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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