Image Processing Reference

In-Depth Information

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

by analogy).

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

fluctuating.

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

suited.

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

[8.84]

where
X
is a variable with possible values in the reference space

S

and
P
is a fuzzy

subset of

S

, 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

and

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

or

Y
is
B

Search WWH ::

Custom Search