HANG et al.: DISCRIMINANT SUBSPACE ENSEMBLE FOR HS IMAGE SPATIAL–SPECTRAL FEATURE FUSION 785
Fig. 2. Flowchart of the proposed method.
in this feature matrix and improve the discriminant ability of
feature representation, we use MDA to map Y into a new
subspace [33], in which the intraclass scatter is minimized
and the interclass scatter is maximized. Assume that L =
[μ
1
,μ
2
,...,μ
r
] ∈ R
l×r
and R =[ν
1
,ν
2
,...,ν
c
] ∈ R
ω
2
×c
are
two transformation matrices. Then, the projection of Y onto the
(r × c)-dimensional space L
R can be expressed as
Z = L
T
YR. (1)
Suppose there are N training pixels and C different classes
to be classified. The jth training pixel is Y
j
,wherej ∈
{1, 2,...,N}. M and M
i
denote the mean of all training pixels
and training pixels in class Π
i
,i∈{1, 2,...,C}, respectively.
N
i
is the number of training pixels in class Π
i
.Togetthe
optimal projection matrices, we maximize the ratio of the
interclass scatter matrix and the intraclass scatter matrix for
MDA as for LDA. Fortunately, the total scatter of the projected
pixels can be measured by the trace of their covariance matrix.
Therefore, the objective function can be described as follows:
J =
S
B
S
W
(2)
where S
B
denotes the interclass scatter matrix, and S
W
denotes
the intraclass scatter matrix with the following definitions:
S
B
=tr
C
i=1
N
i
(
˜
M
i
−
˜
M)(
˜
M
i
−
˜
M)
T
S
W
=tr
C
i=1
Z
k
∈Π
i
(Z
k
−
˜
M
i
)(Z
k
−
˜
M
i
)
T
. (3)
Since
˜
M
i
=
1
N
i
Z
k
∈Π
i
Z
k
=
1
N
i
Y
k
∈Π
i
L
T
Y
k
R = L
T
M
i
R
thus (3) can be rewritten as
S
B
=tr
C
i=1
N
i
L
T
(M
i
− M )RR
T
(M
i
− M )
T
L
S
W
=tr
C
i=1
Y
k
∈Π
i
L
T
(Y
k
− M
i
)RR
T
(Y
k
− M
i
)
T
L
.
The optimal transformations L and R will maximize (2). The
same as in [33], we utilize an iterative algorithm to optimize L
and R. In particular, for a fixed R, (2) can be rewritten as
J =
L
T
S
R
B
L
L
T
S
R
W
L
(4)
where S
R
B
=
C
i=1
N
i
(M
i
− M )RR
T
(M
i
− M )
T
,andS
R
W
=
C
i=1
Y
k
∈Π
i
(Y
k
− M
i
)RR
T
(Y
k
− M
i
)
T
. Hence, the opti-
mal solution of L consists of r eigenvectors corresponding to
r maximal eigenvalues by computing an eigendecomposition
on (S
R
W
)
−1
S
R
B
. Subsequently, the optimal solution of R can be
obtained when L is fixed. The whole process will iterate until a
predefined convergence condition is arrived.
C. Learning Subspace Ensemble With Random Sampling
Ensemble learning has been proved to be an efficient tech-
nique to improve the stability and accuracy of single weak
classifier by training several different classifiers and combin-
ing their decisions [34], [35]. When forming ensemble, the
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