Image Processing Reference
In-Depth Information
with fuzzy set. Fuzzy morphology was first defined by Goecgherian [9]. Since
then, many authors extended the framework in their work.
Here, a method to generate adjoint fuzzy morphological operations using
a conjunctor and implicator [8,16] is discussed.
Mapping
c
(
x
,
y
): [0, 1] × [1, 0] ↦ [0, 1] is a fuzzy conjunction if
c
is increasing
in both
x
and
y
. Each conjunctor satisfies
c
(
x
, 0) =
c
(0,
x
) = 0,
x
∈ [0, 1]
Mapping
i
(
x
,
y
): [0, 1] × [1, 0] ↦ [0, 1] is a fuzzy implication if
i
is increasing in
y
and decreasing in
x
. Each implicator satisfies
i
(
x
, 1) =
i
(0,
x
) = 1,
x
∈ [0, 1]
A conjunctor is a
t
-norm if it is commutative, that is,
c
(
x
,
y
)
= c
(
y
,
x
), and
associative
c
(
c
(
x
,
y
),
z
)
= c
(
x
,
c
(
y
,
z
)) and
c
(
x
, 1) =
x
Deng and Heijmans [7] proposed a number of conjunctor-implicator pairs
to construct morphological operations. Some examples on conjunctor-
implicator are given in the following:
Gödel-Brouwer operation
cby
(,)min(, )
=
b y
(9.8)
x
,
,
for
for
xb
xb
<
≥
ibx
(
,
)
=
1
Lukasiewicz operation
cby
(,)max(,
=
0
b y
+ −
1
)
(9.9)
ibx
(,)min(,
=
1
x b
− +
1
)
Hamacher operation for λ
> 1
y
byby
)
,
for 0
b
>
=
+− +−
λ
(
1
λ
)(
cby
(, )
0
,
for
b
=
0
(9.10)
(
1
−
λ
)
bx
+
λ
x
)
,
for 0
b
>
λ
+
(
1
−−+
λ
)(
1
xbx
ibx
(, )
=
1
,
for
b
0
=
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