IEEE Geoscience and Remote Sensing Magazine - March 2023 - 33

TABLE 1. THE TENSOR DECOMPOSITION-BASED APPROACHES FOR HS RS.
CATEGORY YEAR METHOD
BRIEF DESCRIPTION
2020 SCNMTF [114]
2021 NLR-TV [115]
2021 BUTTDL1 [116]
2021 SeCoDe [117]
2020 WNLTDSU [118]
2021 NL-TSUn [118]
2021 LRNTF [119]
BT decomposition +nonnegative
matrix factorization
TV + NLR
Sparsity + Tucker
decomposition
Convolution operation +
BT decomposition
Weighted NLR + TV
NLR + joint sparsity
BT decomposition
(CONTINUED)
PRIOR INFORMATION
Spectral correlations
Spectral correlations + nonlocal
similarity + spatial-spectral
smoothness
Spectral correlations
Spectral correlations +
spatial-spectral smoothness
Spectral correlation + sparsity +
spatial smoothness
Spectral correlation + sparsity
Spectral correlations
https://gitSUb.com/
danfenghong/IEEE_TGRS
_SeCoDe
https://github.com/
sunlecncom/WNLTDSU
https://gitSUb.com/
LinaZhuang/HSI_nonlinear
_unmixing_LRNTF
LRTA: LR tensor approximation; PARAFAC: Parallel factors; RITD: rank-1 tensor decomposition; LRTR: LR tensor recovery; NTRM: nonconvex tensor rank minimization;
3DTNN: three-dimensional TNN; TDL: tensor dictionary learning; NSNTD: nonlocal similarity-based nonnegative Tucker decomposition; GNWTTN:
global and nonlocal
weighted TTN; NLTA-LSM: nonlocal tensor approximation with Laplacian scale mixture; ITS: intrinsic tensor sparsity; NLR-CPTD: nonlocal LR regularized CP tensor decomposition;
NGmeet: nonlocal meets global; WLRTR: weighted LRTR; NLTR: nonlocal TR; TLR-TV: tensor nuclear norm-based LR approximation with Total Variation regularization;
SSTV-LRTF: spatial-spectral TV regularized LR tensor factorization; MLR-SSTV: multidirectional LR with SSTV; LRTDTV: LR tensor decomposition with TV; TLR-L1-2SSTV: tensor
LR method with L1-2 SSTV; LRTDGS: group sparsity-regularized TV combined with LR Tucker decomposition; SNLRSF: sparse factorization with the nonlocal tensor constraint
of subspace coefficients; LRTF-DFR: double factor-regularized LRTF; DNTSLR: difference continuity-regularized nonlocal tensor subspace LR learning; OLRT: optimal
LR tensor; Tmac: tensor completion by parallel matrix factorization; TNCP: TNN with CP; AWTC: adaptive weighted TC; LRRTC: LR regularization-based TC; TRTV: TR with TV;
TVWTR: TV with weighted TR; GLTSA: global and local tensor sparse approximation; JTenRe3DTV: joint tensor Tucker decomposition with a weighted 3DTV; LRNLTV: LR
approximation and nonlocal total variation; PLTD: patch-based LR tensor decomposition; NTSRLR: nonlocal tensor sparse representation and LR regularization; SNLTR: subspace-based
NLTR; KCHSI: Kronecker compressive sensing-based hyperspectral imaging; T-NCS: tensor nonlinear CS; NBOMP: N-way block orthogonal matching pursuit;
BOSE: beamformed mode-based sparse estimator; TBR: tensor-based Bayesian reconstruction; LRTD-AD: LR tensor decomposition based AD; TenB: tensor decompositionbased
AD; TDCW: tensor decomposition and cluster weighting; TEELRD: tensor-based endmember extraction and LR decomposition; LRA-STD: LR and sparse tensor decomposition
TPCA: tensor principal component analysis; PTA: prior-based tensor approximation; PCA-TLRR: PCA-tensor LR representation; STEREO: Super-resolution TEnsorREcOnstruction,
NCTCP: nonlocal coupled tensor CP decomposition; SCUBA: SR cube algorithm; CSTF: coupled sparse tensor factorization; CT/CB-STAR: coupled Tucker/coupled
block term decompositions for hyperspectral super-resoluTion with vARiability; CNTD: coupled nonnegative Tucker decomposition; SCOTT: SR algorithm based on coupled
