Image Processing Reference
In-Depth Information
Therefore, the (
A
i
,p
i
) form random closed sets, each one included in the next, or
even focal sets in belief function theory. The corresponding mass functions are thus
described as consonant. Let us note that
p
i
is not the probability for
x
(the variable
whose possibility distribution is
π
) to belong to
A
i
, but rather the probability for
A
i
to actually represent the knowledge available regarding
x
. Once again, we are dealing
with the two concepts of imprecision (through the size of
A
i
) and uncertainty (the
value of
p
i
).
The value of
λ
i
can also be interpreted as a lower bound of the probability for the
actual value of
x
to be in
A
i
. The distribution
π
is then equivalent to a family
P
of
probabilities:
=
P, P
A
i
≥
λ
i
,i
=1
...n
.
P
[8.32]
Possibility is then interpreted as the upper probability:
Π(
B
)=
P
∗
(
B
) = sup
P
(
B
)
,P
∈P
,
[8.33]
and necessity as the lower probability:
N
(
B
)=
P
∗
(
B
) = inf
P
(
B
)
,P
∈P
.
[8.34]
Interpretations in terms of likelihood functions involve probabilities of the form
P
(
s
m
|
s
) where
s
m
refers to the measured value of
x
and
s
to its actual value. The
distribution
π
can then be identified with
P
(
s
m
|
s
). For any subset
A
,wehave:
P
s
m
|
s
≤
P
s
m
|
A
≤
P
s
m
|
s
min
s
∈
A
max
s
∈
A
[8.35]
which then makes it possible to interpret Π(
A
) as the upper bound of
P
(
s
m
|
A
).
However, we cannot get very far with this since the information available is usually
lower than
P
(
s
m
|
s
).
8.5. Combination operators
Following Zadeh's original work [ZAD 65], many operators were suggested in
the fuzzy community to combine membership functions or possibility distributions
2
.
These operators are also called connectives, or combination or aggregation operators.
2. Let us note that because a possibility distribution and a membership function have similar
mathematical expressions and because there is a connection between the two, the operators can
apply to either of them. However, they have different origins, meanings and semantics, which
is a point that should not be overlooked.
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