Image Processing Reference
In-Depth Information
From the definition of dilation,
DA x
() sup(
=
i By xAy
−
), ()
⎣
⎦
yS
∈
Substituting Zadeh's
t
-norm, we get
DA x
() supmin
=
B yxAy
(
−
),()
⎣
⎦
yS
∈
(
)
⎣
⎦
Likewise, for erosion,
EA x
() inf
=
u cByxAy
yS
⋅
(
−
), ()
∈
(
(
)
)
=− −
(
)
'
c
' is the complement, so
cByx By x
⋅
−
1
So,
EA x
() infmax (),( )
A yByx
=
⎣
−
⎦
yS
∈
2. Bandler and Kohout [2]
t
-operators
T
-conorms:
T xyxy
*
=+−
T
-norm:
T xy
=
Fuzzy erosion:
EA x
() inf
A yByx By x
( )( )
1
(
)
=
⎣
⋅ −+−−
⎦
yS
∈
Fuzzy dilation:
DAx
A yByx
(() up () (
=
⎣
⋅
−
)
⎦
yS
∈
3. Hamacher
t
-operators [11]
⋅
+−⋅ +−⋅
xy
xyxy
T
=
γ
(
1
γ
) (
)
=
+−⋅−−
−−⋅
xyxy
(
1
γ
)
xy
T
*
11
(
γ
)
xy
Erosion and dilation are defined as
Fuzzy erosion:
⎡
⎤
1
−−+
By xAyAyByx
(
) () () (
−
⋅ −−−⋅
)
(
1
γ
)
A
(()(
y yx
⋅ − −
1
(
))
EA x
() inf
=
⎢
⎥
11
−−⋅
(
γ
) ()(
Ay
⋅ −−
1
By x
(
))
yS
∈
⎣
⎦
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