Image Processing Reference

In-Depth Information

- for the measure
μ
min
defined by

∀

A

⊂S

,A

=

S

,μ
min
(
A
)=0and

μ
min
(

)=1, the integrals
S
μ
min
(
f
) and
C
μ
min
(
f
) are equal to the minimum of the

values taken by
f
;

- for the measure
μ
max
defined by

S

∀

A

⊆S

,A

=

∅

,μ
max
(
A
)=1and

μ
max
(

)=0, the integrals
S
μ
max
(
f
) and
C
μ
max
(
f
) are equal to the maximum of

the values taken by
f
;

- for any two measurable functions
f
and
f
and for any fuzzy measure
μ
,wehave

the following monotonicity property:

∅

S
μ
(
f
)

∀

f
(
x
)
=

S
μ
(
f
)

≤

x

∈S

,f
(
x
)

≤

⇒

C
μ
(
f
)

C
μ
(
f
)

≤

which is also true in the infinite case;

- for any measurable function
f
and fuzzy measures
μ
and
μ
, we have the follow-

ing monotonicity property:

S
μ
(
f
)

∀

μ
(
A
)
=

≤

S
μ
(
f
)

A

⊆S

,μ
(
A
)

≤

⇒

C
μ
(
f
)

≤

C
μ
(
f
)

which is also true in the infinite case;

- the following inequalities are inferred from the previous properties for any mea-

surable function
f
and fuzzy measure
μ
:

f
x
i

f
x
i
;

N

min

i
=1

max

i
=1

≤

S
μ
(
f
)

≤

f
x
i
≤

f
x
i
;

N

min

i
=1

max

i
=1

C
μ
(
f
)

≤

- for any additive measure (or
σ
-additive in the infinite case), the Choquet integral

coincides with the Lebesgue integral; in this regard, fuzzy integrals can be considered

as an extension of Lebesgue integrals;

- for any fuzzy measurement
μ
, the Sugeno and Choquet integrals satisfy the fol-

lowing continuity property: for any sequence of measurable functions
f
n
in

S

such

that:

lim

n
→
+
∞

f
n
=
f

we have:

S
μ
f
n
=
S
μ
(
f
)
,

lim

n
→
+
∞

C
μ
f
n
=
C
μ
(
f
)
.

lim

n
→
+
∞

Fuzzy integrals are applied in particular in multi-criteria aggregation, but also in

data fusion (as part of the mean operators) and in shape recognition.

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