IEEE Geoscience and Remote Sensing Magazine - March 2023 - 37

TENSOR RING DECOMPOSITION
The purpose of TR decomposition [27] is to represent a
high-order X by multilinear products of a sequence of
three-order tensors in circular form. Three-order tensors are
named TR factors { }{ ,, ,},GGG G
() N
n
G R()
nr Ir
n=1 =
nn n 1 ,, ,, ,.nNrrN In this case, the
12 and
f
element-wise relationship of TR decomposition with factors
G can be written as
X^ hhh
N
ii ii ii
i
12
,, ,:,,::,,::,,: i
^^ ^N
ff
=
N =
Tr
TrGG G
_
1
h
2
^
2
d% ^n ^ :, n,:hn
h
G
i =1
where Tr denotes the matrix trace operation.
HYPERSPECTRAL RESTORATION
In the actual process of HS data acquisition and transformation,
the observed HS images are often degraded by different
sources related to external environmental change and internal
equipment conditions. The atmosphere, through absorption,
scattering, clouds, and the reflection of solar radiation,
can have a significant impact on HS RS data. Artifacts and
noise are produced by imaging systems, such as striping in
HS push broom imaging systems. These inevitably lead to
noises, blurs, and missing data (including clouds and stripes)
[136], [137], which degrades the visual quality of HS images
and efficiency of subsequent HS data applications, such
as a fine HS RS classification for crops and wetlands [138],
[139] and the refinement of spectral information for target
B2
X
h × ν × z
B1
h × r1
(a)
B1
G1
X
A1
h × ν × z
(c)
r2
X
h × ν × z
(e)
r0
h
r1
ν
r2
z
h × ν × z
(f)
FIGURE 3. Six tensor decompositions of a third-order tensor: (a) Tucker decomposition, (b) CP decomposition, (c) BT decomposition,
(d) t-SVD, (e) TT decomposition, and (f) TR decomposition.
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
37
X
A2
A3
h × ν × zh × ν × z
(d)
h × ν × z
h × ν × z
r0
r1
h
r2
r1
ν
r0
r2
z
C1
G2
+
B2
C2
G3
+
X
U ∗∗ VT
S
B3
C3
A
z × r3
B3
r1 × r2 × r3 ν × r2
h × ν × z
(b)
X
b1
(1)
b2
(1)
b3
(1)
ALGORITHM 1. TENSOR SINGULAR VALUE DECOMPOSITION.
Input:
X Rhv z
##
!
1: XX=t
fft ( ,[ ], );3
2: for i = f ,, ,
4: end for
5: for
6:
7:
01 :D+
z
1
2
3: [(:,:, ), (: ,:,),(:,:, )] = SVD X
US V
i :D+ 1 f z
=
z + 1
2
(: ,:,)
,,
U =-+
t
S =- +
tt
8: iz ));
9: end for
V =-+
tt
(: ,:,) ((:,:,
(: ,:,)
((:,:,
10: UU ISS VV ;
Output: US V ,, .
= ==
tt t
(,[],),( ,[ ], ),
fft
fft
ifft 33 (,[],)3
i 2
conj U
S
((:,:,
i 2
t iz ));
iz ))
conj V
i 2
tt tii ii
t
((:,:, ));
1hh h
^
^
N
(19)
TXM () SN=+ +
where ,, ,RR RTSXhv zh vz
##
!! !
##
!
(20)
hv z## and N Rhv z
##
represent an observed HS image, the restored HS image, the
sparse error, and the additive noise, respectively. The additive
noise is modeled as an independent signal, usually
() ()
12
f
! ==## +
()
N
where
detection [140], [141]. Figure 4 depicts HS RS degradation
and restoration. Therefore, HS image restoration appears as a
crucial preprocessing step for further applications.
Mathematically, an observed degraded HS image can be
formulated as follows:
b1
(2)
b1
(3)
b2
(2)
b2
(3) b3
++
(2)
b3
(3)
IEEE Geoscience and Remote Sensing Magazine - March 2023

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