IEEE Geoscience and Remote Sensing Magazine - June 2021 - 63

how best to embed well-studied nonconvex regularizers
to the DL architecture should also be further analyzed.
◗ Real application: Until now, most HS image-restoration
methods have been evaluated on simulated experiments.
However, in most cases, the evaluation indices fail to
predict the accuracy of the real HS image-restoration
results. From another side, the noise distribution in the
real, noisy HS images is complex. How best to verify the
related methods on real HS images should also be further
analyzed. From another side, the training samples
in the real application are always limited. The blind and
unsupervised approaches will become the mainstream of
future, real HS image restoration.
DR
HS DR and feature extraction have long been a fundamental
but challenging research topic in HS RS [55], [56]. The main
reasons mainly lie in the following aspects. Due to highly
correlated characteristics among spectral bands, HS images
are subject to information redundancy, which can hurt the
ability to discriminate materials under certain extremely
conditioned cases (the curse of dimensionality). Plus, as
the HS dimension gradually increases along with the spectral
domain, large storage capability and high-performance
computing are needed. Furthermore, these dimensionreduced
features are usually applied for high-level classification
or detection tasks [57], [58]. Recently, many works
based on nonconvex modeling have proved to be effective
for automatically extracting the dimension-reduced features
of HS images. Linking with (2), the DR task can be generalized
to the following optimization problem:
min ff2
1
fU,X
UU-() XX,
Y
2
F
s.t.
!C,
(7)
X =+<<
() tr()
where fU(*) denotes the transformation from the original HS
space to dimension-reduced subspaces with respect to the
variable set
U , and X is the low-dimensional representations
of Y. Revolving around the general form in (7), we review
currently advanced DR methods from three different aspects:
unsupervised, supervised, and semisupervised models.
UNSUPERVISED MODEL
Nonnegative matrix factorization (NMF) [59] is a common
unsupervised learning tool that has been widely applied in
HS DR. These works can be well explained by (7); the NMFbased
DR problem can be then formulated as
Q$$,X00
mi YXQX Q() (),
n 2
1
-+WX+
2
F
(8)
where Q denotes the combination coefficients and X()U and
X() Q are the potential regularization terms for the variables
X and Q, respectively. Prior to the current ones, there have
been some advanced NMF-based works in HS DR. Gillis et al.
[60] used sparse NMF under approximations for HS data
analysis. Yan [61] proposed a graph-regularized orthogonal
JUNE 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
UUYLYU U ,
2
F
while the regularizer with respect to X can be given by
XXLX
W() tr().
<
=
(10)
NMF (GONMF) model with application to the spatial-spectral
DR of HS images. Wen et al. [62] further extended GONMF
by combining multiple features for HS DR. Rasti et al. [63]
designed an orthogonal TV component analysis (OTVCA)
approach for HS feature extraction. Moreover, the HS data are
directly regarded as a high-dimensional tensor structure in
[64], where the low-rank attribute is fully considered in the
process of low-dimensional embedding. In detail, we summarize
the regularization and constraints of the aforementioned
methods as
◗ sparsity [60]: QQ
X =
()
◗ graph regularization [61]:
XXLX
WR==<
()
i
s
=1tr(),s.t. XX I
i
<
XHXHXQQI
◗ low-rank graph [64]: XX XX<
WR=+ =<
()
() () () , s.t.
W =
i
r
=1
hi
22
vi
1
* + tr().L
LD W=- is the Laplacian matrix, where DWij is,,j
ii =R
the degree matrix and W is the graph (or manifold) structure
of X [65].
21,
:: , and
02,1
,
-
: denote the 0
*
, -norm [66],
, norm [67], and nuclear norm [68], respectively.
Another type of unsupervised DR approach is graph embedding,
also known as manifold learning, which can also be
grouped into (7) well (according to [69]):
min 2
1
UX,
XUYX U s.t. XX I,
-+WX+=R
2
F
() ()
(9)
where U denotes the to-be-estimated projection matrix that
bridges the high-dimensional data Y with the low-dimensional
embedding X. The regularization term for the variable
U can usually be expressed as
◗ TV [63]:
W() tr(),s.t.==
<<
XX I
◗ multigraph regularization [62]:
XXLX
(11)
The main difference between these manifold learning-based
DR approaches lies in the graph construction.
W. Ma et al. [70] integrated the k-nearest neighbor (KNN)
classifier with several representative manifold learning
algorithms, i.e., locally linear embedding [71], Laplacian
eigenmaps [65], and local tangent space alignment [72],
for HS image classification. Huang et al. [73] embedded
the sparse graph structure, which is performed by solving
a 1
, -norm optimization problem, for the DR of HS images.
He et al. [74] extended the work of [73] by generating a
weighted sparse graph. Hong et al. [75] developed a new
spatial-spectral graph for the DR of HS images, called robust
local manifold representation (RLMR), by jointly taking
the neighboring pixels of a target pixel in the spatial and
spectral domains into account. An et al. [76] attempted to
learn the low-dimensional, tensorized HS representations
on a sparse and low-rank graph.
63
IEEE Geoscience and Remote Sensing Magazine - June 2021

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