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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