Image Processing Reference
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We can also find less stringent definitions, particularly if we accept an interval of
modal values, i.e. if there are four real numbers a , b , c , d , with a
d such
that μ ( x )=0outside the interval [ a, d ], μ is non-decreasing on [ a, b ], non-increasing
on [ c, d ] and equal to 1 on [ b, c ] [GOE 83, GOE 86].
A fuzzy number can be interpreted as a flexible representation of an imprecise
quantity, which is a more general representation than the traditional interval.
We now turn again to the concept of the cardinality of a fuzzy set, which is defined
above as a number. If the set is not well defined, we can expect for any measure of
this set to be imprecise as well, particularly its cardinality, which should therefore be
defined as a fuzzy number [DUB 80]:
| f ( n )=sup α
= n .
[0 , 1] ,
μ α |
The quantity
| f ( n ) represents the degree to which the cardinality of μ is equal
to n .
A very common class of fuzzy number is comprised of the L - R fuzzy numbers.
They are defined by a parametric representation of their membership function:
L m
if x
( x )=
R x
if x
where α and β are strictly positive numbers referred to as left and right spreads, m is
a number referred to as the mean value, and L and R are functions with the following
,L ( x )= L (
x );
- L (0) = 1;
- L is non-increasing on [0 , +
The function R has similar properties.
One of the main advantages of these fuzzy numbers is their compact representa-
tion, which allows simple calculations.
In fusion, fuzzy numbers are often used for representing knowledge about mea-
surements or observations, or flexible constraints applied to the values they can be
equal to, for example: “the gray level of this structure is roughly equal to 10”. Elements
of knowledge like this one can then be fused with the data or with other elements of
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