Image Processing Reference
InDepth Information
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 nondecreasing on [
a, b
], nonincreasing
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 nonincreasing 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.
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