Image Processing Reference

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where
H
(
μ
) is defined in a similar fashion to Shannon's entropy:

N

μ
x
i
log
μ
x
i
.

H
(
μ
)=

−

K

[8.17]

i
=1

It is easy to see that
E
satisfies all of the axioms of the degree of fuzziness. Fur-

thermore, we have:

H
max(
μ, ν
)
+
H
min(
μ, ν
)
=
H
(
μ
)+
H
(
ν
)
,

[8.18]

and:

E
max(
μ, ν
)
+
E
min(
μ, ν
)
=
E
(
μ
)+
E
(
ν
)
.

[8.19]

Many other measures of fuzziness have been suggested, with similar properties.

Here are the most important ones.

The Hamming distance
to the closest binary set, which is nothing more than the

0.5-cut, is given by [KAU 75]:

N

μ
x
i
−

μ
1
/
2
x
i
.

f
(
μ
)=

i
=1

The Hamming or quadratic distance between μ and its complement
[YAG 79] or

more generally:

f
(
μ
)=
N

p
1
/p

=
N

p
1
/p

μ
x
i
−

μ
C
x
i

2
μ
x
i
−

1

.

i
=1

i
=1

The measure suggested by Kosko
[KOS 90] compares the intersection of
μ
and
μ
C

with their union according to the formula:

min
μ, μ
C

max
μ, μ
C

.

Generalized entropy
is defined based on a generating function, either in additive

or multiplicative form [BEZ 92]:

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