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3 Regularized KLPDA (RKLPDA)
Obviously,
rank
(
S
w
)
n
.If
S
w
is full rank, i.e.,
rank
(
S
w
)=
n
,then
S
w
is nonsingular and there will be no singularity problem when the
matrix (
S
w
)
−
1
S
b
is computed. Otherwise, if
rank
(
S
w
)
<n
, where this is always
true in face recognition, the SSS problem will occur. For this case, eigenvalue
regularization (ER) scheme proposed in [6] is employed to
S
w
.First,perform
the eigenvalue decomposition of
S
w
,
S
w
=
Φ
w
Λ
w
Φ
w
,where
Φ
w
=
rank
(
K
XX
)
≤
≤
ϕ
i
}
n
i
=1
{
is
the eigenvectors of
S
w
corresponding to the eigenvalues
Λ
=
diag
(
λ
i
)
n
{
}
i
=1
,
λ
1
≥···≥
=0,
r
is the rank of the
S
w
. As ER scheme, the
eigenspace of
S
w
is decomposed into reliable face space
FS
=
λ
r
≥
λ
r
+1
=
···
ϕ
k
}
m
{
k
=1
, unstable
ϕ
k
}
r
ϕ
k
}
n
k
=
r
+1
noise space
NS
=
{
k
=
m
+1
,andnullspace
∅
=
{
. The starting
point of noise region
m
is set by
λ
m−
1
=
max
λ
k
|
λ
k
<
(
λ
med
)+
μ
(
λ
med
−
{∀
λ
r
)
,where
λ
med
λ
k
|
is the point near the center of the
noise region,
μ
is a constant, in all experiments of this paper
μ
is fixed to be 1
for simple. Utilizing the spectrum model
}
=
median
{∀
k
≤
r
}
λ
k
=
α/
(
k
+
β
), the eigenvalues are
predicted as
⎧
⎨
λ
k
,
k
≤
m
(
facespace
)
λ
k
=
α/
(
k
+
β
)
,
m<k
≤
r
(
noisespace
)
(13)
⎩
α/
(
r
+1+
β
)
,r<k
≤
n
(
nullspace
)
where the parameters
α
and
β
are given by letting
λ
1
=
λ
1
and
λ
m
=
λ
m
.Then
using the predicted eigenvalues to weight the corresponding eigenvectors, it has
ϕ
k
/
λ
k
}
Φ
w
=
n
k
=1
{
(14)
To obtain more features,
S
t
is adopted instead of
S
b
in discriminant feature
extraction, since only no more than
C
−
1 features will be obtained when utilizing
S
b
, while
n
1 features might be obtained when utilizing
S
t
. The projection of
−
S
t
in the space spanned by the regularized eigenvectors is
S
t
=(
Φ
w
)
T
S
t
Φ
w
(15)
S
t
:
Φ
t
=
The transformation matrix is consisted of the
d
leading eigenvectors of
ϕ
i
}
i
=1
.
Therefore, for a face image vector
x
,
x
{
N×
1
, let its projection in kernel
space be
x
φ
, then the discriminant features by the proposed RKLPDA method
is given by
y
=
A
T
x
φ
=(
X
φ
Ψ
)
T
x
φ
=(
X
φ
Φ
w
Φ
t
)
T
x
φ
=
Φ
t
∈
R
Φ
w
(
X
φ
)
T
x
φ
=
Φ
t
Φ
w
K
Xx
(16)
where
K
Xx
=
i
=1
is kernel function.
{
k
(
x
i
,x
)
}
4 Experiments and Discussions
4.1 Face Databases and Image Preprocessing
In all experiments reported in this work, images are preprocessed following the
CSU Face Identification Evaluation System [11]. The ORL face database [12]
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