IEEE Geoscience and Remote Sensing Magazine - December 2021 - 244

can be obtained from successive physical approximations
of the Hapke model for small-albedo materials [52]. Based
on an estimate of M0
obtained from the observed image,
SU under the ELMM was formulated as a nonconvex matrix
factorization problem in which the model (15) was enforced
by means of an additive penalty in the cost function
[51]. A regularization promoting the spatial homogeneity
of the scaling factors
} was also considered to reduce
n
the ill-posedness of the SU problem [51]. The ELMM has
shown good performance for multitemporal data [209]
and been used to facilitate the interpretation of local unmixing
results [210]. Moreover, it can be derived from a
Taylor series expansion of a general nonlinear mixture
model [211], which introduces SU with spectral variability
(viewed as a locally linear SU problem) as a direct way
to address the general nonlinear SU problem. This shows
that some mixture models originally devised to represent
spectral variability (such as the ELMM) can achieve good
performance in nonlinear SU.
Despite its physical motivation, the ELMM lacks
the flexibility to represent more complex spectral variability,
e.g., nonuniformly affecting the spectra. To address
this limitation, the generalized LMM (GLMM)
was proposed in [212] by introducing an individual
scaling factor for each band, leading to the following
EM model:
MM ,nn 0
= W %
where the matrix W !
for each element of M0
n RLP
#
(16)
contains the scaling factors
and % denotes the Hadamard (elementwise)
matrix product. Note that the amount of spectral
variability produced by the GLMM is proportional to
the amplitude of the reference spectra M0
in each band.
However, the larger number of parameters makes the SU
problem resulting from (16) more ill-posed, with challenging
estimation problems. This motivated the development
of a tensor interpolation framework to estimate the matrices
W
from training hyperspectral data obtained from
n
prior knowledge about the positions of pure pixels in the
hyperspectral image [213]. However, the performance of
the method proposed in [213] strongly depends on the
number of pure pixels available in the image. The GLMM
also has been successfully used in multitemporal SU [214]
and to represent spectral variability when fusing hyperspectral
and multispectral images acquired at different
time instants [215].
Note that the performance of unmixing methods
based on the ELMM and the GLMM heavily depends
on the quality of the reference EM matrix
M ,0 which
must be estimated from the observed image. To reduce
the dependence of the ELMM on ,M0
of [216] proposed to jointly estimate M0
maining variables during SU. Each column of M0
constrained to have a unit norm to obtain EMs as
directional data in the spectral space. Moreover, M0
244
the authors
with the rewas
MM
,dM
nn0
=+
where the matrix dM Rn
!
(17)
LP is an additive pertur#
bation
representing spectral variability. In this case,
the reference EM matrix M0
additive perturbation terms dMn
and the pixel-dependent
were blindly estimated
from the hyperspectral image. However, this
model has a large number of parameters. Thus, to
mitigate the ill-posedness of the SU problem, a regularization
term consisting of the Frobenius norm of
dM ,, ,nN1
n
= f was included in the unmixing cost
function. Besides the simplicity and mathematical
tractability, the use of an additive perturbation in (17)
also makes the problem amenable to an interesting interpretation
when only a single additive perturbation
matrix is considered for all image pixels. In this case,
the SU problem becomes equivalent to a total leastsquares
problem with constraints [218]. Furthermore,
the perturbed LMM (PLMM) has been considered for
robust SU using an outlier-insensitive RE metric with
an L -quasip
norm [219] and for multitemporal and distributed
SU [220], [221].
One difficulty of parametric EM models involves
constructing functions (·)f
that are parsimonious and
still flexible enough to represent complex spectral variations.
To circumvent this issue, a deep generative EM
model was proposed in [222], based on the hypothesis
that the EMs lie on low-dimensional manifolds. Instead
of fixing the EM model a priori, variational autoencoders
with neural networks were employed to learn the
parametric function (·)f
in (13) by using pure pixels
extracted from the observed hyperspectral image. SU
was then formulated as the recovery of the abundances
and representations of the EMs in the learned manifold,
which can be of very small dimension. Despite making
SU more well-posed, the resulting cost functions are
nonconvex and can be difficult to optimize. A different
work proposed to exploit the spatial correlation of the
EMs and abundance maps by proposing a general multiscale
mixing model addressing EM variability [223].
The SU problem was solved using a multiscale representation
of the mixing model, which facilitated the use
of any parametric EM representations, as in (13). This
generated improved results when compared to standard
spatial regularization strategies. Although the formulation
was algebraically involved, an approximate algorithm
with little complexity was derived through some
simplifying assumptions.
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021
was initialized using a simple cosine-based k-means
clustering of the observed data cube, which improved
the robustness of the method to the presence of shadowed
pixels.
A different EM model was proposed in [217] by considering
an additive term to the mean EM matrix, resulting in
the following EM representation:
IEEE Geoscience and Remote Sensing Magazine - December 2021

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