IEEE Geoscience and Remote Sensing Magazine - March 2023 - 39

the spectral curve to be severely distorted. In Figure 5(c),
the Gaussian noise is densely spread over the whole image
of the Salinas dataset, resulting in severe blurring of the
image edge information.
The wealth of spatial and spectral information in HS
images can be extracted by different prior constraints,
such as the LR property, sparse representation (SR), nonlocal
similarity, and total variation (TV). Different LR tensor
decomposition (LRTD) models are introduced for HS
denoising. Consequently, one or two kinds of other prior
constraints are combined with these tensor decomposition
models.
LOW-RANK TENSOR DECOMPOSITION
In this section, LRTD methods are divided into two categories:
1) factorization-based approaches and 2) rank
minimization-based approaches. The former need to
predefine rank values and update decomposition factors.
The latter directly minimize tensor ranks and update
LR tensors.
FACTORIZATION-BASED APPROACHES
Two typical representatives are used in the HS image denoising
literature, namely, Tucker decomposition and CP
decomposition. Renard et al. [23] considered Gaussian
noise and suggested an LR tensor approximation (LRTA)
model to complete an HS image denoising task:
min
s.t. BB B
X
X A## #
TX
=
-
F
2
11 22 33
where A represents a core tensor and 12
BB and B3
,,
(22)
denote
factor matrices. Nevertheless, users should manually
predefine the multiple ranks along all modes before
running the Tucker decomposition-related algorithm,
which is intractable in reality. In (22), the Tucker decomposition
constraint is easily replaced by another tensor
decomposition, such as CP decomposition. Liu et al. [24]
used a parallel factor analysis decomposition algorithm
and still assumed that HS images were corrupted by white
Gaussian noise. Guo et al. [28] presented an HS image
noise reduction model via rank 1 tensor decomposition,
which was capable of extracting the signal-dominant features.
However, the smallest number of rank 1 factors was
served as the CP rank, which has a high computation cost
to be calculated.
RANK MINIMIZATION APPROACHES
Tensor rank bounds are rarely available in many HS noisy
scenes. To avoid the occurrence of rank estimation, another
kind of method focuses on minimizing the tensor rank directly,
which can be formulated as follows:
minrank
s.t.
X
()
X
TX SN=+ +
(23)
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
where rank ( )X denotes the rank of HS tensor X and includes
different rank definitions, such as Tucker rank, CP
rank, TT rank, and tubal rank. Due to the preceding rank
minimizations belonging to nonconvex problems, these
problems are NP-hard to compute. Nuclear norms are generally
used as the convex surrogate of nonconvex rank function.
Zhang et al. [121] proposed a tubal rank-related TNN
to characterize the 3D structural complexity of multilinear
data. Based on the TNN, Fan et al. [16] presented an LR tensor
recovery (LRTR) model to remove Gaussian noise and
sparse noise:
min
s.t.
,,
XS N
TX SN
X
=+ +
)
1
++ 2
mm
2
(24)
where parameter 1m controls the strength of sparse noise
S and parameter 2m is used to adjust the additive Gaussian
noise strength.
The alternating direction method of multipliers (ADMM)
framework has become popular to solve constrained optimization
problems, such as (24), of tensor decomposition
for HS data processing. Auxiliary variables are introduced
in the ADMM. An equivalent problem is derived with a separable
unconstrained function, which is subject to a linear
compatibility constraint between the original and auxiliary
variables [145]. The original variables, auxiliary variables,
and dual variables are alternately updated to solve the converted
problem. Assisted by proper auxiliary variables, each
update step reduces to a simple subproblem whose solution
is often found with closed-form terms. Meanwhile, the
ADMM hardly depends on the smoothness of the optimization
problem and quickly converges to one optimal solution
with moderate accuracy [146]. Therefore, the ADMM
has become an attractive choice for solving large-scale optimization
problems, such as tensor decomposition and HS
data processing.
Xue et al. [29] applied a nonconvex logarithmic surrogate
function to a tensor trace norm (TTN) for tensor
completion (TC) and tensor robust PCA tasks. Zheng et al.
[18] explored the LR properties of tensors along three directions
and proposed two tensor models: a three-directional
TNN and three-directional log-based TNN as its convex
1 SN F
(a)
(b)
(c)
FIGURE 5. HS datasets with different noise types: (a) heavy stripes
and dead lines on the urban dataset, (b) salt-and-pepper noise and
Gaussian noise on the Indian Pines dataset, and (c) heavy Gaussian
noise on the Salinas dataset.
39
IEEE Geoscience and Remote Sensing Magazine - March 2023

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