Image Processing Reference
In-Depth Information
2
++ ++ ++ −− −
++ ++
11 1
(
λ
)
x
(
λ
)
y
(
1
λ
)
xy
(
1
x
)(
1
y
)
*
(,)
Cxy
=
2
11 1
(
λ
)
x
(
λ
)
y
+++ +− −+−−
(
1
λ
)
xy
(
1
x
)(
1
y
)
λ
(
1
x
)(
1
y
)
2
=
++++ +
+
22
xyxy xy
λλλλ
2
xy
=
+++
+++
(
λ
2
21
)(
xy xy
xy
λ
)
22
xy
3
2
λλλ
xy
xy
(
)(
xy
)
+
+
+
λ
λ
xy xy
=
++
+
λ
(3.9)
)
xy
+
1
,
(
1
λ
From this equation, it is seen that
0
1
C
*
(,)
00
=
11
11
1
=
++
λ
2
2
+
+
λ
λ
C
*
(,)
++
=
=
1
1
λ
1
1
C
*
(, )
10
==
1
=
++
=
x
00
1
*
(,)
Cx
0
x
The
t
-conorm is strict and satisfies the criteria.
T
T-norm: The generating function for
t
T-norm: is given by
⎛
⎜
11 1
+− −
(
λ
)(
x
)
⎞
⎟
≤≤
fx
() ln
=
,
0
λ
1
x
which is a decreasing function where
f
(1) = ln((1 + 0)/1) = 0. As
x
approaches 0, the limit of the logarithm tends to be infinity as 0 ≤ λ
≤ 1.
That means it is a decreasing function and
f
−1
(
x
) = (2 − λ)/(
e
x
+ 1 − λ):
−
1
Cxyf fx fy
(,)
=
( () ())
+
(
2
−
λ
)
xy
=
[( )(
11 1
x
)][
1
(
1
)(
1
y
)]
(
1
)
xy
+− −
λ
+ −−+
λ
−
λ
xy
xyxy
(3.10)
=
−+−+−
(
1
)(
)
(
2
)
λ
λ
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