Image Processing Reference
In-Depth Information
1
*
(,)
=
Dxy
1
−
⎛
−
λ
⎞
−
λ
λ
⎛
⎞
⎛
⎜
1
⎞
⎟
+−
1
⎜
⎜
⎟
⎟
1
+−
1
1
⎜
⎟
x
y
⎝
⎠
⎝
⎠
4. Frank [7] proposed logarithmic
t
-norm and
t
-conorms:
y
⎛
⎝
x
⎞
⎠
(
a
−
1
)(
a
−
1
)
Fxy
(,) og
=
1
+
⎜
⎟
a
a
−
1
,
a
∈∞
( ,)
0
(3.14)
⎛
⎜
1
−
x
1
−
y
⎞
⎟
(
a
−
1
)
(
a
−
1
)
*
(,)
Fxy
=− −
1
log
1
a
a
−
1
3.3.3 Negation
A function
n
: [0, 1] → [0, 1] is called negation iff
1.
n
(0) = 1,
n
(1) = 0
2.
n
is non-increasing
Negation is called strict iff
1.
n
is decreasing
2.
n
is continuous
Strict negation is called involution iff
n
(
n
(
a
)) =
a
for all
a
Fuzzy
t
-norm and
t
-conorm are used in many areas of medical image
processing such as in image enhancement, segmentation and morphology.
These are shown later in different chapters.
3.4 Fuzzy Aggregating Operators
In many cases in multi-criteria decision-making problems, the type of aggre-
gation may not be pure 'ANDing' of
t
-norms or pure 'ORing' of
t
-conorms.
Yager [19] introduced an aggregation technique based on the ordered
weighted averaging (OWA) operator that provides an aggregation lying in
the middle of these two extremes (to satisfy all the criteria or at least one of
the criteria). A special case of 'mean' operator may be used. There are many
operators for aggregation information [2,5,10,11,19], and the two common
operators used are the weighted averaging (WA) operator and OWA operator.
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