IEEE Geoscience and Remote Sensing Magazine - March 2022 - 24

Furthermore, [107] proposed collaborative sparse HS unmixing
using an
min{, }
,
20 ,
inspired by an iterative, hard-thresholding
algorithm.
The sparsity regularizers adopted in [1], [7], [8], [93]-
[96], and [205] can effectively promote the sparseness of
abundances and significantly improve the unmixing performance,
especially for 1
, and , norms. However, they fail
05 .
to explore the other intrinsic characteristics of HSIs. Thus,
relevant approaches have been developed by considering
more constraints or extending the structures in [201]-[203].
A weighted -norm1
,
regularizer [97], [98] encourages the
sparsity of the abundance matrix from the column perspective
(i.e., spectral domains), thus greatly enhancing the unmixing
performance. By further considering the sparsity of
abundances in the spatial domains, a double-reweighted
1 -norm
,
regularizer pursues sparser abundances [99], [100].
Nevertheless, they are sensitive to noise contamination.
Collaborative sparsity is an excellent method for enforcing
sparsity among the rows [101], [102] while ignoring spatial
information. Therefore, the authors in [103] recommended
applying the weighted -norm
,
21,
regularizer on blocks and
introducing TV regularization. A group sparsity regularizer
is successful in combining sparsity with a spatial group
structure [104], [105], but how best to effectively group the
abundances remains to be solved. Approaches focus on
producing sparser results based on an 0
, norm [106], [107],
[206], resulting in more efforts for the solution of the model.
LRR AND SPARSE UNMIXING
The assumption that adjacent pixels have similar fractional
abundances relative to the same endmember set implies
that the matrix formed by their corresponding abundance
vectors should be low rank. LRR seeks the lowest rank representation
of the observed data in the term of a suitable
dictionary. Inspired by LRR, researchers have employed it
to the HS unmixing problem to make the abundance estimation
more accurate. Qu et al. utilized LRR for semisupervised
sparse unmixing for the first time [108] and proposed
a soft joint recovery method by LRR, instead of enforcing
strict joint sparsity on all adjacent pixels. Based on the
LMM, the unmixing model with LRR can be described as
the following optimization problem:
XX YAXX
X
)=- * 0,
argminrank s.t (),.
where Y RLN
!
and X)
#
the endmember matrix, X RMN
#
!
= 0,
!
(36)
is the given HSI data matrix, A RLM
#
is
is the abundance matrix,
is the nonnegative lowest-rank solution. Generally,
HS pixels are divided into several groups of superpixels,
then the low-rank constraint is imposed on them to capture
the local spatial correlation in the neighboring pixels.
The formulation (36) is difficult to solve due to the discrete
nature of rank minimization. Hence, the matrix rank
function X)
is replaced by a nuclear-norm ,X )
which provides
an appropriate substitution for it. The nuclear norm of
a matrix is defined as
24
Similarly, Giampouras et al. encapsulated the estimation
of the number of endmembers and unsupervised unmixing
in a multiconstrained optimization framework [207], which
used the tight upper bound to impose the low rankness in
bilinear terms of
PQ .T
The difference is that P and Q represent
the endmember and abundance matrices, respectively.
In addition, some improved methods that address different
weaknesses have also been developed. For instance, as
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2022
XP .QFF
22
) # _
+
i
(38)
XXX
)== v
=
trace () /
T
i
MN
1
i
where iv is a singular value of the matrix X.
In addition, the weighted nuclear norm is employed in
some sparse unmixing methods where the individual singular
values i
v are treated differently by the weights, and
the sparsity is expected to be enhanced. Giampouras et al.
suggested an alternating direction sparse and low-rank unmixing
algorithm [109], which simultaneously imposed
single sparsity and low rankness on the abundance matrix
for pixels in a small sliding window. Instead of single
sparsity, Huang et al.[110] proposed joint sparse blocks and
low-rank unmixing by imposing the joint sparsity-block
structure and low rankness to further exploit the spatial
information in HSIs. Wang et al. advised a method for HS
unmixing by reweighted low rank and TV to better characterize
the global structure of the abundance matrix in [111].
Moreover, some low-rank, constrained sparse unmixing
schemes also consider spectral information to enhance
performance. Wang et al. proposed a group low-rank, constrained
NMF method for unmixing [112], where the lowrank
constraint was applied to the groups of superpixels
to explore the semantic geometry in both the spatial and
spectral domains. Zhang et al. [113] put forward a sparse
unmixing process based on LRR with a space-consistency
constraint to better identify the homogeneous regions,
which appraised both the nearest spatial and spectral distances
when selecting the nearest pixels as real neighbors of
the current pixel. Rizkinia and Okuda advised joint localabundance
sparse unmixing by imposing the nuclear norm
on three dimensions of the abundance maps for the sake of
the low rankness of local spectral signatures in the abundance
dimension [114]. Rizkinia and Okuda also implemented
the local-abundance regularization for 3D local regions
to sparse unmixing via variable splitting augmented
Lagrangian and total variation (SUnSAL-TV) [115].
A common issue in rank minimization with the nuclear
norm is high computational complexity. Giampouras et al.
[116] used the upper bound of the low rank, suggesting
trace norm as an alternative way for minimizing the rank,
and proposed the alternating-minimization sparse, lowrank
unmixing method. Let X be the product of two matrices,
P and Q, i.e.,
XP .QT
=
norm of X is
,
(37)
The upper bound of the nuclear
IEEE Geoscience and Remote Sensing Magazine - March 2022

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