L1/2正则化与稀疏表示：光谱空间高光谱图像分类的新策略
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更新于2024-08-30
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本文主要探讨了"使用L1/2正则化低秩表示和基于稀疏表示的图割进行光谱空间高光谱图像分类"这一主题，发表在2015年6月的《IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing》第8卷第6期。高光谱成像技术因其能收集数百个连续的窄波段信息，对于地表物质识别具有巨大潜力。这些数据集中的空间信息对于提高分类精度至关重要，因为它假设邻近像素很可能属于同一类别。
作者Sen Jia、Xiujun Zhang和Qingquan Li提出了两种主要的方法来利用这种空间信息。首先，他们将每个像素及其周围区域分解为低秩形式，这有助于高效地整合空间特征到光谱特征中。低秩表示通常能够捕捉数据中的潜在结构，但为了更精确地描述这种结构，文章引入了L1/2范数正则化。L1/2范数结合了L1和L2范数的优点，它在保持数据稀疏的同时，对较大的元素给予更强的约束，有助于处理数据的噪声和异常值。
其次，作者开发了一种离散算法，用于解决低秩表示和L1/2正则化的联合优化问题。这种方法的优势在于能够在保持模型简洁性的同时，充分利用空间信息增强光谱分类的鲁棒性和准确性。通过这种方式，图割（Graph Cuts）策略被应用到光谱空间融合中，这是一种全局优化方法，它在处理高光谱图像时可以有效地寻找最优的类别划分。
本文的研究旨在提升光谱空间高光谱图像分类的性能，通过低秩表示和稀疏表示相结合的方式，既考虑了数据的内在结构，又充分利用了空间邻域的信息。这种创新的方法在遥感和地球观测领域具有重要的理论价值和实际应用前景，对于精准的地表物质识别和环境监测具有重要意义。
JIA et al.: SPECTRAL–SPATIAL HYPERSPECTRAL IMAGE CLASSIFICATION USING LHALFLRR AND SR-BASED GRAPH CUTS 2475
II. RELATED WORK
In order to facilitate the following description, some nota-
tions are introduced. Let R ∈ R
X×Y ×L
denote the hyperspec-
tral image cube that contains two spatial dimensions X and
Y (pixels) and one spectral dimension L (wavelength). R
xy
is the reflectance spectral signature at spatial location (x, y),
which provides the ability to identify various materials in the
ground scene. S =
R
uv
∈ R
L×ω
2
, max(|u − x|, |v − y|) ≤
ω
2
is the spatial neighborhood of R
xy
(including R
xy
)using
boxes of predefined spatial window size ω × ω pixels (here
ω is an odd number and greater than or equal to 3, and
ω
2
rounds the elements of
ω
2
to the nearest integers less than or
equal to
ω
2
). Considering supervised hyperspectral classifica-
tion, denote by A
i
=[r
1
, r
2
,...,r
n
i
] ∈ R
L×n
i
the training
set of the ith class, and each column r
j
(j =1, 2,...,n
i
) is
an L-dimensional spectral signature. Suppose that there are K
classes present in the scene, and let A =[A
1
, A
2
,...,A
K
] ∈
R
L×n
be the concatenation of the n training samples from all
classes, where n = n
1
+ n
2
+ ···+ n
K
.
A. Low-Rank Representation
With the increase of the spatial filtering window, it is natural
to expect that more pixels of different classes could be included
in the spatial neighborhood S. Due to the high degree of sim-
ilarity among spectral signatures within the same class, LRR
can be applied to investigate the spatial neighborhood informa-
tion more precisely. That is, for a given spatial adjacency matrix
S ∈ R
L×ω
2
, it can be decomposed into a low-rank matrix D
and an error matrix E, which can be mathematically described
as follows:
min
D,E
rank(D)+λE
s.t. S = D + E (1)
where rank(D) represents the rank of matrix D, E
is a cer-
tain regularization strategy for the error, and λ is a parameter.
