IEEE Geoscience and Remote Sensing Magazine - December 2021 - 237

in a number of works proposing improvements, such as the
use of alternative sparsity-promoting penalties [110], [111]
and different means of spatial regularization [112], [113].
Sparse unmixing methods would merit a more comprehensive
review, which is beyond the scope of this article. Thus,
in the following, we restrict ourselves to modifications of
the sparse SU framework specifically aimed at dealing with
spectral variability and structured libraries.
In [114], L -norm
21,
based group sparsity constraints were
used to favor the selection of abundance vectors containing
many entire material classes with zero proportions. A later
formulation considered a fractional group (p, q)-norm sparsity
constraint as a generalization of the approaches based
on the L ,21
norm [84]. The (p, q)-norm penalty provides
better control of the sparsity within each group of variables
as well as the addition of the sum-to-one constraint. However,
this comes at the expense of making the optimization
problem nonconvex. Another sparse SU formulation
[115] proposed to explicitly represent mismatches between
the library spectra and the hyperspectral image caused by
different acquisition conditions. In this case, the spectral
signatures of the library are also estimated in the SU process.
However, they are constrained to be within a given
Euclidean distance of a corresponding element of the library
known a priori. This enables the estimated signatures
to vary arbitrarily within Euclidean balls centered at the library
elements to compensate for spectral mismatches.
A different approach [116] was to modify the LMM for
unmixing mineral spectra in mining applications by including
an additional term representing the mixture of the
" background " spectrum of the EMs. This background spectrum
was defined as the low-frequency part of the spectral
signatures and estimated a priori from the library as a parametric
function of smooth splines. The performance of an
L1
norm-based sparse SU framework under this model was
reported to be similar to MESMA, albeit at a much smaller
computational cost.
MACHINE LEARNING ALGORITHMS
MACHINE LEARNING ALGORITHMS
The following is true of machine learning algorithms:
◗ They are very flexible approaches that, in principle, can deal with any
effect that is represented in the training data.
◗ Most methods have significant computational complexity or do not
have a clear physical motivation.
◗ The SU quality depends on the representativeness of the training data
(which are usually generated using a spectral library).
◗ They generally do not return EM spectra for each pixel.
Some works propose to address spectral variability using
machine learning methods (Figure 11) by formulating SU
as a supervised regression problem. The objective is to learn
transformations that map the observed (mixed) pixel to the
abundance fractions [117]-[121] using a supervised training
procedure. Mixed pixels with known proportions are employed
as training data for algorithms, such as neural networks,
random forests, and support vector machines (SVMs).
These techniques can be straightforwardly adapted to address
spectral variability by considering multiple spectral
Spectral
Library
Synthetic
Abundances
Mixed
Pixels
Supervised
Learning
Estimated
Abundances and EMs
Training Data
Observed ImageNeural Network
Hidden
Layers
FIGURE 11. A description of machine learning-based SU techniques. The flexibility and representation power of machine learning algorithms
can be exploited to address spectral variability by formulating SU as a supervised learning problem. One simple approach is to learn
a mapping between the mixed pixels and the abundance and EMs based on training data generated synthetically via a spectral library. However,
the incorporation of expert knowledge about the SU problem in the design of the machine learning algorithm is important to obtain
better performance and effectively address spectral variability.
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
237

IEEE Geoscience and Remote Sensing Magazine - December 2021

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