Image Processing Reference
In-Depth Information
6.6 Thresholding Using Type II Fuzzy Set Theory
If we interpret images in terms of Type II fuzzy set, then a question will
arise as how fuzziness is a fuzzy set, that is, to what degree the member-
ship values are certain. If the degrees are defined with no uncertainty, then
this fuzziness diminishes. Tizhoosh [22] defined this fuzziness in terms
of ultrafuzziness where maximum ultrafuzziness is 1. But if the member-
ship value lies in an interval range, then this ultrafuzziness will increase.
Ultrafuzziness is defined as
−
∑
L
1
1
()
=
()
−
() ()
(6.23)
A
MN
hg g
[
g
]
γ
μ
μ
U
L
×
g
=
0
where
μ
U
(
g
) and μ
L
(
g
) are the upper and lower membership values of the interval,
respectively
'
g
' is the grey level of the image
This ultrafuzziness is used in thresholding. The following cases hold for the
ultrafuzziness:
1. If μ
A
is a type 1 or ordinary fuzzy set, then μ
U
(
g
) = μ
L
(
g
) and
then
γ(
A
) = 0.
2. If μ
U
(
g
) − μ
L
(
g
) = 1, then
γ(
A
) = 1.
3. γ(
A
) ≥ γ(
A
)′ if
A
′ is a crisper version of
A
.
Tizhoosh [22] suggested Type II thresholding using the measure of
ultrafuzziness. Type II fuzzy set considers the membership function as
fuzzy and so the membership function lies in an interval. As the member-
ship function is considered to be fuzzy, better results may be expected on
medical images. A measure of ultrafuzziness is used to find the optimal
threshold. The concept of ultrafuzziness aims at capturing/eliminating the
uncertainties within fuzzy systems using type I fuzzy sets. A Type II fuzzy
set may be written as
A
x
x
x xX
Type II
=
{, (), ()|
μμ
∈
}
U
L
and
() () (),
x
x
x
∈
01
( ,]
μ
<
μ
<
μ
μ
U
L
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