Tucker tensor approximation; NLSTF: nonlocal sparse tensor factorization; NLSTF-SMBF: SeMiBlind Fusion; UTVTD: unidirectional TV Tucker decomposition; NRLTD-SU:
NRLTD with SU; SSGRLTD: spatial-spectral graph LRTD; gLCCTD: graph Laplacian-guided coupled tensor decomposition; NN-CBTD: NN coupled nonnegative BT tensor
decomposition; BSC-LL1: bind structured coupled LL1 (rank-(Lr, Lr, 1)) decomposition; GLCBTD: graph LR-guided BT decomposition; LTTR: low TT rank; CTRF: coupled TR factorization;
HCTR: high-order CTR; FSTRD: factor-smoothed TR decomposition; LRTRTNN: LR TR decomposition based on TNN; LTMR: low tensor multirank; FLTMR: fast LTMR;
NPTSR: nonlocal patch tensor SR; TV-TLMR: TV-regularized tensor low multilinear rank; LRTA-SR: low rank tensor approximation with SR; NTF-SU: nonnegative tensor factorization
with SU; ULTRA-V: unmixing with TLR regularization algorithm accounting for spectral variability; MVNTF: matrix-vector NTF; SPLRTF: sparse and LR constrained tensor
factorization; TAC-MVNTF: MVNTF with three additional constraints; SCNMTF: sparsity-constrained coupled nonnegative matrix-tensor factorization; BUTTDL1: blind
unmixing method for HS images based on an L1 regular term and tucker tensor decomposition; SeCoDe: sparsity-enhanced convolutional operation; WNLTDSU: weighted
NLR Tucker decomposition method for HS sparse unmixing; NL-TSUn: nonlocal tensor-based sparse unmixing; LRNTF: LR representation of abundance maps and nonnegative
tensor factorization.
CODE LINKS
if it satisfies Q QQ QI))
)) A three-order tensor S is
== .
f-diagonal if all its slices are diagonal matrices.
DEFINITION 4 (FIRST MODE-k UNFOLDING/
MATRICIZATION [20])
This operator denoted unfold( ,)kX converts a tensor
X RII II
!
11 into a matrix X R()k
ff# +kk N
versely, fold(, )kX()k
a tensor.
!
II II Ikk kN11 1
# ff
-+
. Indenotes
the folding of the matrix into
DEFINITION 5 (SECOND MODE-k UNFOLDING/
MATRICIZATION [27])
For a tensor
X RII II
kk N
!
ing matrix is represented by X12! +-# ff
inverse operation is matrix folding (tensorization).
k RII II I
kk Nk
DEFINITION 6 (MODE-k PERMUTATION [122])
For a tensor
X RII
!
,
12 IN this operator, noted by Xk
## #g
11 its second mode-k unfold.
ff#
+
,
11 1 The iI ,, ,.== +
for ff
12
RrI II rn11 21
##f
= perMARCH
2023 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
X (),GU=
kn 1kn12 jI ,, ,,
induced as 6 @==Pn=1
:.
21 12
"
33
From this, the multilinear product of all the TR factors can be
()
GG GG GG
. The TR decomposition can be rewritten as
where U is a dimensional shifting operator
N nn " 12
12f
(, ,, )()()( )n ,
=
mutation( ,X k), changes its permutation order k times and RRU ## ## #fgrI II rI II
11 nn
,, ,f !
obtains a new tensor X RkI II IkN k11## ## #gg -
erator is defined as X = ipermutation(( ,Xk
!
be written as ,, .
XXXvzh
zhv
RR and
##
##
and G ,()n 1+
!! ! R ##
hv z
DEFINITION 7 (MULTILINEAR PRODUCT [123])
Given two TR factors G()n
product G(, + ##++
nn 1) rI Irnn nn
! R
nk
k
11 is calculated as
GG1
(:,( ),:)-+ =
(, )()( )11 (4)
nn++(:,,:)
Ii ji jk
n (:,,:)G n
k
their multilinear
. The inverse opthree
mode-k permutation of an HS tensor
12 3
k). For example, the
## can
X Rhv z
1
!
https://github.com/danfenghong/IEEE_TGRS_SeCoDe https://github.com/sunlecncom/WNLTDSU https://github.com/LinaZhuang/HSI_nonlinear_unmixing_LR-NTF
IEEE Geoscience and Remote Sensing Magazine - March 2023

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