Image Processing Reference
In-Depth Information
8.6.3.
Modifiers
The meaning of a term of a linguistic variable can be modulated by operators
known as modifiers. If
A
is a fuzzy set, then the modifier
h
allows us to construct a
composite term
h
(
A
) that is a fuzzy set in the same universe
S
. Here are the most
common operators:
- normalization:
μ
A
(
u
)
sup
v
∈S
μ
norm(
A
)
(
u
)=
μ
A
(
v
)
,
where
μ
A
refers to the membership function to
A
and
u
is an arbitrary value in
S
;
- concentration:
μ
con(
A
)
(
u
)=
μ
A
(
u
)
2
;
- dilation
4
:
μ
dil(
A
)
(
u
)=
μ
A
(
u
)
0
.
5
;
- contrast enhancement:
⎧
⎨
2
μ
A
(
u
)
2
if
μ
A
(
u
)
∈
[0
,
0
,
5]
μ
int
(
A
)
(
u
)=
2
1
μ
A
(
u
)
2
⎩
1
−
−
otherwise
.
Typical modifiers defined using these operators are [DUB 80]:
-very
A
=con(
A
),
- more or less
A
=dil(
A
),
- plus
A
=
A
1
.
25
,
- slightly
A
= int[norm(plus
A
and not(very
A
))] where “and” and “not” are de-
fined by a t-norm and a complementation, respectively.
8.7. Fuzzy and possibilistic logic
The development of fuzzy logic is directly related to the specificities of human
reasoning: more flexible than traditional propositional logic, it tolerates imprecision
and can be used to make inferences even in the presence of imperfect data and knowl-
edge. It is capable of dealing with gradual predicates, originating either from the use
4. Note that this is not a dilation in the morphological sense.
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