Image Processing Reference
In-Depth Information
Figure 4.5 Two different filter windows. (a) The 5-point cross and (b) the 3-point asymmetri-
cal window formed by omitting pixels X 3 and X 4 .
ues of X 3 and X 4 . This is carried out by summing N 0 and N 1 for these inputs. The net
effect is that a single output must be allocated to each combination of X 0 , X 1 , and X 2 .
The new table of observations shown in Fig. 4.6(b) has only eight inputs and the
error is now 3006 pixels.
Figure 4.7 shows the source of the errors. Effectively each set of four separate
inputs for the original 5-point window must now all have the same output for the
3-point sub-window. Those outputs that differ from the new combined value result
in an increase in error. The total increase in error is 1544, which corresponds pre-
cisely to the difference between the error for the 5-point window (1462) and the er-
ror for the 3-point window (3006).
This method may be used to compare any two sub-windows within an overall
region of support by omitting certain inputs. In the example shown, these were the
two least significant variables, so the inputs to be combined were adjacent in the ta-
ble. For other inputs the table must be rearranged. Also, it is possible to use this
technique to compare windows at different resolutions though this is beyond the
scope of this topic. For details see Dougherty et al. 1
From all the evidence thus far, it would therefore seem that it is better to use a
large window. This is true provided that the optimum filter for the large window
can be found. This task, however, gets progressively harder as the size of the win-
dow increases.
In Chapter 2 it was shown that a 3-point window had 2 3 = 8 input combinations
and 2 2 3
= 256 possible functions. Therefore for a window with n points, there are 2 n
input combinations and hence 2 2 n
functions. Consider Table 4.1. The number of in-
put combinations and associated functions scale at an alarming rate. Even the sim-
ple 5-point cross used in Chapter 3 is capable of implementing more than 4.29 × 10 9
functions! For window sizes of 17 and 25 points, the number of filter functions is
too large to express in terms of standard floating-point numbers.
So what effect does this rapid increase in the number of possible filters cause
when designing an optimum filter? The key column in the table is the number of in-
put combinations possible. Recalling the design process in Chapter 2, a table of ob-
servations was constructed from the training set. The size of this table corresponds to
the number of input combinations. For each input it is necessary to observe its occur-
rence a sufficient number of times to make a good statistical estimate of the optimum
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