IEEE Geoscience and Remote Sensing Magazine - December 2021 - 224

library augmentation and reduction. Finally, we conclude
with discussions and an outline of possible future directions
for the field.
OVERVIEW
Hyperspectral cameras can sample electromagnetic spectra
at hundreds of contiguous wavelength intervals. The high
spectral resolution of hyperspectral images makes them an
important tool for the precise identification and discrimination
of different materials in a scene. Hyperspectral images
significantly contribute to different fields and are now at the
core of a vast number of applications, such as space exploration
[1], land use analysis, mineral detection, environment
monitoring, field surveillance [2], [3], disease diagnosis,
and image-guided surgery [4]. Despite the advantages of
their high spectral resolution, hyperspectral cameras operate
with a delicate tradeoff between spatial resolution and
the signal-to-noise ratio. This happens because the light
observed at the sensor is decomposed into several spectral
bands, which, in turn, demands the pixel size to be large
enough to attain an acceptable signal-to-noise ratio. When
combined with a large target-to-sensor distance, which is
common in many applications, this leads to images that
have a low spatial resolution [5]. The limited spatial resolution
of hyperspectral images means that each image pixel
is actually a mixture of P different pure materials, whose
spectra are termed EMs, that are present in the scene [6].
This mixing process conceals important information about
the pure materials and their distribution in an image. SU
aims to solve this problem by decomposing a hyperspectral
image into the spectral signatures of the EMs and their fractional
abundance proportions for each pixel [7].
The simplest and most widely used model to represent
the interaction between light and the EMs in the scene is
the linear mixing model (LMM) [6], which represents a given
pixel yn
indexed by n with L spectral bands as
yM ,,ae aa110subjecttoand
nn nn n
=+
where [, ,]Mm m,,P0010
umns are the P EMs, an
= f
=
<
$
(1)
is an L × P matrix whose colis
a vector containing the abundances
of every EM in the pixel y ,n and en
for all pixels ,, ,nNy
n
= f
1
is an additive
noise vector. Traditionally, the LMM assumes that
the signatures M0
of the pure materials are the same
in the image. Although this
assumption leads to a well-posed and computationally
simpler framework, it limits the applicability of the LMM
since it can jeopardize the accuracy of estimated abundances
in many circumstances, due to the spectral variability
of the EMs.
SPECTRAL VARIABILITY IN SPECTRAL UNMIXING
Spectral variability is an effect commonly observed in many
scenes in which the spectral signatures of the pure constituent
materials vary across the observed hyperspectral image,
as illustrated in Figure 1. It can be caused, for instance, by
224
variable illumination and atmospheric conditions. Variability
can also be intrinsic to the very definition of a pure
material, such as signatures of a single vegetation species
that differ significantly due to growing and environmental
conditions [8], [9]. In this context, the use of a single matrix
M0
for all pixels in the LMM (1) leads to problems such as
proportion indeterminacy, where errors in the estimation
of the EM spectra at each pixel propagate to the estimated
abundances. This results in erroneous abundance estimation
and the selection of too many EMs to represent the
spectrum of each pixel yn
[8]-[10]. Due to the significant
impact of EM variability on abundance estimation quality,
a lot of effort has been dedicated to developing algorithms
that are able to obtain better abundance estimates in this
scenario. The most general form of the LMM that considers
spectral variability generalizes (1) to facilitate a different
EM matrix for each pixel, resulting in
yM ,ae aa110subjecttoand
nn nn nn
=+
!
=
for nN1 f= ,, , where M Rn
<
$
(2)
LP is the nth pixel EM
#
matrix.
SU for spectral variability can be generally defined as
two complementary problems related, respectively, to the
recovery of the abundances and EMs. These can be defined
as the following:
◗
P ,1 which mitigates the adverse effects of spectral variability
in the abundance estimation
◗ P ,2 which estimates the spectral signatures of the EMs
present in each pixel of the image.
Both problems have attracted substantial interest. While all
SU methods must deal with P1
tral variability, not all of them take P2
due to the additional difficulty that is involved.
CONTRIBUTION, TAXONOMY, AND ORGANIZATION
Many SU algorithms have been proposed to address
problems P1
while accounting for specinto
consideration,
and P .2 Different algorithms follow various
methodologies to represent the EMs in the scene. Existing
methods employ Bayesian, parametric, and spatially localized
models as well as libraries containing different instances
of material spectra known a priori. This multiplicity
of models gives rise to solutions presenting advantages
and disadvantages in terms of computational complexity,
accuracy, and the amount of user supervision. In this article,
we categorize the methods according to criteria that are
most relevant to the practitioner, such as, e.g., computational
complexity, to provide a comprehensive review that
complements and updates previous summaries [8], [9],
[11], [12]. Since existing SU methods that address spectral
variability have very heterogeneous characteristics, navigating
the field can be difficult, especially when accounting
for classical algorithms and recent developments. This
difficulty motivated the present review, which presents
a novel taxonomy aimed at the practitioner as well as a
comprehensive categorization of existing approaches. The
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021
IEEE Geoscience and Remote Sensing Magazine - December 2021

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