Image Processing Reference
In-Depth Information
5. Similarity using the Hausdorff distance
The Hausdorff distance [14] is a measure of how much two non-
empty sets (closed and bounded)
A
and
B
in a metric space
S
resem-
ble each other with respect to their positions. If
d
(
A
,
B
) is a metric,
one-way Hausdorff measure is defined as
*
(,)max (, ),
HAB
=
d aB aA
∈
aA
∈
The Hausdorff distance between two sets
A
and
B
is
{
}
*
(,),
*
(, )
HAB
(,)max
=
H ABHBA
To define the distance measure between two
IFS
s based on the
Hausdorff distance, consider two
IFS
s
A
and
B
in
X
= {
x
1
,
x
2
,
x
3
, …,
x
n
}
and
I
A
(
x
i
) and
I
B
(
x
i
) are the subintervals on [0, 1] which is denoted as
Ix x
()[(),
=
μ
1
1
−
ν
( ]
x
Ai Ai
Ai
( ]
,
i
=
123 …, ,
n
, ,,
Ix x
()[(),
=
μ
−
ν
x
Bi Bi
Bi
Let
H
(
I
A
(
x
i
),
I
B
(
x
i
)) be the Hausdorff distance between
I
A
(
x
i
) and
I
B
(
x
i
).
Then the distance
d
(
A
,
B
) is defined [8] as
n
1
∑
dAB
(,)
=
HI xIx
A
( (),())
(4.7)
i Bi
n
i
=
1
From Equation 4.2, the Hausdorff metric between two intervals
A
= [
a
1
,
a
2
] and
B
= [
b
1
,
b
2
] in sets
A
and
B
is
{
}
dAB
(,)max
=
⎣
a bab
−
,
−
⎦
1
1
2
2
So, Equation 4.7 is written as
n
(,) ax () (), ()
1
∑
(
)
dAB
=
μ
x x x x
Ai Bi Bi Ai
−
μ
ν
−
ν
(
)
n
i
=
1
And the similarity measure is
S
H
(
A
,
B
) = 1 −
d
H
(
A
,
B
).
6. Dengfeng and Chuntian [5] suggested another similarity measure
n
1
∑
p
SAB
(,)
=−
1
ϕ
(
x
)
−
ϕ
(
x
)
(4.8)
p
IFS
Ai Bi
p
n
i
=
1
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