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defined on the basis of the Gaussian mixture model and the ratio of the between-
class variance to the within-class one of the feature values.
The weak clusters are constructed by using only a few facial feature values with
high usefulness values. That is, the feature vectors
X
i
,
1
,...,
X
i
,
N
}
{
are used as the in-
put of the k-means clustering algorithm.
X
i
,
j
is given as
X
i
,
j
=(
f
i
,
r
1
(
)
,...,
f
i
,
r
m
(
))
x
j
x
j
,
where
f
i
,
r
k
is the
k
-th most useful facial feature value (i.e., the facial feature value
with the
k
-th highest usefulness value). The number of feature values
m
is deter-
mined experimentally. The algorithm to determine the usefulness of a facial feature
value
f
i
,
j
is shown in Algorithm 1.
Algorithm 1.
Determination of the usefulness of a facial feature value
Input:
A set of facial feature values for each frame image
φ
=
{
f
i
,
j
(
x
1
)
,...,
f
i
,
j
(
x
N
)
}
and
the number of weak clusters
K
.
Procedure:
1: Compute the Gaussian mixture model
P
by using φ through the EM algorithm. The
number of Gaussian distributions is
K
.
P
is given by Equation (1).
exp
K
k
=
1
w
k
2
1
2πσ
−
(
f
i
,
j
(
x
)
−
μ
k
)
P
(
f
i
,
j
(
x
)) =
(1)
k
2σ
k
2
k
where,
w
k
is the mixing coefficient of the
k
-th Gaussian distribution.
μ
k
and
σ
denote
the mean and variance of the
k
-th distribution, respectively.
2: Compute the usefulness value
U
using Equation (2).
k
=
1
(
μ
k
−
μ
)
2
U
=
∑
(2)
k
2
k
∑
1
σ
=
where, μ is the mean value of μ
k
(
k
=
1
,...,
K
)
. Note that we do not use the mixture
coefficients in the computation of the usefulness values.
4.4
Construction of Strong Clusters
The eight sets of weak clusters are integrated into a set of strong clusters according
to the CSPA-based ensemble clustering algorithm.
First, a similarity matrix is computed from the sets of weak clusters. An element
in the similarity matrix corresponds to the similarity between the facial feature val-
ues of arbitrary two frame images. The similarity is defined as the number of weak
clusters containing both of the two frame images. In this case, the minimum and
maximum values of the similarity are 0 and 8 respectively, because there are eight
sets of weak clusters.
Next, the initial strong clusters are defined so that a strong cluster contains a sin-
gle frame image. Then, two most similar strong clusters are integrated into a single
strong cluster. The similarity is determined on the basis of the similarity matrix. The
integration step is repeated until the number of strong clusters becomes
K
.
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