IEEE Geoscience and Remote Sensing Magazine - December 2021 - 49

where m is a regularization parameter to balance the sparsity
term and the data fidelity term,
diag C 0=^h
= means that the affine subspace model is adopted.
Although SSC has shown significant potential in hyperC
1T
1
is to avoid the
trivial solution caused by representing each pixel by itself,
and
spectral clustering, due to some shortcomings, e.g., ignoring
the importance of the spatial information and the nonlinearity
of HSIs, the clustering performance is still limited. Based
on this fact, in recent years, many enhanced subspace clustering
algorithms have been proposed to further improve the
clustering performance and exploit the potential of subspace
clustering. On the basis of the working mechanism, such
methods can be coarsely summarized into three main categories:
1) spectral-spatial subspace clustering, 2) multiview
subspace clustering, and 3) kernel subspace clustering.
SPECTRAL-SPATIAL SUBSPACE CLUSTERING
Spectral-spatial subspace clustering methods focus on exploring
the spectral-spatial duality of HSIs within the selfrepresentation
framework to reduce the influence of saltand-pepper
noise and enhance the spatial homogeneity of
the clustering result. By incorporating spatial information
to help spectral analysis in the representation domain, the
piecewise smoothness of the representation coefficient matrix
can be effectively enhanced, and the representation bias
can be reduced to some degree. As a result, the clustering
performance can be effectively improved. In general, with a
certain spatial constraint, the spectral-spatial subspace clustering
model can be formulated as follows:
min G a+= +
C CC subject to YYCN ,
^^
R
hh
(16)
where R $^h denotes the spatial regularization term and a
is a regularization parameter to trade off the importance
between the spectral term and the spatial term.
A typical example is the spectral-spatial SSC SC4^h algorithm
[40]. It promotes the target pixels to be represented
by highly related atoms via a weighting strategy and incorporates
the spatial neighborhood information to generate
an integrated self-representation model by constructing an
eight-neighborhood local average spatial regularization,
based on the assumption that the average coefficient in the
local small window should be close to the coefficient of the
center pixel. Considering that the assumption of the local
average constraint cannot be satisfied in areas with a complex
land cover distribution, a new
, norm regularized SSC
2 -
(L2-SSC) algorithm was proposed [41]. It incorporates spatial
information in a more refined way by constructing an
efficient four-neighborhood
2 -
which further improves the clustering performance.
In addition, in [128], a spectral-spatial SSC based on 3D
edge-preserving filtering (SSC-3DEPF) algorithm was put forward.
It utilizes 3D edge-preserving filtering for the sparse coefficient
matrix obtained by SSC to extract the spectral-spatial
information to generate a more accurate coefficient matrix,
which is favorable for clustering. In [129], a joint SSC (JSSC)
method was proposed to make use of spatial information
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
, norm spatial regularization,
through joint sparse representation. It forces the pixels in a
spatial neighborhood to share the same sparse basis. In [44],
based on the sparse coefficient matrix learned by SSC, two enhanced
methods were put forward to construct a more accurate
adjacency matrix, i.e., cosine-Euclidean (CE) and CE dynamic
weighting (CEDW). These two methods simultaneously utilize
the spectral and spatial information, with the cosine similarity
exploited to measure the spectral similarity and Euclidean distances
utilized to incorporate the spatial information.
Moreover, in [130], a Laplacian-regularized low-rank
subspace clustering (LLRSC) algorithm was proposed. It
incorporates three different Laplacian regularizations into
the low-rank subspace clustering (LRSC) model to explore
the importance of the correlation information of HSIs, and it
achieves good performance in HSI band selection. In [131],
a spectral-spatial LRSC (SS-LRSC) model was developed. It
utilizes a new modulation strategy to incorporate the correlations
into the low-rank representation matrix through weighting
and local spatial bilateral filtering, which performs well for
HSIs. Furthermore, in [132], a Gaussian kernel dynamic similarity
matrix-based SSC (GKD-SSC) method was introduced.
It improves the quality of the adjacency matrix by simultaneously
utilizing the sparse coefficient matrix obtained by SSC
and the Gaussian kernel similarity based on the distances
between pixels after PCA processing.
Considering the large computational complexity of
sparse recovery-based methods, a novel total variation
(TV)-regularized collaborative representation clustering algorithm
with a locally adaptive dictionary (TV-CRC-LAD)
was proposed for HSIs [21]. This approach exploits the collaborative
and competitive relationships among pixels from
all classes in the self-representation process and deals with
the clustering task within the collaborative representation
framework, with less complexity. Then, it reduces the serious
interferences from unrelated atoms in the whole dictionary
by constructing a locally adaptive dictionary for each target
pixel and integrates spatial information to enhance the
piecewise smoothness of the coefficient matrix via the TV
regularization. As a result, the TV-CRC-LAD may perform
better for HSIs.
In addition, to overcome the large computational complexity
and the time and memory cost of the self-dictionarybased
methods, a sketched subspace clustering method was
developed [133]. It conducts the self-representation learning
under a sketched dictionary with much fewer atoms obtained
by random projection, which reduces the computational
complexity and enhances the scalability of the model to a
large degree. Then, the sketched subspace clustering method
was introduced to HSIs, and based on it, through the TV constraint
to incorporate spatial information, an enhanced method
was proposed, i.e., TV sketched subspace clustering [134].
Furthermore, considering that pixel-based clustering
methods generally encounter several obstacles and that they
were easily affected by salt-and-pepper noise and could not
accurately model the spatial neighborhoods of hyperspectral
pixels with various shapes, several object/super-pixel
49
IEEE Geoscience and Remote Sensing Magazine - December 2021

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