Image Processing Reference
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- the additive form is given by:
N
g
μ
x
i
+
g
1
μ
x
i
f
(
μ
)=
−
[8.20]
i
=1
[0
,
1]
,g
(
t
)
<
0. Examples
of generating functions are
g
(
t
)=
te
1
−
t
,
g
(
t
)=
at
+
where
g
is a function of [0
,
1] in
R
such that:
∀
t
∈
bt
2
−
(with 0
<b<a
),
g
(
t
)=
t
log
t
(this last form gives the fuzzy entropy of [LUC 72]).
- the multiplicative form is given by:
−
N
g
μ
x
i
g
1
μ
x
i
f
(
μ
)=
−
[8.21]
i
=1
[0
,
1]
,g
(
t
)
>
0 and
g
(
t
)
<
0.
Two examples of generating functions are
g
(
t
)=
te
1
−
t
,
g
(
t
)=
t
α
.
+
where
g
is a function of [0
,
1] in
R
such that:
∀
t
∈
In fusion problems, these measures of fuzziness can be used for learning mem-
bership functions. It is also possible to infer comparison measures from fuzzy set
measures [BOU 96] which are used, for example, in order to compare an element of
information to a model or a constraint and then are combined in a fusion or multi-
criteria aggregation process.
8.4. Elements of possibility theory
Possibility theory, which is derived from fuzzy set theory, was introduced by Zadeh
in [ZAD 78] and later developed by several researches, particularly Dubois and Prade
in France [DUB 80, DUB 88].
8.4.1.
Necessity and possibility
A possibility measure is a function Π of
C
(whose argument is therefore a crisp
subset of
S
)in[0
,
1] such that:
- Π(
∅
)=0;
S
- Π(
)=1;
∀
⊂
N
∀
A
i
⊆S
(
i
∈
I
)
,
Π(
∪
i
∈
I
A
i
)=sup
i
∈
I
Π(
A
i
).
-
I
,
In the finite case, a possibility measure is a fuzzy measure. It corresponds to the
limit of equation [8.13], which is inferred from the monotonicity of a fuzzy measure.
By duality, a measure of necessity is defined as a function
N
of
C
into [0
,
1] such
that:
Π
A
C
.
∀
A
⊆S
,N
(
A
)=1
−
[8.22]
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