Image Processing Reference
In-Depth Information
membership degree of the element
i
to the cluster
j
, hard clustering corresponds
to setting
u
ji
equal to 0 or 1, whereas in a fuzzy context it can vary from 0 to 1
depending on the growing membership degree. For each data element, we have
K
∑
u
ji
=
1
(16.10)
i
=
1
where
K
is the cluster number. The partition can be found using the fuzzy
c
-means
algorithm [56,57], which consists of minimizing the objective function
N
K
∑
∑
J
=
()
2
u
m
d
(16.11)
m
ji
ji
i
=
1
j
=
1
where
d
ji
is the distance of the
i
th element to the
j
th center and
m
is the mem-
bership degree, which is related to the fuzziness of the algorithm. This function
shows a minimum if
1
u
=
(16.12)
jk
2
/(
m
−
1
)
K
d
d
∑
ik
jk
j
=
1
and
N
∑
()
u
x
m
ji
j
v
=
i
=
1
(16.13)
j
N
∑
()
u
m
ji
i
=
1
where
v
j
is the center of the
j
th cluster. The algorithm starts from an initial set
of membership values for the data set elements, named
fuzzy partition
, collected
in a matrix form and given by
2
2
2
2
U
()
0
=−
1
U
+
V
(16.14)
with
U
[1/
K
] and
V
a matrix of randomly chosen centers. In the second step,
the new fuzzy centers are computed using Equation 16.13, and then the new
membership values are computed using Equation 16.12. These two steps are
repeated until
=
UU
(
l
+
1
)
−≤
( )
l
ε
(16.15)
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