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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