Image Processing Reference
In-Depth Information
5. Dubois and Prade [5]:
xy
xyc
T
-norm:
D xy
(,)
=
max( ,,)
(3.7)
D xy
xyxy
=
+− −
min( ,,
xy
1
−
c
)
*
(,)
T
-conorm:
and
c
∈
(,)
0
1
1
−
min( ,, )
xy c
1
−
6. Sugeno's
t
-norm and
t
-conorm [14]:
Txy
(,)max(,(
=
01
+ +−−
b xy bxy
)(
1
)
)
1
(3.8)
*
(,)min(,
Txy
=
1
x ybxy
+ −
),
b
> −
Algebraic operators do not contain conditional operators and are purely
algebraic in nature. This class of operators does not have conditional func-
tion; rather, they simply follow arithmetic operations:
1. Chaira suggested an algebraic
t
-norm and
t
-conorm that does not
contain min or max operators. The following shows the calculation
for finding
t
-norms and
t
-conorms:
1
++
−
(
λ
1
)
⎛
⎜
x
⎞
⎟
() ln
=
,
0
≤≤
λ
1
gx
1
x
where
g
(·) is an increasing function such that
g
(0) = ln(1) = 0 with
x
gx
e
e
−
++
1
−
1
()
=
x
1
λ
As
x
approaches 1, the limit of the logarithm value tends to be
infinity. That means it is an increasing function:
*
(,)
−
1
Cxyggx gy gx gy
=
( () ( )), () ()
+
+
⎡
⎤
⎛
⎞
1
++
−
(
λ
1
)
x
1
++
−
(
λ
1
)
y
⎛
⎜
⎞
⎟
+
−
1
=
g
ln
ln
⎢
⎢
⎜
⎟
⎥
⎥
1
x
1
y
⎝
⎠
⎣
⎦
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