Image Processing Reference
In-Depth Information
be expected using intuitionistic fuzzy set. There is very little work on thresh-
olding of medical images, and few of these techniques are discussed in this
section.
6.4.1 Intuitionistic Fuzzy Entropy-Based Method
Bustince et al. [2] suggested another thresholding scheme using intuitionistic
fuzzy set theory. The membership function is defined using the restricted
equivalent function.
A function
REF
:[0, 1]
2
→ [0, 1] is called restricted equivalence function if it
satisfies the following conditions [3]:
1.
REF
(
x
,
y
) =
REF
(
y
,
x
) for all
x
,
y
∈ [0, 1].
2.
REF
(
x
,
y
) = 1, if and only if
x
=
y
.
3.
REF
(
x
,
y
) = 0, if and only if
x
= 1,
y
= 0 or
x
= 0,
y
= 1.
4.
REF
(
x
,
y
) =
REF
(
c
(
x
),
c
(
y
)), for all
x
,
y
∈ [0, 1],
c
being a strong negation.
5. For all
x
,
y
,
z
∈ [0, 1], if
x
≤
y
≤
z
, then
REF
(
x
,
y
) ≥
REF
(
x
,
z
) and
REF
(
y
,
z
) ≥
REF
(
x
,
z
).
They defined a restricted equivalence function as a measure of comparison:
(6.4)
REFAB
(,)
=−−
1
x y
Let
c
be a strong negation, such that
c
(
e
) =
e
(
e
is an equilibrium point of
negation) and let
F
: [0, 1] → [
e
, 1] be a function such that
F
(
x
) = 1 iff
x
= 0
F
(
x
) =
e
iff
x
= 1 and
F
(
x
) is non-increasing
In this condition, the membership function at threshold '
t
' is defined as
(
)
(
)
μ
()
t FcREF gm t
=
(, ())
if
gt
≤
B
B
(6.5)
(
)
(
)
t FcREF gm t
gt
, , '
g
is thegreylevel
μ
()
=
(, ())
if
>
O
O
where
m
O
(
t
) and
m
B
(
t
) are the average grey levels of the object and back-
ground regions, respectively, and are given by the following formula:
∑
∑
t
∑
L
−
1
ghg
⋅
()
ghg
⋅
()
(6.6)
g
=
0
gt
=+
1
mt
()
=
,
mt
( )
=
O
B
∑
1
t
L
−
hg
()
hg
()
g
=
0
gt
=+
1
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