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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