Image Processing Reference
In-Depth Information
8.4 Fuzzy Edge Image Using Interval-Valued Fuzzy Relation
The fuzzy edge image visually captures the intensity changes, and the image
can be considered as an image that represents edges in a fuzzy way. This fact
will enable us to better adjust applications where we want to use an edge
detector based on fuzzy edge images. Barrenchea et al. [1] had given an idea
for the generation of fuzzy edges using the interval-valued fuzzy relation. In
the interval-valued fuzzy set, the membership function lies in an interval -
upper bound and lower bound.
Interval-valued fuzzy set
: Let
L
[(0, 1)] denote the set of all subintervals of the
unit interval [0, 1], that is,
L
[(0, 1)] = {[
x
l
,
x
u
]|(
x
l
,
x
u
) ∈ (0, 1)
2
,
x
l
≤
x
u
}, and
x
l
and
x
u
are
the lower and upper levels of the interval-valued fuzzy set [1,11], respectively.
Then
L
[(0, 1)] is a partially ordered set with respect to relation ≤
L
, which is
defined as
(
x
l
,
x
u
) ≤
L
(
y
l
,
y
u
) if and only if
x
l
≤
y
l
and
x
u
≤
y
u
and [
x
l
,
x
u
], [
y
l
,
y
u
] ∈
L
([0, 1])
(
L
[(0, 1)], ≤
L
) is a complete lattice with the smallest element 0
L
= [0, 0] and the
largest element 1
L
= [1, 1].
If
U
is a universe, the interval-valued fuzzy relation is characterized by
mapping
M
:
U
→
L
[(0, 1)].
The membership of each element
u
i
is given as
M
(
u
i
) = [
M
l
(
u
i
),
M
u
(
u
i
)],
where
M
u
and
M
l
are the upper and lower bounds of the membership range,
respectively.
The length of the interval is the difference between the upper and lower
bounds.
To construct an interval-valued fuzzy relation, an interval range is
required. To generate the lower bound of the interval range, the lower con-
structor is used, and likewise, the upper constructor is built from the upper
bound of the interval range.
The lower constructor is built using
t
-norm and upper constructor using
t
-conorm.
A
t
-norm,
T
: [0, 1]
2
→ [0, 1], is an increasing function such that
T
(1,
x
) =
x
for
all
x
∈ [0, 1]. The three basic
t
-norms are as follows:
1. The minimum
t
-norm by Zadeh,
T
M
(
x
,
y
) = min(
x
,
y
)
2. The product
t
-norm by Bandler and Kohout,
T
P
(
x
,
y
) =
x
⋅
y
3. Lukasiewicz
t
-norm,
T
L
(
x
,
y
) = max(
x
+
y
−1, 0)
The associativity extends each
t
-norm to an
n
-ary operation by induction and
for each
n
-tuple {
x
1
,
x
2
, …,
x
n
} ϵ [0,1]
n
as in the following:
n
n
1
⎛
⎜
−
⎞
⎟
=
Tx TTxx Tx xx x
i
=
,
(
,
,
,
…
,
)
i
i n
123
n
=
1
i
=
1
t
-Norm,
T
can take different numbers of arguments.
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