Especially, it has been pointed out that a “dictionary” matrix
should be introduced to better handle the multiple subspaces of
the data (the data itself can be directly taken as the “dictionary”
matrix, and the coefficient matrix is denoted as Z) [19], and
E
2,0
=#{i : [E]
:,i
2
=0} is usually adopted to character-
ize the phenomena that the data vectors are mildly or heavily
corrupted [37]; hence, a more reasonable objective function
might be
min
Z,E
rank(Z)+λE
2,0
s.t. S = SZ + E. (2)
However, the above optimization problem is difficult to
solve due to the discrete nature of the rank function. Let
σ
1
,σ
2
,...,σ
k
be the all sorted singular values of Z, and the
2,1
norm is a good relaxation of the
2,0
norm. So, a low-rank
recovery to S can be obtained by solving the following convex
optimization problem:
min
Z,E
Z
∗
+ λE
2,1
s.t. S = SZ + E (3)
where ·
∗
denotes the nuclear norm of a matrix, i.e., the
sum of the singular values of the matrix. After obtaining an
optimal solution (Z
∗
, E
∗
), the data can be recovered using
SZ
∗
(or S − E
∗
). Since rank(SZ
∗
) ≤ rank(Z
∗
), SZ
∗
is also
a low-rank recovery to the original data S.
Unfortunately, although solving (3) can result in low-rank
solution of the considered problems, it needs some strict con-
ditions (i.e., the restricted isometry property, for short, RIP
conditions) and may yield the matrix with much higher rank
than the real one [38], [39].
B. SR-Based Classification (SRC)
For a test sample y ∈ R
L
, it can be assumed that the training
samples from a single class lie on a subspace, and y from a cer-
tain class will approximately lie in the linear span of the training
samples of the same class. As a generalization of nearest neigh-
bor (NN) and Nearest subspace (NS) classifiers, SR searches
for the most compact representation of a test sample using only
a few basic elements in the training set [30]. According to the
theory of compressed sensing [40], the SR of y can be obtained
by optimizing the following optimization function [41]:
(
0
):
α =argminα
0
subject to Aα = y (4)
where ·
0
denotes the vector
0
-norm (the number of nonzero
elements in a vector), which is the best sparsity measure
available. Solving the optimization equation (4) turns out to
be a nonconvex NP-hard problem, and is computationally
intractable [42]. On the other hand, the theory of compressed
sensing has showed that if the solution of α is sparse enough,
the
0
-norm minimization problem can be relaxed and solved
by optimizing the following objective function [41], [43]:
(
1
):
α =argmin
α∈R
n
{y −Aα
2
2
+ τ α
1
} (5)
where ·
1
denotes the
1
-norm, which is the sum of the abso-
lute value of the vector, and τ is a scalar weight that quantifies
the relative importance of minimizing α. Unlike the
0
-norm,
the
1
-norm is convex and can be cast as a linear program. After
that, the test sample y is assigned to class i if the distance
between y and A
i
α
i
(the approximation of y using only the
coefficients associated with the ith class) is the smallest among
all classes, i.e.,
Class(y)= argmin
i∈{1,2,...,K}
y −A
i
α
i
2
2
. (6)
III.
1/2
REGULARIZED LOW-RANK REPRESENTATION
(LHALFLRR)
A.
1/2
Regularization-Based Relaxation of Rank Minimization
Problem
Compared with the rank and nuclear norm measures in
(2) and (3) respectively, it can be noticed that rank(Z)=
ω
2
i=1,σ
i
=0
σ
0
i
= σ
0
is the number of nonzero elements of
singular values in Z, whereas Z
∗
=
ω
2
i=1,σ
i
=0
σ
1
i
= σ
1
is the sum of the singular elements. Because σ
1
is the con-
vex envelope of σ
0
, (2) can be relaxed as equation (3) and
rank(Z) can be replaced by Z
∗
. It is obvious that model (3)
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