Image Processing Reference
In-Depth Information
Sometimes, exponential operation is very useful in finding the similarity
measure:
If
if
(
x
) =
e
−
x
, then
−
dAB
(,)
−
1
e
−
e
−
1
SAB
(,)
=
,
as
if
() ,()
011
=
if
=
e
−
1
1
−
e
For a logarithmic function
if
(
x
) = ln(1 +
x
), the similarity measure is
ln( ( ,)) ln( )
ln()
ln() ln(
1
+
dAB
−
2
SAB
(,)
=
−
2
,
if
(
0
)
=
ln
( )
1 01 2
=
, () ()
if
=
ln
2
−+
1
dAB
( , )
=
ln()
2
One may choose the inverse function,
if
(
x
) = 1/(1 +
x
); then the similarity rela-
tion between two
IFS
s
A
and
B
is given as:
1
1
2
−
1
dAB
(,)
1
1
−
+
dAB
dAB
(,)
(,)
+
SAB
(,)
=
=
1
2
1
−
4.2.2 Distance Measures
In many practical and theoretical problems, there is a need for many rea-
sons to find the difference between two objects and in that case, the knowl-
edge of distance between two
IFS
is is necessary. Consider two
IFS
s
A
and
B
that take into account the membership degree μ, the non-membership
degree
v
and the hesitation degree (or intuitionistic fuzzy index) π in
X
=
{
x
1
,
x
2
, …,
x
n
}.
A function
D
:
F
(
X
)
2
→ [0, ∞] is called a distance measure between two
IFS
s
A
and
B
if it satisfies the following properties:
1.
D
(
A
,
B
) =
D
(
B
,
A
).
2. For three
IFS
sets
A
,
B
,
C
∀
A
,
B
,
C
∈
F
(
X
), if
A
⊆
B
⊆
C
, then
D
(
A
,
B
) ≤
D
(
A
,
C
) and
D
(
B
,
C
) ≤
D
(
A
,
C
).
3.
A = B
if and only if
D
(
A
,
A
) = 0.
4. 0 ≤
D
(
A
,
B
) ≤ 1 ∀
C
∈
P
(
X
).
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