Image Processing Reference
In-Depth Information
From the earlier properties of Smets-Magrez's axioms, the properties of
Lukasiewicz operators are as follows:
1.
L
(0,
b
) = 1, ∀
b
∈ [0, 1]
2.
L
(
a
, 0) = λ(
a
), ∀
a
∈ [0, 1]
3.
L
(
a
, 1) = 1, ∀
a
∈ [0, 1]
4.
L
(1,
b
) = λ(1 −
b
), ∀
b
∈ [0, 1]
5.
L
(
a
,
b
) = 1 ⇔ λ(
a
) + λ(1 −
b
) ≥ 1, ∀
a
,
b
∈ 1
6.
L
(
a
,
b
) = 0 ⇔ λ(
a
) and λ(1 −
b
) = 0
Lukasiewicz erosion and dilation [5,18] for an image
A
by another image
B
(
A
and
B
are the fuzzy sets of a subset) using inclusion grade
R
.
The inclusion grade
R
∈
F
(
S
)
×
F
(
S
) between fuzzy subsets in some uni-
verse
S
is
{
}
∀∈
RAB
[,] inf min, [()]
=
1
λ
A x
+
λ
[
1
−
B x
( )]
,
A BFS
,
()
⎣
⎦
xS
∈
and it satisfies the following conditions:
RAB
(,)
=⇔ ⊆
1
A B
RAB
(,)
=⇔∃∈
0
x UAx Bx
, () ,()
=
1
=
0
RABRBC AC
(,)
=
(
,
)
RA BC
((
∪≥
), )min((, ,min(, ))
RAC
B C
RABC RAB C
(, )min((, ,min(,))
∩≥
Dilation and erosion are defined as
EABx RBxAx
(,)( ) [(),()]
=
L
(9.19)
{
}
[
]
=
infmin , [()]
1
λ
Bx
+
λ
[
1
−
Ax
()]
xS
∈
DABx RBxAx
(,)( )
=−
1
[(),()]
L
}
(9.20)
{
[
]
=−
1
infmin , [()]
1
λ
Bx
+−
λ
[
1
Ax
()]
xS
∈
Search WWH ::
Custom Search