IEEE Geoscience and Remote Sensing Magazine - December 2021 - 238

signatures for each EM when generating the synthetic training
data set. This has been done by directly applying regression
methods [122] and by converting SU into a classification
problem by quantizing the solution space of abundance
values and using a one-against-all strategy [123]. Another
work modified the SVM cost function to directly minimize
the unmixing RE during the training process [124].
Usually, these approaches result in extremely large training
sets for sizable spectral libraries. Thus, even though some
strategies, such as bootstrap aggregation, have been employed
to speed up the training process [125], [126], the computational
cost is still very high. Although methodologies
to discard irrelevant (regarding the impact on performance)
subsets of the training data [127], [128] could, in principle,
be applied to accelerate training, recent works have focused
on modifying the algorithms to reduce their complexity.
One of the main reasons for this complexity is that the
training data must jointly describe spectral variations due
to changes in the abundances and EMs. Some works have
tried to address this issue by using only pure pixels from a
spectral library as training data. One such approach, which
received considerable attention, consists of extended SVMs.
Extended SVMs employ hybrid soft-hard classification and
regression to address spectral variability. It is assumed that
the spectral space is separable by hyperplanes delimiting
two complementary regions containing only pure and only
mixed pixels, respectively [129]. The extended SVM is then
trained to find a soft-hard classifier containing 1) a hard
classification rule consisting of the hyperplanes delimiting
the regions in which the pixels are considered pure and 2) a
soft classification rule, which determines the abundances of
the pixels considered to be mixed.
Different forms of the extended SVM have been studied,
using either a single kernel [129] or multiple kernels [130],
accounting for the abundance indeterminacy by computing
the maximum and minimum proportion values similarly to
the fuzzy SU procedures [131] and using Fisher discriminant
analysis (FDA) to reduce the within-class spectral signature
variability in the spectral library before training [132]. Although
hybrid soft-hard classification methods can be fast
to train, they lack a clear physical interpretation of the results
since they have no direct relation to the physical mixing
model. Moreover, the influence of spectral variability on the
regions of the spectral space containing mixed pixels is limited
because it comes only from the marginal hyperplanes
that separate the pure from the mixed pixel regions [129].
A related strategy that uses only pure pixels in the training
process consists of modeling the latent function from the
mixed pixel spectra to the abundance maps in a probabilistic
framework as a multitask Gaussian process [133]. In this
case, the abundance means and covariance matrices are obtained
through the posterior distribution of the abundances
conditioned on the training set (i.e., the spectral library) and
the mixed pixels. This strategy was extended to consider spatial
correlation in a two-step procedure by using the Gaussian
process results from [133] as input to the abundances
238
prior information in a maximum a posteriori estimation
problem [134]. Although this strategy has a strong statistical
motivation, the introduction of additional constraints
(e.g., abundance nonnegativity and sum-to-unity) is not
straightforward and results in high computational complexity.
Another work proposed to mitigate the influence of spectral
variability by processing the image using a geodesic SU
method [135] before applying Gaussian process regression to
estimate the final abundances [136]. Although possibly inaccurate,
the preliminary abundances estimated by the geodesic
SU algorithm are not affected by EM variations caused by
differences in illumination and acquisition conditions. The
Gaussian process regression then learns to map the inaccurate
initial abundances to the desired ones. Despite increasing
the robustness to spectral variability, geodesic SU can
introduce significant distortions in the abundances for complex
data manifolds, which may not be trivial to mitigate.
Note that other machine learning techniques have been
employed to perform SU without directly addressing spectral
variability. These include the use of convolutional neural
networks [137], the consideration of neural networks
that are well adapted to learn from fewer samples [138], and
the use of autoencoders to perform unsupervised SU by
identifying the latent codes with the fractional abundances
and the decoder with the mixture model [139]-[142]. Other
works considered specific neural network architectures inspired
by unfolding iterative optimization algorithms [143],
and they employed Hopfield neural networks to optimize
the SU cost functions more efficiently [144]. Machine learning
methods have been applied in different experimental
settings, such as unmixing spectrally similar vegetation
types [145] and urban surfaces [146] and using training data
collected at multiple locations [147]. Given the success that
machine learning methods are achieving with different
problems, particularly in the area of remote sensing [148],
[149], such techniques may bring important advances if
used to address the EM variability problem in SU.
SPECTRAL TRANSFORMATIONS
SPECTRAL TRANSFORMATIONS
Spectral transformations have the following qualities:
◗ They can be seen as a " preprocessing " strategy that can be used with
other library-based SU methods.
◗ They are conceptually simple and have low complexity.
◗ Many of the methods are empirical and require a significant degree of
expert knowledge about the underlying application.
◗ The performance of the less supervised methods depends strongly on
the representativeness/quality of the library.
An approach frequently used to mitigate the effect of
spectral variability in library-based SU consists of selecting
a subspace of the spectral space that is minimally influenced
by the variability of the EMs to be prioritized in
the unmixing process. This idea was introduced to improve
the classification of materials under varying atmospheric
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021
IEEE Geoscience and Remote Sensing Magazine - December 2021

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