Image Processing Reference
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- possibility measures [ZAD 78], which will be introduced in section 8.4.
The links between the various fuzzy measures can be found in [BAN 78] or in
[DUB 80].
8.3.3. Fuzzy integrals
Fuzzy integrals [GRA 92, SUG 74] are the counterpart of Lebesgue integrals when
the integration is performed with respect to a fuzzy measurement. Fuzzy integrals can
be divided into two types.
The Sugeno integral of a measurable function f , defined from
S
into [0 , 1], with
respect to a fuzzy measure μ is defined by:
S μ ( f )= f
min α, μ x
,f ( x ) .
μ = sup
α [0 , 1]
∈S
In the finite case (
= N ), this expression is equivalent to:
S μ ( f )= f
|S|
min f x p ( i ) A i ,
max
i =1
μ =
where p is a permutation of
{
1 , 2 ...N
}
such that:
f x p (1) ≤···≤
f x p ( N )
0
and where A i =
{
x p (1) ,...,x p ( N ) }
.
+ , with
The Choquet integral of a measurable function f , defined from
S
into
R
respect to a fuzzy measurement μ is defined by:
C μ ( f )= fdμ = +
0
μ x, f ( x ) dα.
In the finite case, we get:
C μ ( f )= fdμ =
N
f x p ( i )
f x p ( i 1) μ A i ,
i =1
with f ( x p (0)) = 0.
The properties of these integrals are presented in detail in [GRA 92, MUR 89,
SUG 74]. The major ones in the finite case are the following:
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