Image Processing Reference
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µ j (x p ) = µ (1)
(x p1 )·. . .·µ (n)
(x pn ),
(6.9)
j
j
and µ (n)
j
(·) is the membership function of fuzzy set A (n)
j
. Here, we use triangular
fuzzy sets as in Fig. 6.2.
2. Find
C that has the maximum value of β C (j):
β
ˆ C (j) =
max
C k }(j)}.
(6.10)
1
≤k≤M
1.0
0.0
1.0
Attribute value
FIGURE 6.2: Triangular fuzzy membership function
If two or more classes take the maximum value, the consequent class C j of rule R j cannot
be determined. In this case, C j =∅. Thus, each fuzzy If-Then rule can have only a single
consequent class.
If a single class C takes the maximum value, let C j be a class C. The grade of certainty
CF j is determined as
β C (j)−β
P C β C (j)
CF j =
(6.11)
with
β = P C 6= ˆ C β C (j)
M−1
.
(6.12)
Using this rule generation procedure, we can generate N fuzzy If-Then rules defined by
Eq. 6.7. After both the consequent class C j and the grade of certainty CF j are determined
for all N rules, a new pattern x = (x 1 , . . . , x n ) can be classified by the following procedure
of fuzzy reasoning:
1. Calculate α C (x) for C, j = 1, . . . , M , as
α C (x) = max{µ j (x)·CF j |C j = C}, C∈C (6.13)
2. Find Class C 0 that has the maximum value of α C (x):
α C 0 (x) =
max
1≤k≤M
C k (x)}.
(6.14)
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