Image Processing Reference
of continuous frames of reference, or from concepts of typicality (a situation may gen-
erally be typical of known situations and this has to be accounted for when reasoning
When reasoning with propositions, the uncertainty (in a wide sense) corresponds
to the inability to state whether a proposition is true or false, either because the infor-
mation is incomplete, vague, imprecise, or because the information is contradictory or
In the first case, a possibilistic model makes it possible to take into account this
type of uncertainty, whereas in the second case, a probabilistic model would be well
There is another important distinction between the degree of certainty and the
degree of truth. In fuzzy logic, propositions are assigned a degree of truth, whereas
in possibilistic logic, they are usually assigned degrees of uncertainty.
8.7.1. Fuzzy logic
In fuzzy logic [DUB 80, DUB 91], reasoning is based on elementary fuzzy propo-
sitions of the type:
X is P
where X is a variable with possible values in the reference space
and P is a fuzzy
, with the membership function μ P .
The degrees of truth of such propositions are defined as values in [0 , 1] based on
μ P .
Logical connectives are defined in a very simple way, by using the same operators
as their set equivalents. For example, the degree of truth of a conjunction such as:
X is A
Y is B
is defined based on a t-norm t by:
μ A ∧ B ( x, y )= t μ A ( x ) ,μ B ( y ) .
Likewise, a disjunction such as:
X is A
Y is B