Image Processing Reference
In-Depth Information
- for the measure μ min defined by
A
⊂S
,A
=
S
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
=
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
( 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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