IEEE Geoscience and Remote Sensing Magazine - March 2023 - 41

dataset as a typical example to depict the gradient images
along three directions in Figure 7. The smoothing areas and
edge information of gradient images are much clearer than
the origin.
Inspired by TV applications to gray-level images, the 2D
spatial TV norm of X is easily introduced to an HS image
in a band-by-band manner [13]. This simple band-by-band
TV norm is defined as follows:
XX XhvTV 11
=+
DD
(27)
where Dh and Dv stand for first-order linear difference operators
corresponding to the horizontal and vertical directions,
respectively. These two operators are usually defined as
D X = (
h
1
D X = (
v
1
XX(, ,), 1 # 1
0,
(, +1,)
(,
,)
+1
=
XX(,
,), 1 # 1
0,
=
ij ki jk jv
jv
ijki jk ih
ih
.
(28)
(29)
,
To enforce the spatial piecewise smoothness and
spectral consistency of HS images, a 3DTV norm [149]
and spatial-spectral TV (SSTV) norm [151] are formulated,
respectively:
X DD DXX Xhv z11 1
X XXzh
3DTV=+ +
() ()
zv
where D Xz
spectral direction and defined as
D X = (
z
1
XX # 1
0,
(, ,) (, ,), 1
+1
=
ijkijk
ih
kz
.
(32)
Considering the degraded model with mixed noise,
Chen et al. [39] integrated both 2DTV and the 3DTV
regularizations into the TNN. Fan et al. [17] injected the
preceding SSTV norm into LR tensor factorization. Wang
et al. [152] used an SSTV term in a multidirectional weighted
LR tensor framework. Based on the different contributions
of the three gradient terms to 3DTV regularization, Wang
et al. [41] proposed the TV-regularized LRTD method:
min XS N 2
XS N
,,
s.t.
xm b++
=+ +
3DwTV
TX SN
X
1
== =
1122 33 i
i
where the 3DwTV term is defined as
XX (34)
3DwTV hv z
=+ + 3
wD wD wD 1
1 XX1
1
2
al L SSTV12.
Zeng
et al. [42] integrated the advantages of both a globand
the local patch TNN. Chen et al. [43] exploited
the row-sparse structure of gradient images and proposed
a weighted group sparsity-regularized TV combined
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
(c)
(d)
FIGURE 7. The spatial smooth properties of the WDC dataset:
(a) the original band, (b) the gradient image along the spatial
horizontal direction, (c) the gradient image along the spatial vertical
direction, and (d) the gradient image along the spectral direction.
41
A### ,( ,, )
BB BB BI i 12 3
T
F
(33)
(30)
SSTV 11DD DD=+ (31)
1 is a 1D finite-difference operator along the
st
with LR Tucker decomposition (LRTDGS) for HS mixednoise
removal.
Due to the mentioned TV norms penalizing only large
gradient magnitudes and easily blurring real image edges,
a new l0
image edges [152]. Actually, the l TV1
form of the l0
[40] applied the l0
gradient. Xiong et al. [44] and Wang et al.
gradient constraint in an LR BT decomposition
and Tucker decomposition, respectively. However,
the degrees of smoothness of this l0
gradient form are controlled
by a parameter without any physical meaning. To alleviate
this limitation, Ono [124] proposed a novel l0
gradient
projection that directly adopts a parameter to represent
the smoothing degree of the output image. Wang et al. [45]
extended the l TV0
model into an LR tensor framework to
preserve more information for classification tasks after HS
image denoising [154], [155]. The optimization model of
TLR-l TV0
min / kE XS TX S 2
XS k=1
is formulated as
()
m
.. BDX 10, #
i
am n
c
k
1
where the functions E ( )Xk
k
~++ -- F
,
(35)
~ are set to be X() ~ )
k
,
weighted sum of weighted nuclear norm (WSWNN)-l0TVbased
method and X
~ )
,
in the weighted sum of
weighted tensor nuclear norm (WSWNN)-l0-TVbased
method. Operator B forces boundary values of gradients
to be zero when ih= and
to calculate both horizontal and vertical differences. Compared
with many other TV-based LRTD, TLR-l TV0
achieves
better denoising performance for mixed-noise removal in
jv .= Operator D is an operator
in the
gradient minimization was proposed to sharpen
norm is a relaxation
(a)
(b)
IEEE Geoscience and Remote Sensing Magazine - March 2023

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