Image Processing Reference
In-Depth Information
∑
∑
∑
t
L
−
1
fcount f
⋅
()
fcount f
⋅
()
f
=
0
ft
=+
1
m
=
,
m
=
0
1
∑
1
t
L
−
1
count f
()
coun
ttf
()
f
=
0
ft
=+
The optimal threshold is selected by minimizing fuzzy divergence.
Fuzzy divergence between two images
A
and
B
is written as
M
−
1
M
−
1
(
)
∑
0
μ
() ()
a
−
μ
b
μ
μ
Bij Aij
() ()
b
−
a
21
−− +
(
μ
(
a
)
μ
(
b
))
e
Aij Bij
− −
(
1
μ
(
b
)
+
μ
(
ae
))
Aij
B
ij
Bij
A
ij
i
0
j
=
=
For thresholding purposes, fuzzy divergence between the thresholded image
and the ideally thresholded image, μ
B
(
b
ij
) = 1, is written as
M
−
1
M
−
1
(
)
∑
∑
μ
a
−
1
1
−
μ
a
(6.2)
DAB
(,)
=
22
−
(
−
μ
a
)
⋅
e
A j
−
μ
a
⋅
e
A ij
A j
A j
i
0
j
0
=
=
Divergence is calculated for all threshold grey levels, and the grey level cor-
responding to the minimum divergence is selected as the optimal threshold.
Image thresholding using
measures of fuzziness
and
fuzzy entropy
follows
the same procedure as fuzzy divergence where the measures of fuzziness or
entropy are minimized.
6.3.2 Fuzzy Geometry Method
Rosenfeld [17] introduced the concept of fuzzy geometry of image subsets
such as area, perimeter and compactness. Pal and Rosenfeld [16] used the
concept of fuzzy compactness to obtain an optimal threshold and standard
Zadeh's
S
-function for finding the membership values of the pixels in an
image. The optimal threshold is selected by minimizing the fuzzy compact-
ness. Compactness for a threshold
t
i
is given as
area
per
()
()
μ
μ
Compactness
()
μ
=
2
where area and perimeter corresponding to threshold
t
i
are defined as
∑
∑
()
=
a
μ
μ
( ,
a
ij
,
=
123
,,,
…
,
M
ij
t
i
i
j
Perimeter
P
(μ) is calculated as follows:
M
M
−
1
M
M
−
1
∑
∑
∑
1
∑
()
=
P
() (
a
a
)
() (
a
a
)
μ
μ
−
μ
+
μ
−
μ
ij
ij
,
1
ij
i
1
,
j
+
+
i
=
1
j
=
1
j
=
i
=
1
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