procession. The specific values for threshold selection are
discussed in the experimental section.
Endmember Error Constraint Based on K-SVD
Due to the complexity of the real object distribution, there are
many features in a scene. For each mixed pixel, it is a mixture
of a small number of features, which satisfies the sparsity of
the representation coefficient. Following the objective func-
tion of
K-SVD
, the objective function of endmember extraction
can be expressed as:
min
, ,...,
s
x As
i
i
i F
i
N
≥
−
=
0
2
1 2
for
(7)
First, assume that both
A
and
S
are fixed, and discuss that
only one column in the endmember
a
p
and the
p
th row in
abundance matrix
S
T
P
corresponds to it. The nonzero indices
of
S
T
P
indicated that all those pixels which use the
a
p
end-
member for representation and the coefficient in linear combi-
nation. Therefore, Equation 7 can be rewritten as:
X AS X a s
X a s
a s
E a s
−
= −
= −
−
= −
=
≠
∑ ∑
F
j j
T
j
P
F
j j
T
j p
p p
T
F
p p p
T
F
2
1
2
2
2
(
)
(8)
Equation 8 represents the error between the image and
the linear combination of endmember matrix and abudance
matrix. In the process of Equation 8 decomposition, the error
can be represented by two parts. One is the error when the
endmember
a
p
is not taken into account, and the other error
is from the
A
and
S
reconstruction. The multiplication
AS
can be decomposed to the sum of
P
rank-1 matrices. Then,
P
– 1 terms are assumed fixed, and only one column,the
p
th
in question. Therefore, the problem of minimizing the whole
error transforms to finding a matrix which best approximates
the error matrix
E
p
.The matrix
E
p
denotes the error for all the
N
pixels when the
p
th endmember is removed.
E X
a s
p
j j
T
j
j p
P
= −
= ≠
∑
1
,
where
a
p
is the
p
th endmember, and
S
T
P
i
abundance vector. The
SVD
finds the closest rank-1 matrix (in
Frobenius norm) that approximates
E
p
, in this way, the error
defined in Equation 9 will be minimized.
We obtain
U
and
V
after the
SVD
decomposition of the
constraint error matrix
SVD
(
E
p
) =
U
Δ
V
T
. The column vectors
of these two matrices are orthogonal basis vectors, which are
respectively the left and right singular vectors of the corre-
sponding odd values. The symbol
Δ
denotes a singular value
matrix in which singular values after
SVD
are arranged from
large to small, indicating that the energy
E
p
is distributed from
large to small in several directions. During the
SVD
decompo-
sition, the locations of nonzero values in
S
T
P
are not the same,
creating a divergence problem. Therefore, it is necessary to
define a group of indices
ω
p
= {
i
|1
≤
i
≤
P
,
S
T
P
(
i
)
≠
0} to note the
position of nonzero samples. After discarding of the zeros
entries, resulting with
Ω
p
∈
R
N
×
|
ω
p
|
, with ones on the (
ω
p
(
i
),
i
)
th
entries and zeros elsewhere. In this way, the multiplication
S
R
P
=
S
T
P
Ω
p
creates a vector of length |
ω
p
|. Similarly, multipli-
cation
X
R
P
=
X
Ω
p
creates a matrix of size
N
×
|
ω
p
|
. Meanwhile,
E
R
P
=
E
P
Ω
p
means the error matrix corresponds to samples that
use endmember
a
p
.
The proposed method calculates the error matrix by suc-
cessively stripping the contribution of the endmember to
the observation data
X
, and uses the
SVD
to decompose the
calculated error. The matrix updates the endmembers and
their corresponding coefficient matrices to obtain the optimal
endmembers. While the operation process, it is necessary to
ensure that the column vector of the endmember matrix
A
remains normalized.
Gross Errors Elimination
E
p
consists of three parts: The first part is (
a
p
S
T
p
–
a
pt
S
T
pt
),
which represents model error caused by spectral unmixing
model. The second part is
a
pt
S
T
pt
, which represents endmem-
bers and their corresponding abundances under the condition
without noise. The third part
ε
is the residual term caused by
the remaining factors and can be regarded as noise. Therefore,
E
p
defines as Equation 10.
E a s
a s a s
a s
p p p
T
p p
T
pt pt
T
pt pt
T
=
+ =
−
+
+
µ
µ
(
)
(10)
Our goal is to get true endmember
a
pt
in the second part.
K-SVD
will make the sum of the first part and the third part
tend to
0
. The model error (
a
p
S
T
p
–
a
pt
S
T
pt
) is closely related to
a
pt
and
S
T
pt
. When the model error is
0
, the ideal endmember
and its corresponding abundance can be obtained without
considering the noise. Nevertheless, due to the existence of
noise term
ε
, during the iteration of
K-SVD
, the model error (
a
p
S
T
p
–
a
pt
S
T
pt
) actually tends to –
ε
. Therefore, the value of
ε
will
directly affect endmembers’ optimization accuracy.
The sources of the noise term
ε
, including but not limited
to the following reasons: 1) The number of real endmembers
in the image is often more than the number of extracted end-
members; that is, the extracted endmembers cannot represent
the observed image perfectly. 2) There is noise in the observed
image itself. Sources of noise are coupled to each other, and
cannot be distinguished and modeled accurately. Therefore,
in order to minimize the impact of
ε
endmember optimiza-
tion, we define the residual matrix
R
δ
as:
R X AS r r r
δ
δ δ
δ
= − =
[ , ,
...,
]
1 2
L T
(11)
where
r
i
δ
, (
i
= 1, 2, …,
L
) denotes residual vector for each
band.
R
is different from
E
p
, though they look familiar.
R
δ
of the overall observed data.
E
p
is the
N pixels when the
p
th endmember is
ng –
r
i
δ
and treating it as background noise
in the image, the effect of the residual term on the endmem-
ber optimization can be weakened. According to –
r
i
δ
histogram
curve, the values outside the confidence interval are removed.
Single-band residual mean –
r
i
δ
defined as follows:
r
r
δ
δ
i
i
j
N
N
j
=
=
∑
1
1
( )
(12)
The observed data after removing the background noise
can be expressed as:
X X r
( )
( )
i j
i j
i
,
,
=
−
δ
(13)
The full description of the method is described below as:
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
December 2019
881
