Image Processing Reference
In-Depth Information
- 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
)
>α
.
◦
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
)
,μ
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
)
>α
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:
Search WWH ::
Custom Search