IEEE Geoscience and Remote Sensing Magazine - March 2023 - 49

HYPERSPECTRAL KRONECKER COMPRESSIVE
SENSING METHODS
Unlike the current 1D or 2D sampling strategy, Kronecker
CS (KCS) consists of Kronecker-structured sensing matrices
and sparsifying bases for each HS dimension [66],
[174]. Based on multidimensional multiplexing, Yang et al.
[65] used a tensor measurement and nonlinear sparse tensor
coding to develop a self-learning tensor nonlinear CS
(SLTNCS) algorithm. The sampling process and SR can be
represented as the model based on Tucker decomposition.
Generally, an HS image
X Rnnn
!
## can be expressed as
123
the following Tucker model:
XS 11 22 33
= ## #UU U
where S Rmm m12 3
!
U !
jj
##
(47)
stands for an approximate blocksparse
tensor in terms of a set of three basis matrices
j R ,kk
# with mk ,,.j 123jj
% =
,
trices denoted by ,, ,
In the context of KCS, three measurement and sensing maW
=
j j 123 of size nk ,jj
# with nk ,jj
%
are used to reduce the dimensionality of the measurement
tensor. The compressive sampling model is given as
YX
=
=
## #
## #WW W
S QQQ
where Qjj jUW ,== ,,.j 123
Zhao et al. [65] designed a 3D HS KCS mechanism
to achieve independent samplings in three dimensions.
Suitable sparsifying bases were selected, and the corresponding
optimized measurement matrices were generated,
which adjusted the distribution of the sampling
ratio for each dimension of HS images. Yang et al. [66]
constrained the nonzero number of the Tucker core tensor
to explore the spatial-spectral correlation. To address
the issue of the computational burden on the data
reconstruction of early HS KCS techniques, researchers
have proposed several tensor-based methods, such as the
tensor form greedy algorithm, N-way block orthogonal
matching pursuit [67], beamformed mode-based sparse
estimator [68], and tensor-based Bayesian reconstruction
(TBR) [69]. The TBR model exploited the multidimensional
block-sparsity of tensors, which was more consistent
with the sparse model in HS KCS than conventional
CS methods. A Bayesian reconstruction algorithm was
developed to achieve the decoupling of hyperparameters
by a low-complexity technique.
EXPERIMENTAL RESULTS AND ANALYSIS
An HS data experiment is employed to validate the effectiveness
of tensor-based models on HS CS with four different
sample ratios, i.e., 1%, 5%, 10%, and 20%. The Reno dataset
selected for the experiment has a size of
150 150 100## .
The randomly permuted Hadamard transform is adopted
as the compressive operator. Table 10 compares the reconstruction
results of SLTNLS and JTenRe3DTV. They have
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
FIGURE 17. The inpainting results of different methods under a
10% sampling ratio: (a) an original Reno image, (b) SLTNLS, and
(c) JTenRe3DTV.
49
METHOD
SLTNLS
INDEX
MPSNR
1%
18.7
MSSIM 0.3273
ERGAS
MSAD
JTenRe3DTV MPSNR
MSSIM 0.8116
ERGAS
MSAD
23.3411
22.0035
27.91
8.2422
7.5545
5%
24.44
0.6593
12.1203
11.2031
34.54
0.9443
4.0139
3.5703
10%
27.72
0.8047
8.3119
7.6354
36.28
0.9638
3.299
2.9233
20%
32.14
0.9159
5.0263
4.6003
37.41
0.9709
2.9124
2.5723
11 22 33
11 22 33
(48)
quality decays, with sample ratios decreasing, but SLNTCS
obtains poorer results than JTenRe3DTV in lower sampling
ratios.
In the light of visual comparison, one representative
band in sampling ratio 10% is presented in Figure 17. The
basic texture information can be found in the results of two
HS CS algorithms. As shown in the enlarged area, SLTNLS
causes some artifacts, but JTenRe3DTV produces a more acceptable
result with the smoothing white area than SLTNLS.
FUTURE CHALLENGES
The low acquisition rate of CS inspires a novel development
potentiality for HS RS. Many tensor-based methods have
been proposed to achieve remarkable HS CS reconstruction
results at a lower sampling ratio. However, here, we briefly
point out some potential challenges.
Some novel tensor decomposition approaches need
to be explored. In past research works, Tucker decomposition
has been successfully applied for HS CS. But with
the development of tensorial mathematical theory, many
tensor decomposition models have been proposed and introduced
in other HS applications. Therefore, how best to
TABLE 10. A QUANTITATIVE COMPARISON OF DIFFERENT
SELECTED ALGORITHMS FOR HS CS.
(a)
(b)
(c)
IEEE Geoscience and Remote Sensing Magazine - March 2023

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