IEEE Geoscience and Remote Sensing Magazine - March 2023 - 58

the strict LR constraint of NTF. Various spatial and spectral
structures, such as spatial-spectral smoothness and
nonlocal similarity, are proved to tackle the problem of
pure MVNTF.
Xiong et al. [111] presented a TV-regularized NTF
(NTF-TV) method to make locally smooth regions share
similar abundances among neighboring pixels and suppress
the effect of noises. Zheng et al. [112] offered a
sparse and LR tensor factorization method to flexibly
achieve the LR and sparsity characteristics of the abundance
tensor. Feng et al. [113] installed three additional
constraints, namely, sparseness, volume, and nonlinearity,
in the MVNTF framework to improve accuracy
on impervious surface area fraction/classification maps.
Li et al. [114] integrated NMF into MVNTF by making full
use of the approaches' individual merits to characterize
intrinsic structure information. Besides, a sparsity-enhanced
convolutional operation (SeCoDe) method [117]
incorporated a 3D convolutional operation into MVNTF
for the blind SU task.
MODE 3 TENSOR REPRESENTATION MODEL
Under the definition of the tensor mode-n multiplication,
the LMM (70) is equivalent to
XN A E3
=+#
!
(73)
where A Rhv R## denotes the abundance tensor containing
R endmembers. In [115], the NLR tensor and 2DTV regularization
of the abundance tensor were introduced to extract
the spatial contextual information of HS data. With an
abundance nonnegative constraint (ANC) and abundance
sum-to-one constraint (ASC) [217], the objective function
of NLR-TV for SU is expressed as
min 2
1
A
K
X AA Ak
2
s.t.AA#1
$ ,
-+mm+
= #
#3
Ph v
F
TV
2DTV
01 1
E
NL /
k=1
where 1P is a P-dimensional vector of all one, 1hv#
p
A NL = / LS A()
k
() .
i
i =1
The ANC, ASC, and sparseness of abundance are often
introduced into sparse unmixing models [218], [219],
which produces endmembers and corresponding abundance
coefficients by a known spectral library instead of
extracting endmembers from HS data [220], [221]. Sun et
al. [118] developed a weighted NLR Tucker decomposition
method for HS sparse unmixing by adding collaborative
sparsity and the 2D TV of the endmember tensor into a
weighted NLR tensor framework. The LR constraint and
joint sparsity in the nonlocal abundance tensor were imposed
in a nonlocal tensor-based sparse unmixing algorithm
[222].
58
(75)
NL
(74)
denotes
a matrix of element one, and the NLR regularization is defined
as
NONLINEAR MIXING MODEL
Due to physical interactions among multiple materials in
a scene, NLMMs have been studied in SU by modeling different-order
scattering effects and producing more accurate
unmixing results [223], [224], [225]. As a typical representative
of NLMMs, bilinear mixture models (BMMs) have a
clear physical interpretation and can almost characterize
the radiative transfer process in a scene. To further improve
the description of nonlinear interactions, several BMMbased
models are proposed to characterize macroscopicand
microscopic-scale effects, such as the Nascimento
model [226], fan model [227], polynomial postnonlinear
mixing model [228], and generalized bilinear model [229].
However, traditional BMMs usually transform an HS cube
into a 2D matrix and have the same fault as LMMs [229],
[231], [232].
To effectively address the nonlinear unmixing problem,
Gao et al. [119] expressed an HS cube X Rhv z## based on
!
tensor notation in the following format:
XB A CE N33
=+ +
##
where C R #
!
zR B R ()/hv RR 12
## -
,
!
, and E R ()/
zR R 12
!
# -
(76)
represent
the mixing matrix, nonlinear interaction abundance
tensor, and bilinear interaction endmember matrix, respectively.
A nonlinear unmixing method [119] was first
based on NTF by taking advantage of the LR property of the
abundance maps and nonlinear interaction maps, which
validated the potential of tensor decomposition in nonlinear
unmixing. Note that the optimization of tensor-based
unmixing methods can be effectively solved by the ADMM.
EXPERIMENTAL RESULTS AND ANALYSIS
Urban HS datasets obtained by the Hyperspectral Digital
Imagery Collection Experiment sensor over Texas, USA,
are selected for evaluating the performance of different
unmixing methods qualitatively, including MVNTF [110],
MVNTF-TV [111], a SeCoDe [117], and LR-NTF [119]. For
a fair comparison, HS signal subspace identification by
minimum error [233] and vertex component analysis [234]
algorithms are adopted to determine the number of endmembers
(NOE) and endmember initialization. The urban
data contain 307 307#
pixels and 210 bands ranging from
0.4 to 2.5 μm. Due to water vapor and atmospheric effects,
162 bands remained after removing affected channels.
Four main materials in this scene are investigated, that is,
asphalt, grass, trees, and roofs. Two quantitative metrics are
utilized to evaluate the extracted abundance and endmember
results, namely, the RMSE and SAD.
For illustrative purposes, Figures 23 and 24 display the
extracted abundances and corresponding endmember results
of different tensor decomposition-based SU approaches.
The quantitative results of the urban data are reported in
Table 12, where the best results are marked in bold. MVNTF
yields poor unmixing performance for both endmember
extraction and abundance estimation compared with other
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2023
IEEE Geoscience and Remote Sensing Magazine - March 2023

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