Image Processing Reference
The membership functions and possibility distributions can have different seman-
tics. Here are the major ones:
- a semantics of degree of similarity (the concept of distance);
- a semantics of degree of plausibility for an object for which only an imprecise
description is available to actually be the one we are trying to identify;
- a semantics of degree of preference (a fuzzy class is then the set of “right” choi-
ces), this interpretation being closer to the concept of a utility function.
These three types of semantics are used in signal and image processing as well as
8.4.4. Similarities with the probabilistic, statistical and belief interpretations
Rather than opposing the various formalisms, it is interesting to emphasize the
cases where the interpretations overlap. Several of these similarities are given in
[DUB 99], and we will sum them up here.
A possibility distribution π , representing, for example, knowledge about the possi-
ble values of a variable x , can also be interpreted as a family of subsets
A 1 ,...A n }
each one included in the next with A i ⊂
A i +1 , to which levels of confidence λ i
attributed, which are defined by:
λ i = N A i =1
Π A i .
Since the necessity N is monotonic, we have λ 1 ≤···≤
λ n . This implies that the
set of values of the possibility distribution is finite. Let α 1
=1 ,α 2 ≥···≥
α n be
these values and let α n +1 =0. Then the ( A i ,λ i ) are given by:
A i = s
α i ,
,π ( s )
α i +1 .
Conversely, the least specific possibility distribution (i.e. the most possible) asso-
( A 1 ,λ 1 ) ,... ( A n ,λ n )
that verifies λ i = N ( A i ) is given by:
λ i ,A i ( s ) .
π ( s )=min
α i +1 ,wehave i p i =1and:
π ( s )=
i,s ∈ A i
If we define p i = α i −
p i .