Digital Signal Processing Reference
In-Depth Information
v
(
t
)
}
3. Calculating the fuzzy cluster centers: Compute all cluster centers
{
using
U
(
t
−
1
)
with the equation specified in (
2.9
).
4. Update membership function
U
(
t
)
: Update
U
(
t
)
using
v
(
t
)
with the equation spec-
ified in (
2.8
).
5. Check convergence condition: Check the previous defined convergence behavior
as
by computing
v
(
t
−
1
)
.
v
(
t
)
−
=
(2.45)
6. If
<
ε
or a preset loop count
N
t
is reached then terminate; otherwise set
t
=
t
+
1 and go to Step 3, where
ε
is the preset terminating criterion.
In (
2.45
), the superscript
t
denotes the number of iterations. If the changes of the
class centers are less than a predefined criterion
ε
that means the objective function
(
,
)
J
m
is no longer decreasing. The final segmentation result is achieved.
To improve orientation sensitivity (OS), Schmid in [
33
] suggested a modified
FCM algorithm (OSFCM) by modifying
A
j
described in (
2.42
)as:
U
V
;
X
V
j
L
j
V
j
A
j
=
,
(2.46)
where
L
j
denotes the diagonal matrix containing the inverse of eigenvalues and
V
j
represents the unitary matrix lining up the corresponding eigenvectors of the fuzzy
covariance matrix
C
j
for the
j
th cluster. The fuzzy covariance matrix for the
j
th
cluster
C
j
is given by
q
=
1
u
jq
x
q
x
q
−
v
j
v
j
.
N
1
N
C
j
=
(2.47)
C
j
)
−
1
.
From simple matrix derivations, it is obvious that
A
j
=(
2.4.2
Eigen-Based FCM Algorithms
In this section, we combine the FCM classification with eigen-subspaces projec-
tion together to achieve effective color segmentation. By using eigenvectors, we can
transform the original color space into the modal coordinate system of the desired
color as
T
x
q
=[
φ
q
z
q
=[
w
q
w
2
w
3
]
ϕ
q
ψ
q
]
.
(2.48)
Now, the first-principal elements,
φ
q
for
q
=
1
,
2
,...,
N
specify the signal subspace
whereas the second and the third elements
ψ
q
and build the noise subspaces.
The vector
x
q
is defined as in (
2.40
). With signal and noise subspaces, we de-
velop two eigen-based FCM detection procedures, the separate eigen-based FCM
(SEFCM) and the coupled eigen-based FCM (CEFCM) methods. The main proce-
dures of eigen-based FCM are shown in Fig.
2.20
. First, we compute the covariance
matrix
R
s
of the desired color samples from the RGB color planes followed by
ϕ
q
and
