IEEE Geoscience and Remote Sensing Magazine - December 2021 - 234

UNMIXING METHODS THAT
USE SPECTRAL LIBRARIES
Spectral Mixture Analysis and Its Variants for Small Spectral
Libraries " and " Sparse Unmixing " sections.
Different methods have been proposed to solve the SU
METHODS THAT USE SPECTRAL LIBRARIES
Here, the following characteristics are present:
◗ The approaches are usually conceptually simple and easy to interpret.
◗ The quality of the SU results strongly depends on the spectral
library.
One of the main approaches to addressing spectral variability
in SU is to consider large libraries of spectra acquired
a priori. These libraries contain different instances of each
material in a scene, and the unmixing problem becomes generally
equivalent to finding which signatures can best represent
each pixel in the scene. Different algorithms have been
proposed for this task, which we review. The spectral libraries
used by these methods are sometimes called bundles, and,
in principle, they should account for all possible variations of
each material. Mathematically, they are represented as
M mm pP1
uu
pp,,pMp
1
where Mp
spectral signatures m RL
u
pi,
==
is a library/bundle containing Mp
!
{, ,},, ,,ff (4)
reference
of the pth material and P is the
number of materials in the scene. The spectral signature of
each material in the nth pixel yn
age is then represented as an unknown element m Mnp,
of the hyperspectral im!
p
belonging
to this bundle.
Those sets can be readily used to constrain the EM matrices
of the LMM for the N pixels to belong to a new set
,
M Mn
! withnN1 f= ,, , where
M {[ ,, ], ,, ,}== (5)ff!mm pPm
1 Pp Mp
1
is the set of all possible EM matrices, with Mp
P
P =1
p elements.
This definition assumes that only one signature from each
library, ,, ,pP1Mp
M ,p
= f , is present in each pixel. However,
other representations of the EM signatures as, e.g., sparse
and convex combinations of the elements in
can also
be considered to obtain more flexibility (see, e.g., [82]-[84]).
Such strategies are discussed in the " Multiple-Endmember
problem by using spectral libraries. These can be roughly
divided into four groups of formulations: multiple-EM
spectral mixture analysis (MESMA), sparse SU, machine
learning, and spectral transformations. The MESMA algorithm
and its variants formulate SU as a computationally
demanding optimization problem and achieve good quality.
Sparse SU formulations use mathematical relaxations
to the MESMA problem that are computationally easier to
solve. Machine learning algorithms provide more flexible
ways to perform SU but also at a large computational complexity.
Spectral transformations are empirically oriented
techniques that can be employed to improve methods from
the first three categories.
Although these families use spectral libraries to address
spectral variability in SU, the reasoning underlying each of
them can be quite different, leading to varying degrees of
required user supervision, computational complexity, and
abundance estimation quality, as illustrated in Figure 2.
Moreover, additional prior knowledge can be considered
in different ways, including, e.g., the design of principled
neural network architectures and the manual specification
of the robustness of particular spectral bands to variability.
We review each family of approaches in the following.
MULTIPLE-ENDMEMBER SPECTRAL MIXTURE
ANALYSIS AND ITS VARIANTS FOR SMALL
SPECTRAL LIBRARIES
MULTIPLE-ENDMEMBER SPECTRAL MIXTURE
ANALYSIS AND ITS VARIANTS
This algorithm and its modifications have the following qualities:
◗ They generally provide good SU results.
◗ They are easy to set up, with few or no parameters.
◗ They have very high computational complexity.
◗ Their results depend strongly on the quality of the available spectral
library.
×10-3
0.8
1.2
1.6
2
0.4
(a)
(b)
(c)
FIGURE 9. The spatial behavior of EM variability. (a) The soil subregion of the Samson image (highlighted by a red square). (b) The Euclidean
distance and (c) the spectral angle between each pixel and the average spectra of the region.
234
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE DECEMBER 2021
0.05
0.04
0.03
0.02
0.01

IEEE Geoscience and Remote Sensing Magazine - December 2021

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2021