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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