IEEE Geoscience and Remote Sensing Magazine - December 2021 - 242

A different line of work attempted to relax the assumption
of the connectedness of the local spatial regions, performing
SU in different subsets of the hyperspectral image
that were not necessarily spatially adjacent. For instance,
the piecewise convex model proposed in [189] considered
a set of different EM matrices, all estimated from the entire
image. Each pixel was then assigned to one of these
EM matrices through a (fuzzy) membership function,
which was estimated with the other variables in a nonconvex
matrix factorization problem. Other works extended
this approach by considering cluster validity indices [190]
and sparsity-promoting priors [191] to estimate parameters
such as the number of EM matrices and material
classes in each segment and using spatial constraints to
encourage neighboring pixels to have similar membership
values [192].
Similar work considered the estimation of multiple EM
matrices in a nonnegative matrix factorization framework
by using abundance sparsity constraints instead of employing
(fuzzy) membership functions while also penalizing
the mutual coherence between the signatures of different
material classes to improve interclass separability [193]. A
related strategy considered a self-dictionary model, where
multiple EM signatures were directly selected as the hyperspectral
image pixels that could best reconstruct most of
the remaining pixels in the scene as a sparse linear combination
[194]. Another approach with even more flexibility
considered an individual EM matrix for each image pixel
in a nonnegative matrix factorization formulation [195]. A
regularization term penalizing the trace of the covariance
matrix of the estimated spectral signatures for each class
was also considered to reduce the ill-posedness of the estimation
problem.
PARAMETRIC MODELS
PARAMETRIC ENDMEMBER MODELS
Parametric EM models possess the following traits:
◗ The SU algorithms are computationally efficient.
◗ They are very flexible and physically motivated models to represent
any kind of variability.
◗ It is easy to incorporate prior information.
◗ Determining a good EM model might require some degree of expert
knowledge.
◗ They require significant user supervision for tuning the free model
parameters.
◗ Estimating the parameters of the EM models (along with the abundances)
can be challenging due to the presence of nonconvex optimization
problems and sensitivity to parameter choice and initialization.
A flexible and physically reasonable way to address spectral
variability consists of employing parametric models to
represent EM spectra; see Figure 13. These strategies provide
great freedom to incorporate constraints and information
from the underlying application. They are generally based
on representing the EM spectra as
MMf (, ),nn0
=
i
n .
(13)
where (·)f is a function of an average or reference EM matrix
M0
and a vector of parameters i The number of parameters
in ni is usually small, which enables one to confine the
EM spectra to a low-dimensional manifold. The SU problem
is then formulated as the recovery of the abundances and parameters
n
i for all pixels in the image. The model in (13)
can be defined based on the underlying physics describing
material spectra as a function of numerous geometric and
Parametric EM Model
Estimated
Abundances and EMs
{θn,an}N
min
n = 1
n = 1
N
Σ
, A
min
yn
- f(M0, θn)an2 + R ({θn}n = 1, A)
N
Optimize
Cost Function
φ gφ(Y - M0A) +R (φ, A)
Observed Image
EM-Model-Free
Objective Function
Regularization
Terms
FIGURE 13. A description of SU techniques based on parametric EM models and EM-model-free SU approaches. Parametric EM models
represent the (variable) signatures of EMs as a function of a low-dimensional vector of parameters. The abundances and vector of EM
parameters for each pixel are then recovered by solving an optimization problem. EM-model-free methods, on the other hand, generally
attempt to indirectly mitigate spectral variability through the design of robust cost functions using, e.g., additive residual terms. The use
of regularization terms is important in both cases to incorporate a priori knowledge about the problem.
242
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IEEE Geoscience and Remote Sensing Magazine - December 2021

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