Image Processing Reference
In-Depth Information
The practical fuzzy systems use the standard Zadeh's fuzzy negation as
c ·
μ(
x
) = 1 − μ(
x
)
(1.3)
This is the simplest and is extensively used in fuzzy set theory.
Fuzzy complement is computed from the fuzzy complement functional,
which is defined as
−
1
Nx gg gx
(())
μ
=
(()
1
−
( ())
μ
(1.4)
where
g
(
·
) is an increasing function with
g
(0) = 0.
Sugeno-type fuzzy complement [15] is generated using the same fuzzy
complement function with an increasing function, which is given as
1
gx
(())
μ
=
log(
1
+
λμ
())
x
(1.5)
λ
From Equation 1.4,
⎡
⎢
1
1
⎤
⎥
−1
NX g
(( ))
μ
=
log(
1
+
λ
)
−
log(
1
+
λμ
( ))
x
λ
λ
or
⎛
⎜
⎞
⎟
1
1
λ
λμ
+
−1
Nx g
(())
μ
=
log
λ
1
+
()
x
From Equation 1.5,
gx
e
λμ
()
x
1
−
−
1
(())
μ
=
λ
Then by induction method,
⎛
⎜
1
1
λ
λμ
+
⎞
⎟
λ
⋅
log
λ
1
+
.()
x
⎛
⎜
1
+
+⋅
1
λ
λμ
⎞
⎟
=
e
−
1
−
1
g
log
λ
1
()
x
λ
1
+
λ
−
1
1
+⋅
λμ
λ
()
x
1
−
+⋅
μ
λμ
()
()
x
=
=
1
x
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