Image Processing Reference
In-Depth Information
degrees and π
A
(
x
) and π
B
(
x
) are the hesitation degrees, with π
A
(
x
) = 1 − μ
A
(
x
) −
ν
A
(
x
) and π
B
(
x
) = 1 − μ
B
(
x
) − ν
B
(
x
). The interval is due to the hesitation or the
lack of knowledge in assigning the membership values.
In an image of size
M
×
M
with
L
distinct grey levels having probabilities
p
0
,
p
1
, …,
p
L
−1
, the exponential entropy is defined as
L
−
1
∑
0
−
p
i
Hp e
i
=
1
i
=
In fuzzy cases, the fuzzy entropy of an image
A
of size
M
×
M
is defined as
−
∑
0
M
1
M
−
1
1
1
−
μ
()
a
μ
()
a
⎣
⎦
HA
()
=
μ
()
ae
⋅
Aij
+
(
1
−
μ
(
ae
))
⋅
Aij
−
1
(
)
Aij
A
ij
ne
−
1
i
=
0
j
=
where
n
=
M
2
i
,
j
= 0, 1, 2, …,
M
− 1
μ
A
(
a
ij
) is the membership degree of the (
i
,
j
)th pixel
a
ij
in the image
A
For two images
A
and
B
, at the (
i
,
j
)th
pixels (i.e. at pixels
a
ij
and
b
ij
), the
amount of information between the membership degrees of images
A
and
B
is given as follows:
1. Due to
m
1
(
A
) and
m
1
(
B
), that is, μ
A
(
a
ij
) and
μ
B
(
b
ij
) of the (
i
,
j
)th pixels
μ
()
a
e
e
A ij
μ
() ()
a
−μ
b
or
e
Aij Bij
μ
()
b
Bij
2. Due to
m
2
(
A
) and
m
2
(
B
), that is, μ
A
(
a
ij
) + π
A
(
b
ij
) and μ
B
(
a
ij
) + π
B
(
b
ij
) of the
(
i
,
j
)th pixels
μ
() ()
a
+
π
a
e
e
A ij Aij
μ
() ()
b
+
π
b
B
ij Bij
Corresponding to the fuzzy entropy, the divergence between images
A
and
B
due to
m
1
(
A
) and
m
1
(
B
) may be given as
∑
∑
(
)
μ
() ()
a
−
μ
b
μ
()
b
−
μ
(()
a
DAB
(,)
=
11
− −
(
μ
( )
a
⋅
e
Aij Bij
−
μ
(
a
)
⋅
e
Bij A
ij
(4.14)
1
Aij
Aij
i
j
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