Fuzzy Sets and Possibility Theory - Information Fusion in Signal and Image Processing - page 129

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].

≤ b ≤ c ≤

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 .

| μ ∈ [0 , 1] , | μ α |

The quantity | μ | 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

⎧
⎨

− x

if x ≤ m

α

∀ x ∈ R ,μ ( x )=

R x

if x

⎩

− m

≥ m

β

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
properties:
-

∀ x ∈ R ,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
knowledge.