Image Processing Reference
InDepth Information
8.3. Fuzzy measures
The definitions and a few examples of fuzzy measures are presented here. For a
more detailed presentation, see [DUB 80, SUG 74].
8.3.1.
Fuzzy measure of a crisp set
A fuzzy measure is a function
f
from
C
, which is the set of subsets of
S
(hence
defined on crisp sets), into [0
,
1] that satisfies the following conditions:

f
(
∅
)=0;

f
(
)=1;
 monotonicity:
S
2
,
A
∀
(
A, B
)
∈C
⊆
B
⇒
f
(
A
)
≤
f
(
B
);
 continuity:
∀
i
∈
N
,
∀
A
i
∈C
,A
1
⊆
A
2
···⊆
A
n
···
or
A
1
⊇
A
2
···⊇
A
n
···
f
A
i
=
f
lim
i
→∞
A
i
.
=
⇒
lim
i
→∞
Some notable properties of fuzzy measures are:
max
f
(
A
)
,f
(
B
)
,
2
,f
(
A
∀
(
A, B
)
∈C
∪
B
)
≥
[8.13]
min
f
(
A
)
,f
(
B
)
.
2
,f
(
A
∀
(
A, B
)
∈C
∩
B
)
≤
[8.14]
This definition assumes no additivity constraint. They could simply be called non
additive measures, since the link with fuzzy set theory which was presented earlier is
relatively weak.
8.3.2.
Examples of fuzzy measures
Several families of fuzzy measures can be found in other works, the most common
of which are:
 probability measures;
 fuzzy
λ
measures, obtained by relaxing the additivity constraint for probability
measures:
2
,A
∀
(
A, B
)
∈C
∩
B
=
∅
=
⇒
f
(
A
∪
B
)=
f
(
A
)+
f
(
B
)+
λf
(
A
)
f
(
B
)
[8.15]
with
λ>
1;
 belief and plausibility function in belief function theory [SHA 76] (see Chapter
−
7);
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