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images, we execute the Generalized fuzzy algorithm with two membership func-
tions taken from the possible combinations of the membership functions (there are
8
2
=
64 combinations). Then we execute the ignorance based algorithm using the
expression given in Example 2 for all the cases, so we obtain 64 different solutions.
To interpret the results of this experiment we study the graphic in Fig. 12.8. This
graphic is obtained in the following way:
1. We arrange all the cases from the smallest to the biggest percentage of badly
classified pixels (error) in the solution of the fuzzy algorithm.
2. For each pair of membership functions, we calculate the error obtained with the
ignorance based algorithm.
The crosses represents the error we get with the ignorance based algorithm, and the
dotted line, the error we get with the fuzzy algorithm. (If the dots are under the
cross, it means that for that pair of membership functions the error of the ignorance
based algorithm is smaller than the error of the fuzzy algorithm). Observe that:
1. For the combinations of membership functions such that the fuzzy algorithm
solution is good (small error), the ignorance based algorithm does not provide
better results.
2. If the error we get with the fuzzy algorithm increases (i.e., if we have used bad-
chosen membership functions), then the result of the ignorance based algorithm
improves the fuzzy algorithm.
We observe that with the solutions obtained by the fuzzy algorithm, around 40% of
possible combinations of membership functions do not represent correctly areas of
the image and a very high error is obtained. However, the ignorance based algorithm
gets good solutions for almost all the cases.
90
80
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
Fig. 12.8
Percentage of error in all cases of first prostate ultrasound image in Fig. 7 with
ignorance function using the geometric mean
