Image Processing Reference
In-Depth Information
4.2 Similarity Measure
4.2.1 Similarity/Distance Measure
A function
S
:
F
(
X
)
2
→ [0, ∞] is called a similarity measure between two
IFS
s
A
and
B
if it satisfies the following properties:
1.
S
(
A
,
B
) =
S
(
B
,
A
).
2. For three
IFS
sets
A
,
B
,
C
∀
A
,
B
,
C
∈
F
(
X
), if
A
⊆
B
⊆
C
, then
S
(
A
,
B
) ≥
S
(
A
,
C
) and
S
(
B
,
C
) ≥
S
(
A
,
C
).
3.
S
(
A
,
B
) = 1 if
A
=
B
.
4. 0 ≤
S
(
A
,
B
) ≤ 1 ∀
D
∈
F
(
X
).
When the distance is small, nothing can be said about the similarity based on
pure distance, when the complement of the object is not taken into account.
So, in some situations, pure distance is not a proper measure of similarity.
IFS
is take into account the non-membership degree, which is used when the
distance between the two objects is small, but actually the objects are not
similar.
We know that the distance measure and similarity measure are dual con-
cepts. Therefore, one may use the distance measure than the similarity mea-
sure [8]. If
f
is a monotonically decreasing function and since 0 ≤
d
(
A
,
B
) ≤ 1,
f
(1) ≤
f
(
d
(
A
,
B
)) ≤
f
(0) may be written as
fdAB f
f
(( , ) ( )
() ()
−
1
0
≤
≤
1
0
−
f
1
Thus, the similarity measure between
A
and
B
is given as
fdAB f
f
(( , ) ( )
() ()
−
1
SAB
(,)
=
0
−
f
1
So,
f
should be defined to obtain a reasonable similarity measure. The sim-
plest form of expressing
f
is
f
(
x
) = 1 −
x
and in that case
S
(
A
,
B
) = 1 −
d
(
A
,
B
) as
f
(0) = 1,
f
(1) = 0
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