IEEE Geoscience and Remote Sensing Magazine - December 2021 - 247

in the scene (e.g., atmospheric effects) [244] and the explicit
representation of the higher correlation between adjacent
spectral bands to introduce spectral smoothness to the signatures,
leading to a well-posed model that is fast to compute
[245]. An alternative approach, which has been used
to simplify the unmixing process associated with the NCM,
is to estimate the EM means and covariance matrices a priori
based on spectral libraries extracted from the observed
image. This has been performed by considering libraries
obtained through pure pixel-based EM bundle extraction
methods [246] and on multiple EM matrices estimated by a
piecewise, convex, blind SU algorithm [191]. However, these
methods suffer from the limitations of image-based EM
bundle extraction techniques, which is discussed in detail in
the " How to Construct Spectral Libraries " section.
Other works considered a piecewise convex model that
uses a set of different Gaussian distributions to model EMs.
Afterward, during SU, each image pixel is assigned to one of
these distributions through a membership function represented
by a Dirichlet random variable. The unmixing problem
in this model was solved by considering an alternating
optimization method in a maximum a posteriori framework
[247] and a Markov chain Monte Carlo sampling approach
providing an estimate of the posterior distribution of interest
[248]. Although the Dirichlet prior distribution is frequently
used to represent abundances, many works have considered
variations that incorporate useful information from the underlying
practical problem. Examples include the enforcement
of abundance sparsity through a sparse Dirichlet prior
[249] and the encouragement of spatial homogeneity by dividing
abundance maps into a finite number of classes sharing
the same Dirichlet distribution parameters. This division
has been performed blindly by means of a classification
prior by using the Potts model [242] and through an a priori
segmentation of the hyperspectral image in a latent Dirichlet
allocation framework [250].
More recently, the NCM has been applied to problems
unrelated to linear unmixing and spectral variability. For
instance, it has been considered for representing the uncertainties
in EM estimation instead of the intrinsic variability
of the material classes, which changes the problem by introducing
statistical dependence between the different image
pixels [251]. Other works applied the NCM to problems
such as nonlinear SU with a bilinear mixing model [252]
for the linear unmixing of sediment grain size distribution
(where EMs represent the grain sizes of constituent materials)
to study the transport and deposition of sediments
[253] and represent the variability of the EMs across multiple
images in multitemporal SU by using additional spatially
sparse terms accounting for potential abrupt spectral
changes between the different images [254].
OTHER ENDMEMBER DISTRIBUTIONS
Despite its popularity, the NCM does not have a strong
physical motivation, which led to the consideration of more
accurate distributions to represent EMs. For instance, a beta
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
distribution was considered in [255] to constrain reflectance
values to physically meaningful ranges and enable possible
skewness in the distribution. Unfortunately, a direct solution
to the SU problem cannot be obtained. Thus, a piecewise
constant model was assumed for the abundances, which
enabled the parameters of the distribution to be estimated
using a combination of a clustering algorithm and a variant
of the method of moments. A Gaussian mixture model was
also considered in [256] to facilitate possibly multimodal
EM distributions. The SU problem was solved as a maximum
a posteriori estimation problem using a generalized
expectation-maximization approach. However, since learning
the parameters of Gaussian mixture models can be difficult,
the distributions were estimated before performing SU,
based on spectral libraries assumed to be known a priori.
Another approach proposed to represent EM spectra as
a sum of an average spectral signature known a priori and
a spatially and spectrally smooth function representing EM
variability to provide a model that is physically more reasonable
[257]. Bilinear mixing models were also considered
with an additive residual term to account for mismodeling
effects and outliers. A different technique has been offered
that does not make an explicit assumption about the distribution
of EM spectra and, instead, relies only on some of
their statistics. This is the case in [258], which formulates
the SU problem similarly to the method of moments by trying
to find the abundance values that match the mean and
covariances obtained through the LMM to those of the observed
mixed pixels. A similar work applied the same idea
by using transformed statistics constructed from the ratio
between the means and covariances of the pixels and EMs
in different spectral bands [259]. This strategy increases the
robustness of the method since band ratios are invariant to
illumination variations. However, similar to [255], a piecewise
constant abundance model is employed to estimate the
covariance matrix of the observed pixels. Moreover, the covariance
matrices of the EMs are assumed to be known a priori.
SPECTRAL LIBRARIES
A large number of the SU techniques discussed in the
" Unmixing Methods That Use Spectral Libraries " section
address spectral variability by using spectral libraries and
bundles known a priori. The performance of these methods
is often heavily impacted by how well the libraries can
represent the EMs actually present in the scene. Moreover,
in many practical situations, it is either very costly or even
impossible to obtain laboratory and in situ measurements
of EM spectra. Another issue with many methods presented
in the " Unmixing Methods That Use Spectral Libraries " section
(such as MESMA) is that the approaches' computational
complexity increases very quickly with the library size,
which can make the problem intractable for large libraries.
Thus, the challenges of removing redundant and irrelevant
spectra before SU and, especially, extracting spectral libraries
directly from observed hyperspectral images are of central
importance to enable the techniques discussed in the
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IEEE Geoscience and Remote Sensing Magazine - December 2021

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