Image Processing Reference
In-Depth Information
∈ℜ
( )
is a relation,
T
1
and
T
2
are two
t
-norms, and
n
and
m
are two
values with
n
≤ (
P
− 1)/2 and
n
≤ (
Q
− 1)/2, then the lower constructor using
T
1
and
T
2
may be defined as
If
RXY
m
n
LRxy TTRx iyjRxy
,
[ (,)
=
( (( )(
−
−
), (,))), (,)
∀
xy
∈∈
XY
(8.7)
TT
12
12
i n
j m
=−
=−
where the values of
i
and
j
are such that 0 ≤
x
−
i
≤
P
− 1 and 0 ≤
y
−
i
≤
Q
− 1.
The values of
n
and
m
indicate that the window size is (2
n
+ 1) × (2
m
+ 1)
which is centred at (
x
,
y
). An example on how the lower constructor operation
works is shown. Considering
n
=
m
= 1 with window size 3 × 3,
T
1
=
T
2
=
T
M
(min
t
-norm by Zadeh), the value at (
x
1
,
y
1
) is computed as
(0, 0)
0.67
(0, 1)
0.77
(1, 0)
0.74
(1, 1)
0.78
(
x
− 1,
y
− 1)
0.76
(
x
− 1,
y
)
0.56
(
x
− 1,
y
+ 1)
0.51
(
x
,
y
− 1)
0.78
(
x
,
y
)
0.81
(
x
,
y
+ 1)
0.88
(
x
+ 1,
y
− 1)
0.62
(
x
+ 1,
y
)
0.56
(
x
+ 1,
y
+ 1)
0.54
1
L Rxy
T
M
,
[ (,)min(min(. ,. ),(. ,. ),(. ,. )
=
076081 056081 051081
, , (. ,. ),
078081
(. ,. ),(. ,. ),(. ,. ),(. ,.
081081 088081 062081 056081 054081
),(. ,. ))
=
051
.
In this way, the values at each coordinate are computed by shifting the
window point by point and a matrix is formed. One may use different
t
-norms for
T
1
and
T
2
. It is to be noted that on applying the lower con-
structor, the value of
R
is reduced. The lower constructor darkens the
image as it takes lower values (shown in the example). The smaller the
t
-norm, the more reduction in the intensity of the pixels, thereby darken-
ing the image.
Likewise, for the upper constructor, to-conorm is used.
A
t
-conorm,
S
: [0, 1]
2
→ [0, 1], is an increasing function such that
S
(0,
x
) =
x
for all
x
∈ [0, 1]. The three basic
t
-conorms are as follows:
1. The maximum
t
-conorm by Zadeh,
S
M
(
x
,
y
) = max(
x
,
y
)
2. The product
t
-conorm by Bandler and Kohout,
S
P
(
x
,
y
) =
x
+
y
−
x
⋅
y
3. Lukasiewicz
t
-conorm,
S
L
(
x
,
y
) = min(
x
+
y
, 1)
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