IEEE Geoscience and Remote Sensing Magazine - March 2023 - 44

the differences among competing denoising methods, we
calculate four quality indices and include them in Table 6,
with the best results in bold.
where wi is a reweighting factor inversely proportional to
singular values of
L ,i with L A BB B ,
ii 11 22 33
TLR- L T0 V obtains the highest
denoising performance among all the approaches. For
MPSNR, LRTDTV, and LRTDGS, the values are slightly larger
than those of three-directional TNN, whereas the SSIM
and ERGAS values of LRTDTV and LRTDGS are better than
those of three-directional TNN LRTR and LRTDGS are the
first- and second-fastest, but they hardly handle the complex
mixed-noise case, with some dead lines remaining.
HYPERSPECTRAL DEBLURRRING
Atmospheric turbulence and the fundamental deviation
of some imaging systems often blur HS images during
the data acquisition process, which unfortunately damages
the high-frequency components and edge features
of HS images. HS deblurring aims to recover sharp latent
images from blurred ones. Chang et al. [49] discussed LR
correlations along HS spatial, spectral, and nonlocal similarity
modes and proposed a unified optimal LR tensor
(OLRT) framework for multiple HS restoration tasks. But
a matrix nuclear norm is used to constrain the LR property
of unfolding nonlocal patch groups. Consequently,
Chang et al. [37] proposed a weighted LRTR (WLRTR)
algorithm with a reweighted strategy. Considering spectral
correlation and nonlocal similarity, the HS deblurring
optimization problem can be formulated as follows:
min
X Bij
,,A
+- BBB 2
i
hv
-
2 M
1
/^h+
(37)
TX
RX
ii## #%i ii 1
AA
2
11 22 33 F
w
TABLE 7. A QUANTITATIVE EVALUATION OF THE OLRT FOR
DIFFERENT BLUR CASES.
BLUR CASE
Gaussian blur
(8 × 8; v = 3)
Gaussian blur
(17 × 17; v = 7)
Uniform blur
MPSNR MSSIM
0.9912
0.9807
0.9784
43.5
39.63
39.39
ERGAS MSAD
1.591
2.5407
2.9332
1.9486
3.0819
3.8355
TIME (S)
314.5
305.7
314.28
X
() F
2
2
-
= ### and
HOSVD is applied to see the different sparsities of higherorder
singular values, i.e., the LR property. The last term,
YXM () ,F
by -- -
TXM () SN F
2
is a data fidelity item, which can be replaced
for HS inpainting, destriping,
and denoising problems.
An experimental example is given to display the deblurred
performance of the OLRT for the Gaussian blur
with different levels and uniform blur in the WDC dataset.
In the first case, the Gaussian blur kernel has a size of
88 ,# with a standard deviation of
v 3 .= The second case
tests the heavy Gaussian blur kernel with a size of 17 17#
and
v 7 .= Figure 11 provides visual results under different
blur cases. The specific texture information is hardly
distinguished in the three blurred images in Figure 11(b),
(d), and (f). The prior knowledge of the OLRT reliably
reflects the intrinsic structural correlation of HS images,
which benefits the recovery of structural information and
image edges. Quantitative results in different blur cases are
reported in Table 7.
HYPERSPECTRAL INPAINTING
In this section, we introduce and discuss LR tensor-based
methods for HS inpainting. These methods are also suitable
for missing data recovery of high-dimensional RS (HDRS)
images. RS images, such as HS, MS, and multitemporal images,
often have missing data problems, such as dead pixels,
thick clouds, and cloud shadows. Figure 12 presents several
real examples of missing data problems. The goal of inpainting
is to estimate missing data from observed images,
which can be regarded as a TC problem.
LRTC theory has been successfully applied for HS inpainting
[37], [49], [50], [51], [53], [125], [157]. Liu et al. [51]
suggested a trace norm-regularized CP decomposition for
missing data recovery. Ng et al. [52] learned from high-accuracy
LRTC (HaLRTC) [125] for recovering the missing data
of HDRS and proposed an adaptive weighted TC (AWTC)
method. The proposed AWTC model is expressed as
min Mw2 TX X() )
h
-+/ i
i =1
2
i
() F
3
(38)
where wi is a well-designed parameter related to the singular
values of
X .()i Xie et al. [53] proposed an LR regularization(a)
(b)
(c)
FIGURE
12. The missing information problems of RS data: (a) dead
lines in Aqua Moderate Resolution Imaging Spectroradiometer
band 6, (b) the scan-line-corrector-off problem in the Landsat 7
Enhanced Thematic Mapper Plus, and (c) thick cloud obscuration in
a Landsat image.
44
based TC, fusing the logarithm of the determinant with a
TTN. With the definitions of a new TNN and its t-SVD [54],
Wang et al. [55] and Srindhuna et al. [158] proposed new
low-tubal-rank TC methods to estimate missing values in
HDRS images. Consequently, a novel TR decomposition
is formulated to represent a high-dimensional tensor by
circular multilinear products on a sequence of third-order
tensors [27]. Based on TR theory, He et al. [56] fused spatial
TV into the TR framework and developed two solving algorithms:
the augmented Lagrangian method and alternating
least squares (ALS). Similarly, Wang et al. [57] incorporated
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
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