IEEE Geoscience and Remote Sensing Magazine - March 2023 - 48

lower than the Nyquist rate, the limitation of high cost
caused by the sheer volume of HS data will be alleviated. A
contradiction usually exists between the massive HS data
and limited bandwidth of satellite transmission channels.
HS images can be compressed first to reduce the pressure
on channel transmission. Therefore, the HS CS technique
is conducive to onboard burst transmission and real-time
processing in RS [163].
The CS of HS images aims to precisely reconstruct HS
data X Rhv z## from a few compressive measurements
y Rm
!
! through effective HS CS algorithms. The compressive
measurements y can be formulated by
( )
y XW=
(42)
where W is a measurement operator instantiated as
·· ,
hv zm## "
W=DH P where D is a random downsampling operator,
H is a random permutation matrix, P is a Walsh-
Hadamard transform, and the mapping of W is RR
(the sampling ratio is m = hvz). The strict reconstruction of
X from y will be guaranteed by CS theory when W satisfies
the restricted isometry property. This compressive
operator has been successfully adopted for various HS CS
tasks [164], [165], [166], [167]. However, operator W can be
replaced with real demands. Apparently, it is an ill-posed
inverse problem to directly recover X from (42). Extra prior
information needs to be investigated to optimize the HS CS
problem. The HS CS task can be generalized for the following
optimization problem:
min y Xh
X
-+W^
F
2
m ^F Xh
(43)
where F X^h denotes the additional regularization term to
use different types of HS prior information, such as spectral
correlation, spatial and spectral smoothness, and nonlocal
similarity.
TENSOR DECOMPOSITION-BASED
HYPERSPECTRAL COMPRESSIVE SENSING
RECONSTRUCTION METHODS
Recently, CP decomposition, Tucker decomposition, and
tensor-based multilinear SVD have been used for HS compression.
For example, Fang et al. [168] decomposed an HS
cube into rank 1 tensors and utilized the sparsity of those
components to compress the original one. Das et al. [169]
developed Tucker decomposition for HS image and video
compression, which gained better efficiency in terms of
the compression ratio and SNR. Renu et al. [170] extended
a 3D multilinear SVD for compressing HS images spatially
and spectrally.
Subsequently, Tucker decomposition-based methods
have drawn wide attention for HS CS. Tucker decomposition
was first introduced into the compression of HS images
to constrain the discrete wavelet transform coefficients
of spectral bands [171]. Most of the following works try to
study the Tucker decomposition-based variants for HS CS
[61], [63], [66], [164], [172].
48
In [173], the LR tensor constraint of (44) was replaced by
the TNN.
TUCKER DECOMPOSITION WITH
NONLOCAL SIMILARITY
Tucker decomposition methods with nonlocal similarity
either cluster similar patches into a 4D group or unfold 2D
patches into a 3D group. Du et al. [62] represented each local
patch of HS images as a 3D tensor and grouped similar
tensor patches to form one 4D tensor per cluster. Each tensor
group can be approximately decomposed by a sparse
coefficient tensor and a few matrix dictionaries. Xue et al.
[63] unfolded a series of 3D cubes into 2D matrices along
the spectral modes and stacked these matrices as a new 3D
tensor. The spatial sparsity, nonlocal similarity, and spectral
correlation were simultaneously employed to obtain
the proposed model:
P
, ,, ,
s.t.
x BBBpppp/ 1
A 12 3 p=1 2 X ###-A BB B
min
m
yx BBB
BB I
++
==
U X ###p pppp
( ,,)
mm X3L ()
,
2 A 1
A
ip
T
where pP1= ,...
ip==i 123
!
11 22 33
(45)
and P denotes the group number, x Rhvz
denotes the vector form of X, and L X^h is the TTN of X .
TENSOR RING-BASED METHODS
Unlike Tucker decomposition methods [62], [63] that
directly capture the LR prior in the original image space
at the cost of high computation, a novel subspace-based
NLTR (SNLTR) decomposition approach projects an HS
image into a low-dimensional subspace [64]. The nonlocal
similarity of the subspace coefficient tensor is constrained
by a TR decomposition model. The SNLTR model
is presented as
EZ LG,, ,ii
min 2
1
s.t.EE I LG
T
yEZZ L
i
-+ -
== ^
W ()
2
,.
ii
F
U 6 @h
m/`
2
1
ii F
2j
(46)
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
p ppp p F
pp
11 22 33
2
TUCKER DECOMPOSITION WITH TOTAL VARIATION
In one earlier work [164], a 2D TV norm was penalized in an
LR matrix framework, which robustly recovers a large HS image
when the sampling ratio is only 3%. A spectral LR model
is rarely enough to depict the inherent property of HS images.
Joint tensor Tucker decomposition with a weighted 3DTV
(JTenRe3DTV) [61] injected a weighted 3DTV into the LR
Tucker decomposition framework to model the global spectral
correlation and local spatial-spectral smoothness of an
HS image. Considering the disturbance ,E the JTenRe3DTV
optimization problem for HS CS can be expressed as
XE UC i
,, ,
min 2
1
s.t yB BB
EX
F
2
+ m
==
W^h
.,
3DwTV
XX A## #
11 22 33+E.
(44)
IEEE Geoscience and Remote Sensing Magazine - March 2023

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