IEEE Geoscience and Remote Sensing Magazine - March 2023 - 53

TENSOR FACTORIZATIONS FOR SPARSE
REPRESENTATION
CANONICAL DECOMPOSITION/PARALLEL FACTOR
ANALYSIS DECOMPOSITION MODEL
Initially, Kanatsoulis et al. [78] employed a coupled CP decomposition
framework for HS SR. The CP decomposition
of an HR-HS tensor X can be expressed as
R
[[ ,, ]]ABC
X ab c
=
=
= 1
/ %%
r=1
r
rr
(56)
where the latent LR factors are [,...,],[ ,...,
and [,...,].Cc cr
==
11
gave the following assumption:
[[ ,, ]];[[, ,]]
PA PB C
12 Z
ABP C
3
where ,, 3 !
PP
12
!!
RR
hH
##
vV and P RbB
#
Aa aB bb ],
rr
In [78], the coupled CP decomposition
Y== (57)
are three linear
degradation matrices. The identifiability of HS SR based
on the algebraic properties of CP decomposition is guaranteed
under relaxed conditions. However, LR properties
of different dimensions are treated equally, which is
rarely suitable for real HS SR. Subsequently, Kanatsoulis
et al. [80] introduced an SR cube algorithm that combined
the advantages of CP decomposition and matrix
factorization. Xu et al. [79] improved the CP decomposition-based
method by adding a nonlocal tensor extraction
module.
ZW
W 12 3
=
=
## #
## #
==
12
12 33AB ()
AB C)
PC
(60)
where ,,AP AB PB and CP C3
))
) =
as follows:
min
AB C W,, ,
+The
l1
YW
ZW
-
## #
## #
12 3
AB C
AB C
))
12 3
2
F
) 2
F + m W 1
norm in (61) was replaced by an l2
.
(61)
norm in [84].
Prévost et al. [85] assumed that HR-HS images assessed
approximately low multilinear rank and developed an SR
algorithm based on coupled Tucker tensor approximation
with HOSVD. In the Tucker decomposition framework and
are the downsampled
dictionaries along three modes. Taking the sparsity
of core tensor
W , Li et al. formulated the fusion problem
TUCKER DECOMPOSITION MODEL
Li et al. [81] extended a coupled sparse tensor factorization
(CSTF) approach in which the fusion problem was
transformed into the estimation of dictionaries along three
modes and corresponding sparse core tensors. When a tensor
X is decomposed by Tucker decomposition,
XW A BC12 3
= ## #
ten as
YW PA PB C
W 12 3
=
=
###
## #
11 22 3
))
() ()
AB C
(59)
.
(58)
The LR-HS and HR-HS degradation models are rewrit(a)
(b)
(c)
FIGURE
20. HS-MS fusion: (a) a low-spatial-resolution HS image, (b) a low-spectral-resolution MS image, and (c) a high-spatial-spectralresolution
HS image.
MARCH 2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
53

IEEE Geoscience and Remote Sensing Magazine - March 2023

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