Image Processing Reference
In-Depth Information
Let
A
= {(
x
, μ
A
(
a
ij
), ν
A
(
a
ij
))|
a
ij
∈
A
} and
B
= {(
x
, μ
B
(
b
ij
), ν
B
(
b
ij
))|
b
ij
∈
B
} be two
intuitionistic fuzzy images. If μ
A
(
a
ij
) and ν
A
(
b
ij
) are the membership and non-
membership values of (
i
,
j
)th element of image
A
, respectively, then while
assigning the membership value of an element, there may be some hesitation
degree π
A
(
a
ij
), where π
A
(
a
ij
) = 1 − μ
A
(
a
ij
) − ν
A
(
a
ij
). In that case, the membership
value will lie in the range {μ
A
(
a
ij
), μ
A
(
a
ij
) + π
A
(
a
ij
)}. Similarly for image
B
, the
membership value will also lie in a range.
So, the divergence between pixels
a
ij
and
b
ij
of images
A
and
B
is given as
DAB
(,)
IFS
∑
∑
(
)
⋅
() ()
⎣
μ
a
−
μ
b
=
21
−
−
μ
() ()
a
+
μ
b
e
Aij Bij
Aij
B
ij
i
j
(
)
⋅
μ
() ()
b
−
μ
a
−−
1
μ
() ()
b
+
μ
a
e
Bij Aij
BBij
A
ij
(
)
⋅
μ
() () () ()
a
+
π
a
−
μ
b
−
π
b
+− −
21
μ
() () (
a
−
π
a
+
μ
b
)
+
π
(
b
)
e
Aij Aij Bij Bij
Aij
A
ij
B
ij Bij
(
)
⋅
() () ()
() () () (
μ
b
+
π
b
−
μ
a
−
π
a
j
)
⎦
1
()
−−
μ
b
−
π
b
+
μ
a ae
Bij Bij Aij Ai
+
π
Bij
Bij
A
ij Aij
(6.17)
For each threshold, the thresholded image is compared with an ideally
thresholded image where the background and foreground regions are pre-
cisely segmented. In an ideally segmented image, the membership, non-
membership and hesitation degrees of all pixels are 1, 0 and 0, respectively.
Thus, with μ
B
(
b
ij
) = 1 and π
B
(
b
ij
) = 0, the IFD in Equation 6.17 is reduced to
∑
∑
⎣
μ
()
a
−
1
1
−
μ
(
a
)
(,)
21
(
() )
1
(
1 1
( )
DAB
=
− −
μ
a
+ ⋅
e
Aij
− −+
μ
a
⋅
e
A
ij
IFS
A
ij
Aij
i
j
μ
() ()
a
+
π
a
− −
10
+−−
21
(
μ
(
a
)
−
π
(
a
)
++ ⋅
10
)
e
Aij Aij
Aij
A
ij
10
+−
μ
() ()
a
−
π
a
⎦
−
(
110
−−+
μ
(
a
)
+
π
(
a
))
⋅
e
Aij Aij
Aij
A
ij
⇔
(6.18)
∑
∑
⎣
μ
()
a
−
1
1
−
μ
()
a
DAB
(,)
=
222
−− ⋅
(
μ
(
ae
))
Aij
−
μ
()
ae
⋅
A ij
IFS
Aij
Aij
i
j
μ
() ()
a
+
π
a
−
1
+− −
22
(
μ
(() ( )
a
−
π
a
⋅
e
Aij Aij
A
ij Aij
1(() ()
−
μ
a
a
ij Aij
−
π
⎦
−
(()
μ
a
+
π
(
a
))
⋅
e
A
Aij
A
ij
The value of π is calculated from π
A
(
a
ij
) = 1 − μ
A
(
a
ij
) − ν
A
(
a
ij
). ν
A
(
a
ij
) is computed
using Equation 6.15.
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