<s>
Fuzzy	B-Algorithm
clustering	I-Algorithm
(	O
also	O
referred	O
to	O
as	O
soft	O
clustering	B-Algorithm
or	O
soft	B-Algorithm
k-means	I-Algorithm
)	O
is	O
a	O
form	O
of	O
clustering	B-Algorithm
in	O
which	O
each	O
data	B-Application
point	I-Application
can	O
belong	O
to	O
more	O
than	O
one	O
cluster	O
.	O
</s>
<s>
Clustering	B-Algorithm
or	O
cluster	B-Algorithm
analysis	I-Algorithm
involves	O
assigning	O
data	B-Application
points	I-Application
to	O
clusters	O
such	O
that	O
items	O
in	O
the	O
same	O
cluster	O
are	O
as	O
similar	O
as	O
possible	O
,	O
while	O
items	O
belonging	O
to	O
different	O
clusters	O
are	O
as	O
dissimilar	O
as	O
possible	O
.	O
</s>
<s>
In	O
non-fuzzy	O
clustering	B-Algorithm
(	O
also	O
known	O
as	O
hard	O
clustering	B-Algorithm
)	O
,	O
data	O
are	O
divided	O
into	O
distinct	O
clusters	O
,	O
where	O
each	O
data	B-Application
point	I-Application
can	O
only	O
belong	O
to	O
exactly	O
one	O
cluster	O
.	O
</s>
<s>
In	O
fuzzy	B-Algorithm
clustering	I-Algorithm
,	O
data	B-Application
points	I-Application
can	O
potentially	O
belong	O
to	O
multiple	O
clusters	O
.	O
</s>
<s>
For	O
example	O
,	O
an	O
apple	O
can	O
be	O
red	O
or	O
green	O
(	O
hard	O
clustering	B-Algorithm
)	O
,	O
but	O
an	O
apple	O
can	O
also	O
be	O
red	O
AND	O
green	O
(	O
fuzzy	B-Algorithm
clustering	I-Algorithm
)	O
.	O
</s>
<s>
Membership	O
grades	O
are	O
assigned	O
to	O
each	O
of	O
the	O
data	B-Application
points	I-Application
(	O
tags	O
)	O
.	O
</s>
<s>
These	O
membership	O
grades	O
indicate	O
the	O
degree	O
to	O
which	O
data	B-Application
points	I-Application
belong	O
to	O
each	O
cluster	O
.	O
</s>
<s>
One	O
of	O
the	O
most	O
widely	O
used	O
fuzzy	B-Algorithm
clustering	I-Algorithm
algorithms	O
is	O
the	O
Fuzzy	O
C-means	O
clustering	B-Algorithm
(	O
FCM	O
)	O
algorithm	O
.	O
</s>
<s>
Fuzzy	O
c-means	O
(	O
FCM	O
)	O
clustering	B-Algorithm
was	O
developed	O
by	O
J.C.	O
Dunn	O
in	O
1973	O
,	O
and	O
improved	O
by	O
J.C.	O
Bezdek	O
in	O
1981	O
.	O
</s>
<s>
The	O
fuzzy	O
c-means	O
algorithm	O
is	O
very	O
similar	O
to	O
the	O
k-means	B-Algorithm
algorithm	I-Algorithm
:	O
</s>
<s>
Choose	B-Algorithm
a	I-Algorithm
number	I-Algorithm
of	I-Algorithm
clusters	I-Algorithm
.	O
</s>
<s>
Assign	O
coefficients	O
randomly	O
to	O
each	O
data	B-Application
point	I-Application
for	O
being	O
in	O
the	O
clusters	O
.	O
</s>
<s>
For	O
each	O
data	B-Application
point	I-Application
,	O
compute	O
its	O
coefficients	O
of	O
being	O
in	O
the	O
clusters	O
.	O
</s>
<s>
K-means	B-Algorithm
clustering	I-Algorithm
also	O
attempts	O
to	O
minimize	O
the	O
objective	O
function	O
shown	O
above	O
,	O
except	O
that	O
in	O
K-means	B-Algorithm
,	O
the	O
membership	O
values	O
are	O
either	O
zero	O
or	O
one	O
,	O
and	O
cannot	O
take	O
values	O
in	O
between	O
,	O
i.e.	O
</s>
<s>
In	O
the	O
limit	O
,	O
the	O
memberships	O
,	O
,	O
converge	O
to	O
0	O
or	O
1	O
,	O
and	O
the	O
Fuzzy	O
C-means	O
objective	O
coincides	O
with	O
that	O
of	O
K-means	B-Algorithm
.	O
</s>
<s>
Using	O
a	O
mixture	O
of	O
Gaussians	O
along	O
with	O
the	O
expectation-maximization	B-Algorithm
algorithm	I-Algorithm
is	O
a	O
more	O
statistically	O
formalized	O
method	O
which	O
includes	O
some	O
of	O
these	O
ideas	O
:	O
partial	O
membership	O
in	O
classes	O
.	O
</s>
<s>
This	O
membership	O
