IEEE Geoscience and Remote Sensing Magazine - June 2022 - 209

multiple linear regression proposed in [56] assumes that
every band is a linear combination of the others. Hence,
the ith band can be approximated using least-squares estimation.
HSI denoising has been considerably improved
by exploiting spatial and spectral information. 3D modeling
and filtering, including 3D wavelets [57] and 3D
(blockwise) NL sparse restoration methods [58], can be
mentioned as the earliest attempts for spatial-spectral
approaches. Later, penalized least squares that exploited
spatial [59], spectral [60], and spatial-spectral penalties
[61], [62] were suggested for HSI denoising. Those approaches
proposed a different prior ()Xz to capture spatial
and spectral correlations.
For example, in [60], the first-order spectral roughness
denoising (FORPDN) was suggested in the wavelet domain
to capture spectral correlations:
W=- ml
W
t
argmin 2
1
where D2
^h // Rw ,
l==1
HD WΩ /-12 2
2
F +
2
1
p
1
j
(15)
wavelet coefficients. Note that Rp
contains 2D wavelet bases and W denotes the 2D
is a () difference
pp1 #matrix.
In [62], a combination of a group lasso (least absolute
shrinkage and selection operator) 2
, as a spatial penalty
with the first-order spectral roughness penalty (GLASSORP)
was proposed. A spatial-spectral mixing prior was
suggested for HyDe in [61]. An efficient edge-preserving
denoising approach is obtained with a prior ()Xz
corresponding to the TV of the signal [63]:
in (14),
argmin 2
XH1
X
t=- m+
XXF
2
TV
,
(16)
j
l
2
2
L
p
where X TV is the isotropic TV quasi-norm of the matrix
X. In the case of HSIs, X contains spatial and spectral
information, and therefore the conventional isotropic TV
quasi-norm cannot effectively capture spectral and spatial
correlation. Thus, several TV denoising approaches
were developed for HSI denoising.
Cubic TV (CTV), proposed in [64] and [65], was adapted
to incorporate spectral variations for HSI denoising.
The various denoising techniques assign different weights
to spatial variations, compared to spectral variations. For
instance, CTV utilizes
argmin 2
XH1 XDXDXXR
++ +
X
t=- mb 1
2 22 2
F vh p
T
() () () ,
(17)
where the tradeoff between spectral and spatial variations
is controlled using parameter ;b Dh
and Dv
are the
matrix operators for calculating the first-order vertical
and horizontal differences of a vectorized image, respectively.
Spatiospectral TV (SSTV) was proposed in [66],
which exploits anisotropic spatial-spectral TV penalties
for HyDe.
LOW-RANK APPROACHES
Due to the high spectral dimension of HSIs, low-rank modeling
has been found to be more efficient for HyDe [67]. In
the additive noise model, it is assumed that X is low rank.
The low-rank property can be applied in two ways: 1) via a
low-rank model
HF ,VNT
=+
(18)
Full-Rank Conventional
3D Wavelets - 2012
SSMP - 2012
CTV - 2012
Nonlocal SR - 2013
GLASSORP - 2013
FORPDN - 2014
SSTV - 2016
Low-Rank Conventional
PARAFAC - 2012
SVDSRR - 2013
WSRRR - 2014
NAIRLMA - 2015
HyRes - 2017
FastHyDe - 2018
Local LRSR - 2019
DL
HDnTD - 2017
HSI-DIP - 2019
HSID-CNN - 2019
HSI-DeNet - 2019
HSI-SDeCNN - 2019
Zero Shot - 2019
QRNN3D - 2021
FIGURE 5. A glance at the hyperspectral denoising (HyDe) literature from the past decade. CTV: cubic total variation; FastHy De: fast hyperspectral
denoising; FORPDN: first-order spectral roughness penalty; GLASSORP: combined group lasso and first order spectra roughness
penalty denoising; HDnTD: HSI denoising using trainable deep learning; HSID: HSI deep; HSI-DeNet: HSI denoising network; HSI-DIP: HSI
deep image prior; HSI-SDeCNN: HSI single model DeCNN; HyRes: hyperspectral restoration; Local LRSR: local low-rank, sparse representation;
NAIRLMA: noise-adjusted image recovery using low-rank matrix approximation; Nonlocal SR: nonlocal sparse restoration; PARAFAC:
parallel factor analysis; QRNN3D: 3D quasi-recurrrent NN; SSMP: spatial spectral mixing prior; SSTV: spatiospectral TV; SVDSRR: singular
value decomposition-based sparse regularization with reduction; WSRRR: wavelet-based sparse reduced rank regression.
JUNE 2022 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
209
IEEE Geoscience and Remote Sensing Magazine - June 2022

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