IEEE Geoscience and Remote Sensing Magazine - December 2021 - 243

photometric parameters, such as Hapke's [48], [49] and
Shkuratov's [196] models for packed particles and the PROSPECT
and PROSAIL models for vegetation [68], [69]. However,
the model (13) can also be inspired by physics but chosen
to facilitate more flexibility and mathematical tractability.
We review these approaches in the following.
PHYSICS-BASED METHODS
The first SU approaches using parametric models aimed
to obtain fractional abundances from intimate mineral
mixtures by inverting the Hapke model [197]. With perfect
knowledge about the viewing geometry, the scattering
properties of the different materials, and the single
scattering albedo of the EMs, the SU problem using
Hapke's model becomes linear in the albedo domain
[198]. However, since these variables are hardly available
in practice, many works attempted to blindly invert
Hapke and related models. This inversion is mathematically
and computationally very difficult, in general, and
requires hyperspectral images acquired at multiple viewing
geometries [197]. Thus, subsequent works proposed
simplifications of the scattering characteristics of the
materials in the model (13) to improve its mathematical
tractability [199]. These methods have been successfully
applied to estimate abundance maps of different scenes,
including the Cuprite mining district in Nevada [200]
and the moon [201], [202].
This approach has also been applied to the SU of
vegetation mixtures based on the inversion of radiative
transfer models. The first works simplified the problem
by assuming external knowledge of biophysical parameters.
For instance, a model for mixtures of vegetation
and shadowed and illuminated soil was proposed for
SU by approximating plant geometry with spatially distributed
cylinders containing layers of leaves [203]. Although
spectral variability was supported by means of
changes in biophysical parameters, these were assumed
to be known a priori to solve the SU problem. Another
approach considered the SU of soil and vegetation by
using a simplified mixing model as a function of NDVI
values instead of the full spectral signatures [204]. In
this case, a physical model was employed to represent
the variability of the NDVI values as a function of parameters,
such as the viewing geometry, leaf density,
and clumping effect. However, the NDVI " EMs " for each
pixel had to be estimated before SU by using multiangled
observations and assuming prior knowledge of the
leaf biophysical parameters.
A later technique for the SU of soil and vegetation
mixtures proposed to blindly estimate the biophysical
parameters from the hyperspectral image using the
PROSAIL model for vegetation spectra [205]. The SU
problem was formulated as the recovery of both the
abundances and the two parameters of the PROSAIL
model and solved through an alternating optimization
procedure. Note that the other parameters of the
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
PROSAIL model had to be fixed a priori. Although these
models possess a strong physical motivation, their use
in SU leads to computationally intensive and mathematically
challenging (i.e., nonconvex and significantly
ill-posed [206]) problems. This occurs because physics
models were originally devised as forward representations
that accurately describe reflectance spectra based
on a set of parameters and were not designed to be inverted,
which limits their use for SU in practical problems
[207].
PHYSICALLY MOTIVATED AND
NONPHYSICS-BASED METHODS
The low mathematical tractability of physics-based models
has motivated recent studies leading to more flexible
and parsimonious models that are inspired only by the
underlying physics. Although these models are not as precise
as those presented in the " Physics-Based Methods "
section when representing physical phenomena underlying
spectral variability, they enable more efficient SU algorithms
that estimate the involved parameters n
i from
the observed image. Moreover, although models inspired
by physics can be ill-posed, EM spectra are often confined
to a low-dimensional manifold since they depend only on
a small number of physicochemical variables. This property
can be exploited to design parsimonious models with
possible constraints and reduce the ill-posedness of the
SU problem.
Several parametric models have been proposed with
these objectives. One of the resulting SU algorithms is the
scaled constrained least-squares method [18], which attempts
to represent uniform illumination variations in each
pixel by introducing an additional scaling factor
}
into the EM matrices as
MM .nn 0
=}
(14)
SU can be performed using the model (14) by solving a
simple nonnegative least-squares problem, which is convex
and computationally efficient. However, this representation
lacks the ability to represent the more complex spectral
variability that has been observed in practical scenes, motivating
the search for more flexible models.
A version of the LMM, the extended LMM (ELMM),
was proposed in [51] and [208] by enabling each EM
in a pixel to be individually scaled by a constant factor,
resulting in the following representation for the
EM matrices:
MM diag (),nn0
=
n RP
where vector } ! +
}
(15)
contains the scaling factors for each
of the P materials. The ELMM can represent more complex
variability originated from variations in the illumination
and topography, which can affect each material
hyperspectral image differently. Furthermore, the ELMM
in the
243
n R!
+
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