputs are not seen very often or the differ-
ence between N 0 and N 1 is very small.
It is interesting to observe that given no other information, the total number of
pixels in error in the noisy image prior to filtering I n may also be determined from

ψ