Image Processing Reference
In-Depth 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 non-additivity. 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]
Search WWH ::

Custom Search