IEEE Geoscience and Remote Sensing Magazine - March 2023 - 42

HS images. In particular, HS classification accuracy is improved
more effectively after denoising by
TLR-l TV0
.
SUBSPACE REPRESENTATION
As Figure 8 illustrates, an unfolding matrix X of a denoised
HS image can be projected into an orthogonal subspace;
i.e.,
X EZ .= Here, E Rzl
#
!
space Sl, and Z Rlhv
#
!
T = [156].
where Zi
!
0 is divided into three steps: 1) reshape the reduced-dimensional
coefficient image Z Rlhv
,
!
i !
#
as a tensor
Z Rhv l## 2) segment the tensor Z as an overlapped
pp l
a neighborhood area by computing the Euclidean distance.
From one side, a spectral LR tensor model is explored
represents the basis of the subdenotes
the representation coefficient
of X with respect to E. Also, E is reasonably assumed
to be orthogonal; i.e., EE I
Cao et al. [46] combined an LR and sparse factorization
with the nonlocal tensor constraint of subspace coefficients
(SNLRSF). Each spectral band of an observed HS image
T Rhv z
!
!
matrix T Rzhv
#
## is reshaped as each row of an HS unfolding
. The spectral vectors are assumed to lie in
a l-dimensional subspace Sl (),lz% and the optimization
model can be written as
argmin 2
1
EZ SLi,, ,
+- +=a ZEEILLi F
2
m / 1
i
1
di
2 Ri
i TTN
k
s.t.
T
(36)
TEZS F-- + m2 S 1
2
according to the fact that spectral signatures of HS images
lie in a low-dimensional subspace. From another side, an
NLR factorization is employed to take the nonlocal similarity
along the spatial direction into consideration. Following
the line of SNLRSF, Zheng et al. [47] employed LR matrix
factorization to decouple spatial and spectral models. The
group-sparse structure of HS images was introduced on
spatial difference images (SpatDIs). A continuity constraint
was applied in the spectral factor to promote the group
sparsity of SpatDIs and spectral continuity of HS images.
Sun et al. [48] projected noisy HS images into a nonlocal
tensor subspace spanned by a spectral difference continuous
basis. The continuity of the restored HS data is significantly
promoted by this difference regularization.
EXPERIMENTAL RESULTS AND ANALYSIS
Subspace
Representation
Z
X
E
FIGURE 8. The subspace representation.
TABLE 5. A QUANTITATIVE COMPARISON OF DIFFERENT
SELECTED ALGORITHMS FOR GAUSSIAN NOISE REMOVAL.
INDEX
PSNR
SSIM
ERGAS
MSAD
Time (s)
LRTA
32.14
0.9097
5.7044
6.672
1.48
TDL
GAUSSIAN NOISE REMOVAL
ITS
LLRT
34.54
0.9484
4.3392
5.0701
13.77
34.38
0.9466
4.3981
5.0912
650.49
35.96
0.9637
4.0462
4.2402
506.84
NGmeet
37.06
0.9707
3.2344
3.7804
29.58
GAUSSIAN NOISE CASE
An HS subimage is selected from the Pavia University dataset,
which can be found on and downloaded from the website
in Table 2. Zero-mean Gaussian noise of noise variance
0.12 is added to each band and shown in Figure 9(b). Five
effective denoising methods are selected, including LRTA
(Tucker decomposition), TDL (Tucker decomposition with
dictionary learning), ITS (CP and Tucker decomposition),
Hyper-laplacian regularized unidirectional low-rank tensor
recovery (LLRT) (with nonlocal similarity), and NGmeet
(with subspace representation). These denoising algorithms
are based on the Gaussian noise degradation model.
MIXED-NOISE CASE
The same Gaussian noise is adopted. Each band is corrupted
by salt-and-pepper noise, with a proportion of 0% -20%
Dead lines are randomly added from band 61 to band 80,
with the width of the generated stripes ranging from one
to three, and the number of stripes is randomly selected
from three to 10. In addition, bands 61 70- are corrupted
by stripes, with the number randomly selected from 20 to
patch tensor Z R ,## and 3) cluster d similar patches in
(a)
(b)
(c)
(d)
(e)
(f)
(g)
FIGURE 9. The different methods of Gaussian noise removal: (a) the original HS image, (b) Gaussian noise, (c) LRTA, (d) TDL, (e) ITS,
(f) LLRT, and (g) NGmeet. LLRT: Hyper-laplacian regularized unidirectional low-rank tensor recovery.
42
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
IEEE Geoscience and Remote Sensing Magazine - March 2023

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