coefficient	O
of	O
each	O
corresponding	O
data	B-Application
point	I-Application
is	O
represented	O
by	O
the	O
inclusion	O
of	O
the	O
y-axis	O
.	O
</s>
<s>
In	O
fuzzy	B-Algorithm
clustering	I-Algorithm
,	O
each	O
data	B-Application
point	I-Application
can	O
have	O
membership	O
to	O
multiple	O
clusters	O
.	O
</s>
<s>
The	O
following	O
image	O
shows	O
the	O
data	O
set	O
from	O
the	O
previous	O
clustering	B-Algorithm
,	O
but	O
now	O
fuzzy	O
c-means	O
clustering	B-Algorithm
is	O
applied	O
.	O
</s>
<s>
Next	O
,	O
new	O
membership	O
coefficients	O
for	O
each	O
data	B-Application
point	I-Application
are	O
generated	O
based	O
on	O
clusters	O
centroids	O
,	O
as	O
well	O
as	O
distance	O
from	O
each	O
cluster	O
centroid	O
.	O
</s>
<s>
As	O
one	O
can	O
see	O
,	O
the	O
middle	O
data	B-Application
point	I-Application
belongs	O
to	O
cluster	O
A	O
and	O
cluster	O
B	O
.	O
the	O
value	O
of	O
0.3	O
is	O
this	O
data	B-Application
point	I-Application
's	O
membership	O
coefficient	O
for	O
cluster	O
A	O
.	O
</s>
<s>
Clustering	B-Algorithm
problems	O
have	O
applications	O
in	O
surface	O
science	O
,	O
biology	O
,	O
medicine	O
,	O
psychology	O
,	O
economics	O
,	O
and	O
many	O
other	O
disciplines	O
.	O
</s>
<s>
In	O
the	O
field	O
of	O
bioinformatics	O
,	O
clustering	B-Algorithm
is	O
used	O
for	O
a	O
number	O
of	O
applications	O
.	O
</s>
<s>
Use	O
of	O
clustering	B-Algorithm
can	O
provide	O
insight	O
into	O
gene	O
function	O
and	O
regulation	O
.	O
</s>
<s>
Because	O
fuzzy	B-Algorithm
clustering	I-Algorithm
allows	O
genes	O
to	O
belong	O
to	O
more	O
than	O
one	O
cluster	O
,	O
it	O
allows	O
for	O
the	O
identification	O
of	O
genes	O
that	O
are	O
conditionally	O
co-regulated	O
or	O
co-expressed	O
.	O
</s>
<s>
Thus	O
,	O
fuzzy	B-Algorithm
clustering	I-Algorithm
is	O
more	O
appropriate	O
than	O
hard	O
clustering	B-Algorithm
.	O
</s>
<s>
Fuzzy	O
c-means	O
has	O
been	O
a	O
very	O
important	O
tool	O
for	O
image	O
processing	O
in	O
clustering	B-Algorithm
objects	O
in	O
an	O
image	O
.	O
</s>
<s>
In	O
the	O
1970s	O
,	O
mathematicians	O
introduced	O
the	O
spatial	O
term	O
into	O
the	O
FCM	O
algorithm	O
to	O
improve	O
the	O
accuracy	O
of	O
clustering	B-Algorithm
under	O
noise	O
.	O
</s>
<s>
Image	B-Algorithm
segmentation	I-Algorithm
using	O
k-means	B-Algorithm
clustering	I-Algorithm
algorithms	I-Algorithm
has	O
long	O
been	O
used	O
for	O
pattern	O
recognition	O
,	O
object	O
detection	O
,	O
and	O
medical	O
imaging	O
.	O
</s>
<s>
However	O
,	O
due	O
to	O
real	O
world	O
limitations	O
such	O
as	O
noise	O
,	O
shadowing	O
,	O
and	O
variations	O
in	O
cameras	O
,	O
traditional	O
hard	O
clustering	B-Algorithm
is	O
often	O
unable	O
to	O
reliably	O
perform	O
image	O
processing	O
tasks	O
as	O
stated	O
above	O
.	O
</s>
<s>
Fuzzy	B-Algorithm
clustering	I-Algorithm
has	O
been	O
proposed	O
as	O
a	O
more	O
applicable	O
algorithm	O
in	O
the	O
performance	O
to	O
these	O
tasks	O
.	O
</s>
<s>
Given	O
is	O
gray	O
scale	O
image	O
that	O
has	O
undergone	O
fuzzy	B-Algorithm
clustering	I-Algorithm
in	O
Matlab	O
.	O
</s>
<s>
Depending	O
on	O
the	O
application	O
for	O
which	O
the	O
fuzzy	B-Algorithm
clustering	I-Algorithm
coefficients	O
are	O
to	O
be	O
used	O
,	O
different	O
pre-processing	O
techniques	O
can	O
be	O
applied	O
to	O
RGB	O
images	O
.	O
</s>
