Image Processing Reference
InDepth Information
8.3.4.
Fuzzy set measures
Up until now, measures have been applied to crisp sets. If we now consider fuzzy
sets, we need measures that provide quantitative evaluations of such sets. These mea
sures are referred to as fuzzy sets [BOU 96] or evaluation measures [DUB 92b]. There
is no real consensus over the definition of such measures. Here is the least strict of
them, given in [BOU 96].
+
such that:
A fuzzy set measure is a function
M
from
F
in
R
1)
M
(
∅
)=0;
2
,μ
2)
∀
(
μ, ν
)
∈F
⊆
ν
⇒
M
(
μ
)
≤
M
(
ν
).
Other conditions can be added depending on the applications, for example:

M
has its values in [0
,
1];

M
(
)=1;

M
(
μ
)=0
S
⇔
μ
=
∅
;

M
(
μ
)=1
⇔
μ
=
S
.
Simple examples of fuzzy set measures are fuzzy cardinality, the cardinality of
the support of
μ
, the supremum of
μ
, etc. Other examples are given in the following
section: measures of fuzziness.
8.3.5.
Measures of fuzziness
An important question regarding the evaluation of a fuzzy set involves the degree
of fuzziness of the set. De Luca and Termini [LUC 72] suggested defining a degree of
fuzziness as a function
f
of
+
such that:
F
into
R
1)
∀
μ
∈F
,f
(
μ
)=0
⇔
μ
∈C
(crisp sets are nonfuzzy and are the only ones to
satisfy this property);
2)
f
(
μ
) is maximum if and only if
∀
x
∈S
,μ
(
x
)=0
.
5;
2
,f
(
μ
)
3)
∀
(
μ, ν
)
∈F
≥
f
(
ν
) if
ν
is more contrasted than
μ
(closer to a binary
set), i.e.:
,
ν
(
x
)
≥
μ
(
x
)
if
μ
(
x
)
≥
0
.
5
∀
x
∈S
ν
(
x
)
≥
μ
(
x
)
if
μ
(
x
)
≤
0
.
5
,f
(
μ
)=
f
(
μ
C
) (a fuzzy set and its complement are both as fuzzy).
4)
∀
μ
∈F
De Luca and Termini also defined the entropy of a fuzzy set [LUC 72], as a degree
of fuzziness, in the finite case:
E
(
μ
)=
H
(
μ
)+
H
μ
C
,
[8.16]
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