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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