Image Processing Reference
In-Depth Information
EAB
(,) nf
=
u cByxAy
yS
⎣
⋅
(
−
),()
⎦
∈
where
i
is a
t
-norm
u
is a
t
-conorm
So,
DAB
(,) up(max(, () (
=
0
A yByx
+ −+
)
1
))
yS
∈
EAB
(,) inf(min( () (
=
1
A ycBy x
+ ⋅ −
)))
yS
∈
=
inf(min( ,()
1
Ay
+ −
1
B
(
yyx
−
)))
yS
∈
=
inf(min( ,
11
+ −−
Ay By x
( )
(
)))
yS
∈
The image is initially fuzzified using any membership function. Then ero-
sion and dilation operations are performed using the structuring element.
The difference of the eroded and dilated image gives a gradient image.
9.5.2 Intuitionistic Fuzzy Approach
Another approach to find the edge image is by using Hamacher
t
-norm and
t
-conorm using intuitionistic fuzzy set theory. The image is converted to an
intuitionistic fuzzy image, and then using morphological operators, the edge
image is obtained.
From Sugeno's fuzzy complement or intuitionistic fuzzy generator, the
intuitionistic fuzzy membership function is computed as
ʹ
=−
−
+⋅
1
μ
λμ
=
+⋅
+⋅
(
1
1
λμ
λμ
)
mn
mn
mn
mn
μ
1
mn
1
Using Sugeno-type fuzzy negation,
1
−
+⋅
μ
λμ
mn
mn
ϕ
()
x
=
1
The non-membership function of an intuitionistic fuzzy image is computed as
λμ
λμ
λ μ
(
1
1
+
+⋅
)
mn
1
−
1
− ʹ
+⋅ʹ
μ
λμ
(
)
(
)
=
mn
νϕμ
ʹ =
ʹ
=
mn
m
mn
mn
⋅ +
(
1
)
1
n
1
+
mn
(
1
+⋅
λμ
)
mn
1
12
μ
λμ λμ
−
+⋅⋅ +
mn
=
2
mn
mn
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