Information Technology Reference
In-Depth Information
(
G
u
3
)
If x
=
0
.
5
and y
=
0
.
5
,thenG
u
(
x
,
y
)=
1
;
2
;
(
)
[
.
,
]
G
u
4
G
u
is decreasing in
0
5
1
2
.
(
)
[
,
.
]
G
u
5
G
u
is increasing in
0
0
5
In this work a construction method of ignorance functions using t-norms and auto-
morphisms is also presented. For example:
Example 1.
Using the t-norm minimum
2
·
min
(
1
−
x
,
1
−
y
)
if
min
(
1
−
x
,
1
−
y
)
≤
0
.
5
G
u
(
x
,
y
)=
1
otherwise
2
·
min
(
1
−
x
,
1
−
y
)
)=
√
x
for all
x
Example 2.
Using the automorphism
ϕ
(
x
∈
[
0
,
1
]
2
(
1
−
x
)
·
(
1
−
y
)
if
(
1
−
x
)
·
(
1
−
y
)
≤
0
.
25
G
u
(
x
,
y
)=
1
2
√
(
1
−
x
)
·
(
1
−
y
)
otherwise
We call
I
G
the total ignorance of the image
L
−
1
q
=
0
G
u
(
μ
Q
B
(
q
)
,
μ
Q
O
(
q
))
·
h
(
q
)
1
I
G
=
(12.1)
L
−
1
(
)
∑
0
h
q
q
=
where
L
is the number of gray levels of the image and
h
is the number of pixels
with intensity
q
.
I
G
represents the total influence of the ignorance in the construction
of fuzzy sets
Q
Bt
and
Q
Ot
.
When
I
G
tends to zero, then
G
u
(
μ
Q
Bt
(
(
q
)
q
)
,
μ
Q
Ot
(
q
))
→
0forall
q
∈{
0
,···,
L
−
1
}
.
By the property (
G
u
2) we have
μ
Q
Bt
(
q
)
→
1or
μ
Q
Ot
(
q
)
→
1. Therefore:
1. If
1, then the pixels with intensity
q
are such that their intensity is
very close to the average intensity of the pixels that represent the background.
This fact enables us to assure that the pixel in question belongs to the back-
ground.
2. If
μ
Q
Bt
(
q
)
→
1, then the pixels with intensity
q
are such that their intensity is
very close to the average intensity of the pixels that represent the object. This
fact enables us to assure that the pixel in question belongs to the object.
μ
Q
Ot
(
q
)
→
Experiment
In this experiment we compare the results of a classical fuzzy algorithm with re-
spect to the algorithm that minimizes ignorance functions. The development tries
to show that ignorance functions outperforms the results when the expert choose
membership functions that do not fit the problem correctly.
We take 8 different membership functions to model the background and the ob-
ject, following the expression
)
|
λ
with
μ
(
)=
−|
−
(
λ
=
.
,
.
,
.
,
,
.
,
,
q
1
q
m
z
t
0
1
0
3
0
8
1
1
3
2
,
6
15, where
z
could denote the object or the background. For all the ultrasound
