IEEE Geoscience and Remote Sensing Magazine - June 2013 - 17

The bilinear model is valid when the scene can be partitioned in successive layers with similar scattering properties. Fig. 6, in the middle, schematizes a two-layer scene and
shows the expression for the measured light. The sum on
the left hand side accounts for the single scattering and is
similar to the LMM; the sum on the right hand side accounts
for the double scattering, where the vectors m i 9 m j ! R n b
(symbol 9 stands for elementwise multiplication) account
for pairwise interactions.
Fig. 6, right hand side, illustrates an intimate mixture,
meaning that the materials are in close proximity and the
mixture occurs at a microscopic level. The Hapke approximation [40] for intimate mixtures models the reflectance as a nonlinear function of a convex combination of
the individual endmember albedos. The coefficients of
the linear combination are the relative geometric crosssections of the components. When the endmember particle sizes and densities are similar, the coefficients are
good approximations for the mass fractions of the different endmembers. However, in general, one needs information concerning the particle sizes of the components to
relate the mass fractions and the relative geometric crosssections [59].
Several strategies have successfully applied the bilinear
model to treat the double scattering problem, such as Bayesian algorithms, where prior models are chosen to satisfy the
positivity and sum-to-one constraints [60]. On the other
hand, kernel-based methods can design flexible kernels to
handle the problem of intimate mixtures. Linear kernels,
radial-basis functions, polynomial, and physics-based
kernels were proposed [61]. To cope with both scattering
and intimate mixture problems simultaneously, machine
learning technologies have been proposed, where training
samples were used to train artificial neural networks for
nonlinearities (see [62] and references therein). Polynomial
functions can be applied as well to model the nonlinearities [63].
A disadvantage of the above methods is that they
require the knowledge of the endmember signatures.
Fully unsupervised nonlinear unmixing methods have
only very recently been explored. One possibility is to
work directly on the nonlinear data manifold on which
it can be shown that the concepts of convex geometry still
hold. The geometry of the data manifold is described by
graph-based methods, and geodesic distances between
spectra are approximated by shortest-path distances on
this graph. If then endmember extraction and unmixing methods can be completely rewritten in terms of
distance geometry, a complete nonlinear unmixing version is obtained [5]. Although this method is completely
data-driven, geodesic distances can be calculated as well
on manifolds induced by nonlinear models, such as the
bilinear model [64].
Other methods utilize the LMM and Hapke's approximation to model macroscopic and intimate mixtures,
respectively. The mixtures are estimated directly from
june 2013

ieee Geoscience and remote sensinG maGazine

the data without the need for a priori knowledge of the
mixture types. In addition, the explicit modeling of both
mixture types allows for direct estimation of the endmembers [65].
g. unMixing viA spARse RegRession
HU via SR has recently been introduced with the objective
of coping with data sets not fulfilling the geometrical or
statistical assumptions on which the HU methodologies
presented in the previous sections rely. In the SR formulation, it is assumed that the measured spectral vectors
can be expressed as linear combinations of a small number of
pure spectral signatures known
spaRsE REGREssIon
in advance [6] (e.g., spectra coltEcHnIqUEs allow
lected on the ground by a field
tacklInG tHE UnmIxInG
spectro-radiometer). Unmixing
pRoblEm UsInG
then amounts to finding the
(potEntIally vERy
optimal subset of signatures in
laRGE) GRoUnD
a (potentially very large) specspEctRal lIbRaRIEs.
tral library (dictionary in the
SR jargon) that can best model
each mixed pixel in the scene.
In practice, this is a combinatorial problem, which calls for efficient linear SR techniques based on sparsity-inducing regularizers. Linear
SR is an area of very active research with strong links to
compressed sensing [66].
Let us assume that we are given a spectral library
A ! R n b # m containing m spectral samples. Usually, we have
m 2 n b and, therefore, the linear problem in hand is underdetermined. Let x ! R n denote the fractional abundance
vector with respect to the library A. With these definitions
in place, we can now write our SR problem as
min
< x < 0 subject to < Ax - y < 2 # d, x $ 0,
x

(4)

where < x < 0, termed the , 0 norm, denotes the number of nonzero components of x, and d $ 0 is the error tolerance due to
noise and modeling errors. Problem (4) is NP-hard [67] and
therefore there is no hope in solving it in a straightforward
way. Greedy algorithms such as the orthogonal matching
pursuit (OMP) [68] and convex relaxations replacing the , 0
norm with the , 1 norm are alternative approaches to compute the sparsest solution [69].
Contrary to problem (4), there are efficient solvers to
solve the convex approximations of it conceived to HU
applications [70]. What is, perhaps, totally unexpected is
that sparse fractional abundances vectors can be exactly
reconstructed by the convex relaxations, provided that the
columns of matrix A are incoherent in a given sense [69].
The applicability of sparse regression to HU was studied in
detail in [6]. Two main conclusions were drawn:
a) spectral signatures tend to be highly correlated what
imposes limits to the quality of the results provided by
solving the convex relaxations of (4).
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Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2013