IEEE Geoscience and Remote Sensing Magazine - December 2021 - 254

the OpenRemoteSensing website, which aims to share and
disseminate code and papers, has an increasing number of
SU methods, some of which consider spectral variability.
EXPERIMENTAL SETUP AND RESULTS
We now present a simulation to illustrate the application
of some of the algorithms reviewed in this work. Note that
this is merely illustrative and not a comprehensive performance
evaluation. We generated a synthetic hyperspectral
image containing vegetation, soil, and water as constituent
materials. Spatially correlated abundances with 50 50#
pixels were first sampled from a Gaussian random field.
Then, we followed the procedure described in the " Generating
Synthetic Endmember Variability " section to generate
different EM spectra for each material in the scene.
The PROSPECT-D model was used to produce vegetation
spectra, while the simplified Hapke and atmospheric models
were used to create dirt and water spectra, respectively,
at different viewing geometries. The generated synthetic
signatures, containing L 198=
bands, can be seen in Figure
16. The EMs contained in each pixel were randomly
sampled from this set of synthesized signatures, and the
pixel spectra were then generated following the LMM with
variability in (2), with white Gaussian noise added to the
image to obtain a signal-to-noise ratio of 30 dB.
To evaluate the SU results, we considered as quantitative
quality measures the root-mean-square error (RMSE) and
the SAM. The RMSE between two generic variables X and
Xt
is defined as
X
N XX1
RMSE =-t
X
where
$
F
2
F
,
is the Frobenius norm and NX
and the estimated pixel-dependent EMs Mn
t
(22)
denotes the
number of elements in X. We used the RMSE to evaluate
the estimated abundances ,At
the reconstructed images ,Yt
(for the cases
when this estimate was available). The SAM was also used
to evaluate the estimated EMs as
0.1
0.2
0.3
0.4
0.5
0.6
0.51 1.5 2
(a)
Wavelength (µm)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0.51 1.5 2
(b)
Wavelength (µm)
SAM = LPN n=1 p=1
M
1 / / eo (23)
<
N
P
arccos
t
mm
mm
t
pn pn
pn pn
,,
,
,
,
where N is the number of pixels and P the number of materials
in the hyperspectral image. We assessed the complexity
of the algorithms through their execution times,
measured on an Intel Core i7 processor with 4.2 GHz and
16 Gb of random-access memory. Finally, to increase the
reliability of the results, we executed the simulation for 10
independent Monte Carlo realizations and reported the average
values for all metrics.
ALGORITHM SELECTION AND SETUP
For illustrative purposes, we considered the recovery of the
abundance maps following four different paths in the decision
tree in Figure 2 (selected according to the algorithm implementations
available in Table 3). These included the following:
1) small spectral libraries extracted directly from the image,
with no expert knowledge available
* less user supervision: MESMA and its variants [37]
* lower computational cost: sparse unmixing (fractional
sparse SU [84])
2) spectral libraries not available a priori
* less user supervision: Bayesian methods [NCM published
by Eches et al. (NCM-E) [239] and beta compositional
model (BCM) [255], [317]]
* lower computational cost: parametric models [ELMM
[51] and deep generative unmixing (DeepGUn)
[222]] and EM-model-free methods [robust unmixing
by variable splitting and augmented Lagrangian
(RUSAL) [227]].
We additionally considered the FCLS solution as a baseline,
using a single set of EMs extracted from the image using
the vertex component analysis (VCA) algorithm [278].
The EMs extracted by VCA were also used as initialization
and reference/mean signatures for some of the algorithms
(ELMM, DeepGUn, RUSAL, and NCM-E). For MESMA and
sparse SU, the spectral libraries were extracted from the
0.15
0.1
0.05
0.5 1 1.5 2
(c)
Wavelength (µm)
FIGURE 16. Generated spectral signatures used in the synthetic hyperspectral image. (a) Vegetation. (b) Soil. (c) Water.
254
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021
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IEEE Geoscience and Remote Sensing Magazine - December 2021

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