Digital Signal Processing Reference
In-Depth Information
U
≡
=(
)
|
≤
≤
,
,
M
fcm
u
jq
0
u
jq
1
for all
j
q
;
q
=
1
u
jq
<
N
,
for all
j
c
j
=
1
u
jq
=
1
,
for all
q
;0
<
N
.
If
dist
(
x
q
,
v
j
)
is specified as the Euclidean distance then it can be expressed as
k
α
=
1
(
x
q
α
−
v
j
α
)
1
2
dist
(
x
q
,
v
j
)=
.
(2.41)
where
x
q
α
and
v
j
α
are the elements in the vector of
x
q
and
v
j
. If the distance
(
,
)
dist
is an inner product norm that is called Mahalanobis distance, then, it
is expressed as
x
q
v
j
dist
2
T
A
j
Q
j
A
j
Q
j
.
(
x
q
,
v
j
)=
x
q
−
v
j
x
q
−
v
j
=
(2.42)
In (
2.42
),
A
j
is a
k
×
k
positive defined matrix derived from the
j
th cluster. When
A
j
=
I
,(
2.42
) is equal to the Euclidean norm as specified in (
2.41
). For
m
>
1and
x
q
=
v
j
, the objective function
J
m
(
U
,
V
;
X
)
may lead to a minimum if the following
equations hold:
−
2
m
(
dist
jq
)
−
1
u
jq
=
m
−
1
∀
j
,
q
.
(2.43)
−
2
c
i
=
(
)
∑
dist
iq
1
and
q
m
x
q
(
u
jq
)
=
∑
=
1
v
i
∀
i
.
(2.44)
q
m
(
u
jq
)
∑
=
1
The similarity measure terms
dist
jq
and
dist
iq
specifiedin(
2.43
) can be defined as
either (
2.41
)or(
2.42
)with respective cluster center
v
i
or
v
j
. Unlike traditional clas-
sification algorithms, the FCM algorithm assigns all object patterns to each cluster
in fuzzy fashions. Each pattern associated with a belonging specified by member-
ship grades between 0 and 1. The fuzzy membership value describes how close
or accurate a sample resembles an ideal element of a population. The imprecision
caused by vagueness or ambiguity is characterized by the membership value. In-
clusive of the concept of fuzziness, the FCM algorithm computes each class center
more precisely and with higher robustness to the noise. The procedures of the FCM
algorithm [
30
,
31
] are enlisted as follows:
1. Initialization: Fix the number of cluster
c
and feature coefficient
m
, set iteration
loop index
t
0, and select initial cluster centers.
We are randomly select
c
initial cluster centers from the space as
v
(
0
)
j
=
,for
c
. Initialized
U
(
0
)
.
2. Sampling: Choose total
N
data samples
x
q
for
q
=
,
,...,
j
1
2
=
,
,...,
1
2
N
from the image. It
is performed by clicking the mouse on the image.
