IEEE Geoscience and Remote Sensing Magazine - June 2021 - 60

models to exploit the spatial, spectral, or nonlocal low-rank
properties of the original clean HS image. The properties of
these five methods are summarized in Table 1. The five methods
are briefly described in the following sections.
LOW-RANK TENSOR APPROXIMATION
On the basis of the observation model [175] (3) with ignoring
sparse noise S, low-rank tensor approximation (LRTA)
Nomenclature
X 3D HS image to be estimated
Y Observed 3D HS image
NG
NS
3D Gaussian noise
3D sparse noise
O Core tensor
r
Rank of matrix
X Unfolded 2D matrix of X
xi
The ith pixel (1D vector) of X
Y Unfolded 2D matrix of Y
NS
Unfolded 2D matrix of NS
H First-order difference matrix
C Model constraint set
U Variable set to be estimated
Transformation functions
Q Combination coefficients of NMF
W Graph or manifold structure
L
Laplacian matrix
D Degree matrix
U Subspace projection matrix
I
Identity matrix
[39] tries to restore the HS image from the following objective
function:
mins.t
X
F
= ## #
!
!
YX XO2
where XO AB C12 3
!
B RNr2# , C RBr3
#
-= ## #AB C ,
12 3
!
(4)
O Rrr r123 stands for the core tensor, and A RMr1
##
is the Tucker decomposition,
#
,
are the factors related to different dirr
r123
mensions. With the rank ( ,,) of Tucker decomposid
Distance or similarity matrix
v Standard derivation
Ck
Sample set of the kth class
M One-hot encoded matrix
P
E
Regression matrix
Endmember matrix
E0
Reference endmember matrix
A Abundance matrix
S
J
f
Scaling factors (matrix)
V Spectral variability dictionary
(matrix)
Coefficients corresponding to V
R Spatial degradation function
G Spectral-response function
NH
HS noise
NH MS noise
Z
c
High-spatial-resolution MS image
m The number of the considered
modality
Scaling constant
tion set in advance, the LRTA model (4) can simultaneously
capture the global spatial and spectral low-rank properties.
Equation (4) provides a simple, general model for different
kinds of low-rank matrix/tensor decomposition-based
HS image-denoising methods; that is, we can change the
Tucker decomposition constraint of X to different kinds of
matrix/tensor decomposition, such as canonical polyadic
(CP), tensor train, tensor ring, and so on.
NAILRMA
The noise-adjusted iterative low-rank matrix approximation
(NAILRMA) [30], [176] method assumes that the spectral
low-rank property is more important than that of spatial
ones; therefore, simply a spectral low-rank regularizer
is utilized to restrict the original spectral image .X From
other works [40], it also indicates that the spatial TV regularizer
is more important than the spatial low-rank regularizer.
In HS images, the similar signatures representing
the same class also appear in the nearby spatial location.
To enhance the spectral low-rank property, NAILRMA segmented
the HS image into spatial overlapping patches and
processed each patch individually. The noise intensity in
different bands of the HS image is dissimilar, a big challenge
in HS image Gaussian-noise removal; this is mitigated
by the noise-adjusted iterative strategy [30]. Finally, the randomized
singular value decomposition (SVD) is utilized to
solve the nonconvex, low-rank approximation problem.
TABLE 1. PRIOR PROPERTIES OF THE SELECTED METHODS.
METHODS
LOW RANKNESS
LRTA
· ·
NAILRMA -
TDL
-
FastHyDe -
NGmeet
-
LRMR
LRTV
LRTDTV
LRTDGS
LRTF-FS
-
-
·
·
-
· ·
· ·
·
· ·
· ·
· ·
· ·
· ·
· · ·
· · ·
-
-
-
· · ·
·
· ·
-
-
-
-
-
· · ·
· · ·
· ·
· ·
-
-
-
-
-
-
-
-
-
-
· ·
-
· ·
., .., ...: Low, medium, and high intensity of prior information, respectively.
LRTA: low-rank tensor approximation; NAILRMA: noise-adjusted iterative low-rank matrix
approximation; TDL: tensor dictionary learning; FastHyDe: fast HS denoising; NGmeet:
nonlocal meets global; LRTV: low-rank TV; LRTDTV: low-rank tensor decomposition with
TV; LRTDGS: low-rank tensor decomposition with group-sparse regularization; LRTF-FS:
factor-regularized low-rank tensor factorization.
60
-
LOCAL SMOOTHNESS
SPATIAL SPECTRAL NONLOCAL SPATIAL SPECTRAL
-
TENSOR DICTIONARY LEARNING
Tensor dictionary learning (TDL) [177] combines nonlocal
regularization and low-rank tensor approximation.
The noisy HS image is first segmented into spatial overlapping
patches, and similar patches are clustered together
to formulate a higher-order tensor. In this way, the
nonlocal spatial information is collected. Then the higher-order
tensors are denoised, similar to that of (4), and,
finally, the denoised higher-order tensors are utilized to
formulate the final denoised HS image. TDL is the first
method that exploits nonlocal, low-rank properties, and
the subsequent methods, hyper-Laplacian regularized
unidirectional (L LRT) [38], Kronecker-basis representation
[32], and nonlocal tensor ring [34], also achieve remarkable
HS image Gaussian-noise-removal results.
FAST HS DENOISING
The main difference between HS images and color/MS images
is the number of spectral bands [178]. To eliminate this
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE JUNE 2021
IEEE Geoscience and Remote Sensing Magazine - June 2021

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