Image Processing Reference
In-Depth Information
⎡
⎢
⎤
⎥
⋅ −
+−⋅ +−−
Ay By x
Ay By xA
() (
)
Fuzzy dilation:
DA x
() sup
=
yByx
γ
(
1
γ
) () (
)
(() (
⋅ −
)
yS
∈
4. Lukasiewicz
t
-operators
T
=
max( ,
0
x
+ −
y
1
)
T
*min(,
=
1
x y
− +
1
)
(
)
Fuzzy erosion:
EABx
(,)( ) nf max,
=
01
+ −−
Ay By x
() (
)
yS
∈
Fuzzy dilation:
DABx
(,)( ) upmin( ,()
=
1
Ay By x
+ −−
( ))
1
yS
∈
9.4 Opening and Closing Operations
Fuzzy opening and closing operations are defined using fuzzy erosion and
dilation:
OABx DEABxBx
(,)( )
=
( (,)( ), ())
(
)
=
D
infmin[,
11
+
A xBxBx
() ( )], ()
−
xS
∈
CABx EDABxBx
(,)( )
=
( (,)( ), ())
⎛
⎜
⎞
⎟
(
)
=
E
supmax[, () () ], ()
0
A xBx x
+
−
1
usingSinha andDougherty
xS
∈
Using Lukasiewicz operator, opening and closing of an image are defined as
LL
⎣
⎦
OABx DEABxBx
(,)( )
=
( , (),()
L
⎡
⎢
⎤
⎥
L
=
DL Bx Ax Bx
inf [(),()], ()
xS
∈
L
⎣
L
⎦
CABx EDABxBx
(,)( )
=
( , (),()
L
⎡
⎢
⎤
⎥
L
=
E
1
−
inf [(),()],
L Bx Ax B
(
xxL
)
is aLukasiewicz operator
xS
∈
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