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:

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