IEEE Geoscience and Remote Sensing Magazine - December 2021 - 239

illumination conditions [17], [20]. The majority of these
methods are based on affine transformations of the observed
pixels, defined as
Wy bW Ma We b ,
mm m
nn
+= ++nn nn
where the matrix Wm and the affine term bn
(10)
are determined
to minimize the effects of EM variability in the subsequent
SU process. Besides modifying the observed pixel
spectrum ,yn
this transformation is applied to the elements
of the spectral library, yielding
m ++m
W {: }
M bWmb m M
pn np
_!
(11)
for pP1 f= ,, . Different cases of this model have been
considered in the literature, most notably with Wm
being
a diagonal matrix with positive real (band weighting) and
binary (band selection) elements. Note that although traditional
dimensionality reduction [e.g., principal component
analysis (PCA)] and band selection methods used to
compress the hyperspectral image could be implemented
through this transformation with
,
LIBRARY-BASED SPECTRAL TRANSFORMATIONS
Spectral transformations proposed more recently leverage
information contained in the spectral library to compute
the terms Wm
and bn of the affine transformation. This
circumvents one of the main downsides of the previous
approaches by making the process automated instead of
delegating the choice to the user. These techniques can be
further divided in three groups, namely, band selection,
band weighting, and more general spectral transformations.
b 0n = the direct application
of compression techniques does not necessarily improve
the robustness to spectral variability [150]. Spectral
transformation approaches can be generally divided into
two major groups: those defined a priori based on the user's
expert knowledge and those constructed automatically by
incorporating information in a spectral library. We review
each case in the following.
USER-DEFINED SPECTRAL TRANSFORMATIONS
The first user-defined spectral transformations were proposed
to normalize the effects of illumination and brightness
variations and to emphasize useful spectral features.
These approaches include subtracting the reflectance value of
a selected (specific) spectral band from all remaining bands
[99], subtracting from each EM its mean value in the spectral
dimension to reduce the variability due to differences in
brightness [151], and normalizing/dividing the reflectance
value at each wavelength by the corresponding value of the
convex hull of the spectral signatures [152]. Other examples
include using the first or second derivatives [153], [154] and
the wavelet transform of the spectral signatures [150] for SU.
A spectrum-based approach that has become very popular
for solving this problem consists of using band selection
methods. These techniques basically work by performing SU
via selected wavelength intervals in which there is little spectral
variability between different spectral signatures of the
same material [9], [99]. Although many of these approaches
rely on expert knowledge about the specific underlying application,
they are simple and easily interpretable and also
help in reducing the computational cost of the SU problem.
Examples of band selection methods defined a priori by the
user include the use of the short-wave infrared 2 spectral region
(2,100-2,400 nm) for unmixing soil and vegetation in
arid and semiarid environments [99] and the combination
of various spectral regions, such as visible, near infrared, and
short-wave infrared, for other applications [100].
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
BAND SELECTION
Newer band selection methods seek to identify robust spectral
regions based on samples in the spectral library. Different
strategies have been proposed. One of the first approaches
is based on the analysis of the spectral residuals
obtained by performing a preliminary unmixing of the image
using the LMM with an average EM matrix [155]. Only
the spectral bands with minimal residual variance are then
used for SU, based on the empirical observation that they
correspond to more robust spectral zones. Another method,
called stable zone unmixing, proposes to select spectral
bands that are robust to spectral variability by minimizing
an instability index defined as the ratio between the
intra- and interclass EM variances (computed based on a
spectral library) [156]. This method was later extended to
minimize the instability index and the correlation between
signatures of different EM classes at the same time, aiming
to improve the numerical conditioning of the SU problem
[157]-[159]. The work [160] proposed to improve the
separability between classes by employing the stable zone
unmixing framework to select an individual set of spectral
bands for each possible subset of EM/material classes that
could be tested with MESMA when considering EM models
with fewer than P signatures in the SU process.
BAND WEIGHTING
Band weighting methods are more flexible techniques that
enable one to prioritize the spectral bands in the unmixing
process according to their reliability and significance by using
a continuous weight term. This is usually done by weighting
the RE of each band in the SU cost function. Different approaches
have been proposed to compute the weight to be
applied to each band. For instance, a weighting strategy based
on two terms was offered in [161]. One term normalizes the
energy of the reflectance spectra to equalize the contributions
to low- and high-reflectance bands, and another term accounts
for the robustness of each band to spectral variability
through its instability index (i.e., the ratio between the intraand
interclass EM variance). This approach was later applied
to monitor the level of defoliation in Eucalyptus plantations
[162] and invasive plant species via multitemporal data [163].
It was later extended in [164] to consider SU that integrated
reflectance and derivative spectra. Band weights based on the
instability index were used to prioritize the more stable spectral
bands when designing spectral filters that were robust to
spectral variability, which are low-complexity alternatives
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IEEE Geoscience and Remote Sensing Magazine - December 2021

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