IEEE Geoscience and Remote Sensing Magazine - March 2023 - 54

the BT of decomposition framework [named coupled Tucker/
coupled block term decompositions for hyperspectral SuperresoluTion
with vARiability (CT/CB-STAR)] [82], an additive
variability term was admitted for the study of general identifiability
with theoretical guarantees. Zare et al. [83] offered a
coupled nonnegative Tucker decomposition method to constrain
the nonnegativity of two Tucker spectral factors.
NONLOCAL TUCKER DECOMPOSITION
Wan et al. [86] grouped 4D tensor patches by using the spectral
correlation and similarity under Tucker decomposition.
Dian et al. [87] offered a nonlocal sparse tensor factorization
(NLSTF) method, which induced core tensors and corresponding
dictionaries from HR-MS images and spectral
dictionaries from LR-HS images. A modified NLSF_ SeMiBlind
Fusion (SMBF) version was developed for the semiblind
fusion of HS and MS [88]. However, the dictionary
and core tensor for each cluster are estimated separately by
NLSTF and NLSF_SMBF.
TUCKER DECOMPOSITION PLUS MANIFOLD
Zhang et al. [91] suggested a spatial-spectral-graph-regularized
LRTD (SSGLRTD). In SSGLRTD, the spatial and spectral
manifolds between HR-MS and LR-HS images are assumed
to be similar to those embedded in HR-HS images.
Bu et al. [92] presented a graph Laplacian-guided coupled
tensor decomposition model that incorporated global spectral
correlation and complementary submanifold structures
into a unified framework.
TUCKER DECOMPOSITION PLUS TOTAL VARIATION
Xu et al. [89] presented a Tucker decomposition model with
unidirectional TV. Wang et al. [90] advocated an NLR Tucker
decomposition and SU-based approach to leverage spectral
correlations, nonlocal similarity, and spatial-spectral
smoothness. Consequently, several fusion approaches are
proposed to provide useful hints, including simultaneously
making use of SU and transfer learning [192], the visualization-oriented
fusion of HS image bands [193], and the
motion cue in superresolving a scene [194].
BLOCK TERM DECOMPOSITION MODEL
Zhang et al. [93] discovered identifiability guarantees in [78]
and [81], at the cost of a lack of physical meaning for the latent
factors under CP and Tucker decomposition. Therefore,
they employed an alternative coupled nonnegative BT tensor
decomposition (NN-CBTD) approach for HS SR. The NNCBTD
model with rank (, ,)LL 1rr
for HS SR is given as
R
,, .
mi ()n
A, ,CB
s.t. $$ $
A0 B0 C0
PA PB c Z
Y / 12
r=1
-
rr
F
2
R
^h<%%3
<
r +- AB Pcr
r=1
/ ()
r
r
F
(62)
Compared with a conference version [93], the journal
version [94] additionally gave more recovery ability analysis
54
2
and a more flexible decomposition framework by using an
advocated rank-(Lr, Lr, 1) model and a block coordinate descent
algorithm. Jiang et al. [95] introduced a graph manifold,
the graph Laplacian, into the CBTD framework.
TENSOR TRAIN DECOMPOSITION MODEL
Dian et al. [96] proposed a low TT rank (LTTR)-based HS SR
method. An LTTR prior was designed for learning correlations
among the spatial, spectral, and nonlocal modes of 4D
FBP patches. The HS SR optimization can be obtained as
K
min YX RZ GX-+ -+x / X TT
=1
2
X()3
() ()
33
()
33
()
2
FF k
k
is defined as
(63)
where K denotes the number of clusters. The TT rank of tensor
Zk
3
Zkt
t
LSAAi
TT = a/ LS ()Z
=1
kt
i log(( )),
GH
(64)
and ()=+R vf with a small positive value
f. Li et al. [97] presented NLR tensor approximation and
NLR SR that formed the nonlocal similarity and spatial-
spectral correlation by the TT rank constraint of 4D nonlocal
patches.
TENSOR RING DECOMPOSITION MODEL
The TR decomposition of an HR-HS tensor
represented as
XG [ ,,]
= U
G R()
,].
2 !
23
23 and
##
,
rr
() GG
() ()
12 3
where three TR factors are denoted by
rV r
G R() rB r
3 31
!
1
21
!
(65)
G R() rH r
1 12
##
,
## , with TR ranks r [,r1
=
Based on TR theory, an LR-HS image is rewritten as
[, ,]PP
YG GG()
()
2
22
= U ## 3
and an HR-MS image can be expressed as
ZG GG() () ()
123
= U[, ,]P .
#23
(67)
He et al. [98] presented a coupled TR factorization
(CTRF) model and modified CTRF version [nuclear norm
regularized CTRF (NCTRF)] with the nuclear norm regularization
of third/spectral TR factor. The NCTRF model is
formulated as
min
GGG
() () ()12 3
,,
+-U
Y
Z
-U PP F
2
[, ,]
[, ,] .
GGG
GG G
()
1
() () ()
123 PG (68)
##
#
21
()
2
22
23 F + m
()
()
2
3
)
Equation (68) becomes the CTRF model when removing
the final term. In [98], the benefit of TR decomposition
for SR is elaborated via theoretical and experimental
proof related to a low-dimensional TR subspace. The relationship
between the TR spectral factors of LR-HS images
and HR-MS images was explored in [99] with a high-order
representation of the original HS image. The spectral
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
()
3
2
()
(66)
X RHV B
##
!
is
IEEE Geoscience and Remote Sensing Magazine - March 2023

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