IEEE Geoscience and Remote Sensing Magazine - December 2021 - 246

THE NORMAL COMPOSITIONAL MODEL
The first statistical model considered for the representation
of EM spectra was a multivariate Gaussian distribution, in
the so-called normal compositional model (NCM), given by
m (),N+ inp,
pn,
where ND / and {, } contain the
i = mean covariance
np,
mean vector and covariance matrix for the pth EM of the nth
pixel. The NCM has been widely used due to its mathematical
tractability [233], [234]. The first works employing
it for SU considered expectation-maximization strategies
in which the abundances and the mean EM values
and their covariance matrices were iteratively estimated
[233]. However, due to the nonconvexity of the estimation
problem, a direct application of expectation-maximization
approaches is unable to decide whether variations
observed in the mixed pixel spectra yn
(19)
are due to different
abundances or to EM variability. This might result in the
EMs absorbing all the variation in the observed scene with
nearly constant abundances [234]. Some techniques proposed
to address this problem by considering the use of
diagonal covariance matrices and empirical strategies to
estimate the EM data more easily from the observed mixed
pixels. For instance, EM means and covariances were estimated
a priori by using pure pixels selected from the hyperspectral
image [235], and they were iteratively based
on large regions of observed pixels with homogeneous
abundances (obtained from the segmentation of estimates
of the abundances available a priori) [236].
Other works attempted to improve different aspects of
this method by using a particle swarm optimization algorithm
to solve the (usually intractable) integrals involved
in the estimation of the abundances in the " expectation "
step of the algorithm [237] and by incorporating a priori
information in the form of additional constraints penalizing
the nuclear norm of the abundances in groups of pixels
determined through image segmentation methods (to promote
spatial homogeneity) [238]. Despite these advances,
the susceptibility of expectation-maximization-based methods
to converge to poor local minima of the nonconvex cost
function prevented their large-scale applicability for this
problem. Instead, most recent approaches rely on more robust
(although costly) techniques based on Markov chain
Monte Carlo methods to sample the posterior distribution.
Although the works that adopt this approach share the same
underlying idea, they differ significantly in the way in which
EMs and abundances are represented and the amount of
user supervision that is required.
For example, different strategies have been proposed to
represent the mean and the covariance matrices of the EMs
in the NCM. One of the first considered the EM mean values
to be known a priori and their covariance matrices to be
multiples of the identity matrix [239] while employing conjugate
distributions to make the estimation of the parameters
easier. Later works attempted to add more flexibility
by considering, for instance, a single full covariance matrix
shared by all EMs [240] or a positive definite matrix defined
a priori and multiplied by EM-dependent scaling parameters
[241]. Diagonal covariance matrices were employed in
[242], which also considered the estimation of the EM mean
values in a hierarchical Bayesian framework, using hyperpriors
to directly estimate the distribution parameters from the
observed hyperspectral image. The Bayesian framework was
also applied in [243] to blindly estimate the number of EMs
in the scene via a uniform discrete prior.
Other works attempted to address physically motivated
cases of the general NCM. These included consideration of
the statistical dependence between different EMs to represent
spectral variability that may equally affect all materials
Estimated Abundances
and EM Distributions
Observed Image
Bayesian
Inference
p(YA, θM) p (A, θM) ∝ p (A, θMY)
Likelihood Prior
Posterior
FIGURE 14. A description of Bayesian SU techniques. Bayesian SU methods represent the EM signatures at each pixel as a realization of a
statistical distribution. Statistical distributions are first attributed to the EMs and abundances and, possibly, other variables or to hyperparameters
of these distributions. Using the Bayes rule, the SU results are then derived from the posterior distribution in a Bayesian inference
problem. The abundances and EM distributions can be computed as, e.g., the mean or mode of the posterior distribution.
246
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021

IEEE Geoscience and Remote Sensing Magazine - December 2021

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