Image Processing Reference
InDepth Information
This duality means that if an event is necessary, its opposite is impossible.
A measure of necessity verifies the following properties:

N
(
∅
)=0;

N
(
S
)=1;

∀
I
⊂
N
,
∀
A
i
⊆S
(
i
∈
I
)
,N
(
∩
i
∈
I
A
i
)=inf
i
∈
I
N
(
A
i
).
Conversely, any measure satisfying these properties is, by duality, a possibility
measure.
Possibility and necessity measures also have the following properties:

,
max(Π(
A
)
,
Π(
A
C
)) = 1, which expresses the fact that one of the two
sets
A
and
A
C
∀
A
⊆S
is completely possible;
,
min(
N
(
A
)
,N
(
A
C
)) = 0, which expresses the fact that two opposite
events cannot be simultaneously necessary;


∀
A
⊆S
∀
A
⊆S
,
Π(
A
)
≥
N
(
A
): an event has to be possible for it to be necessary;
Π(
A
C
)
<
1 and

∀
A
⊆S
,N
(
A
)
>
0
⇒
Π(
A
)=1(since
N
(
A
)
>
0
⇒
max(Π(
A
)
,
Π(
A
C
)) = 1);

∀
A
⊆S
,
Π(
A
)
<
1
⇒
N
(
A
)=0;
∀
⊆S
,N
(
A
)+
N
(
A
C
)
≤

A
1;
∀
⊆S
,
Π(
A
)+Π(
A
C
)
≥

A
1.
The last two properties reflect nonadditivity. Knowing Π(
A
) is not enough to
completely determine Π(
A
C
), unlike with probability measures. The uncertainty
related to an event is expressed by two numbers instead of one as before.
8.4.2.
Possibility distribution
A possibility distribution is a function
π
of
S
in [0
,
1] with the following normal
ization condition:
sup
x
∈S
π
(
x
)=1
.
[8.23]
This condition corresponds to a closed world hypothesis, in which at least one
element of
S
is completely possible. This condition can be relaxed in an open world
hypothesis.
In the finite case, a possibility distribution makes it possible to define a possibility
measure using the formula:
,
Π(
A
)=sup
π
(
x
)
,x
A
.
∀
A
∈C
∈
[8.24]
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