Image Processing Reference
In-Depth Information
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
{β
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
{α
C
k
(x)}.
(6.14)
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