Image Processing Reference
In-Depth Information
3. Definition by Sinha and Dougherty [16,17]
DAx
3
() supmax ,()
=
0
A yByx
+ −−
(
)
1
⎣
⎦
yS
∈
(9.16)
EAx
3
() infmin ,
=
⎣
1 1
+
A y
(
)
−−
By x
(
)
⎦
yS
∈
4. Another definition by Sinha and Dougherty [16,17]
DAx
4
() supmax
=
0 1
,
−
λ
( ())
A y yx
−
λ
( (
−
))
⎣
⎦
yS
∈
(9.17)
EAx
4
() infmin
=
⎣
1
λ
(
1
−
Ay
( )) (( ))
+
λ
By x
−
⎦
yS
∈
where λ is a function from [0, 1] to [1, 0] and satisfies the following conditions:
a.
λ(
x
) is a decreasing function of
x
.
b.
λ(0) = 1 and λ(1) = 0.
c. The equation λ(
x
) = α has a single solution, ∀
x
∈ [0. 5, 1].
d.
λ(
x
) = 0 has a single solution.
e.
λ(
x
) + λ(1 −
x
) ≥ 1, ∀
x
∈ [0, 1].
When λ(
x
) = 1 −
x
,
D
4μ(
x
) and
E
4μ(
x
) reduce to
D
3μ(
x
),
E
3μ(
x
).
9.3.2 Fuzzy Morphology Using Lukasiewicz Operator
Lukasiewicz generalized operator maps
L
: [0, 1] × [0, 1] → [0, 1] such that
Lab
(,)min ,()
=
1
λλ
a
+
(
1
−
b
), ,
⎦
∀∈
a b
[ ,]
01
(9.18)
⎣
where λ: [0, 1] → [0, 1] and λ(0) = 1 and λ(1) = 0.
The properties of Smets-Magrez's axioms [1] for implication operators are
used for the properties of Lukasiewicz implication. These are as follows:
1. The value
L
(
a
,
b
) =
L
(1 −
b
, 1 −
a
), ∀
a
,
b
∈ 1.
2.
L
(1,
b
) =
b
, ∀
b
∈ [0, 1].
3.
L
(
a
,
b
) is continuous.
4.
L
(·,
b
) is non-increasing ∀
b
∈ [0, 1] and
L
(
a
, ·) is non-decreasing
∀
a
∈ [0, 1].
5.
a
≤
b
⇔
L
(
a
,
b
) = 1, ∀
a
,
b
∈ 1.
6. The value
L
(
a
,
b
) depends on the value of
a
and
b
.
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