IEEE Geoscience and Remote Sensing Magazine - December 2021 - 235

The MESMA algorithm [37] and its variants (sometimes
also referred to as iterative mixture analysis cycles) are among
the most widely used for this task. These methods (Figure
10) enable the EM signatures to vary on a per-pixel basis
while following the model in (4). The unmixing problem
is solved by searching for the EM and abundance combinations
that result in the smallest reconstruction error (RE)
for each observed pixel, i.e.,
argmin yM a
M aa
2
-
nn n
subject to 011M
,
aM
nn
nn n
! ,, .
$
<
=
(6)
The EM matrices Mn constructed by taking spectra from
the bundles are sometimes called EM models.
The MESMA algorithm has been employed in a wide
variety of situations, including natural, urban, and extraterrestrial
environments [9, p. 1,607] as well as single and
multidate scenarios [85]. However, even though MESMA
is very amenable to parallelization [86], it consists of a
combinatorial optimization problem whose associated
computational cost can become very high. More specifically,
its computational cost scales as the product of the
sizes of the individual libraries, as it consists of solving a
fully constrained least-squares (FCLS) problem,
P =1
p M
P
p
times [87]. This can make the unmixing complexity unrealistic
for large library sizes. Furthermore, the problem
(6) can become ill-posed when there are many EMs in
the bundles since different material combinations can
lead to very similar REs. To circumvent these limitations,
several modifications to the original MESMA algorithm
have been proposed.
Many variants of MESMA aim to provide computationally
efficient approximate solutions to (6). The simplifications
consist of stopping the exhaustive search
optimization procedure (6) early by selecting the first EM
model that presents an RE that is below a threshold and
well distributed across spectral bands [37]. Another proposed
approach is to solve (6) approximately by performing
unconstrained least squares with every possible EM
model and then select the solution that yields positive
abundances and the smallest RE [88]. Although these simple
modifications successfully reduce the computational
complexity of MESMA, the approximations involved can
also negatively impact the abundance reconstruction results
[89], which imposes practical limitations on the selection
of the thresholds and tolerances. This motivated
the consideration of more elaborate strategies to provide
better complexity reductions without impacting the unmixing
performance.
An alternative approach to MESMA attempted to
lessen the computational complexity by solving an angle
minimization problem with respect to each library [87],
[90]. Although not guaranteed to converge to the optimal
solution of (6), this strategy performed similarly to MESMA
on practical experiments, and it scales linearly with
library sizes, leading to computational improvements for
large numbers of signatures ,, ,,
M pP1= f
p
in the EM
bundles. Another work considered a mixed integer linear
program reformulation of the MESMA problem. This approach
enables a more efficient computation of an exact
solution to (6) for small- to medium-scale problems [91].
A simple technique that is largely employed to reduce
the computational complexity of MESMA is to perform a
careful pruning of the spectral libraries ,, ,.
Mp pP1= f
This process attempts to remove redundant and irrelevant
spectra from the libraries before unmixing. These
strategies are described in detail in the " Library Reduction
Techniques " section.
Sparse
Unmixing
min
A Y- MLibAF,
2
Subject to
Sparsity Constraints
Physical Constraints
Structuring Constraints
Optimize
Spectral
Library
Cost Function
Estimated Abundances
and Endmembers
Observed Image
MESMA
Perform SU With
Every Possible
Combination of EMs
Select Abundances and
EMs With the Smallest
Reconstruction Error
FIGURE 10. The MESMA, fuzzy, and sparse SU techniques. MESMA and sparse SU are the main methods based on spectral libraries. The basic
principle behind MESMA is to iteratively search for the combination of EM signatures in the library that, among all possibilities, enables
the closest reconstruction of each observed pixel under the LMM. Sparse SU, on the other hand, performs EM selection and abundance estimation
in a single optimization problem by using sparsity and structuring constraints and penalties, which facilitates faster processing times.
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
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IEEE Geoscience and Remote Sensing Magazine - December 2021

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