IEEE Geoscience and Remote Sensing Magazine - June 2013 - 15

From an optimization point of view, the MV based
unmixing algorithms are formulated as
min < Y - MA < F2 + mV (M)
M, A

subject to: A $ 0, 1 Tp A = 1 n,

(3)

where V (M) is a volume regularizer, promoting mixing
matrices of "minimum volume" and m 2 0 is a regularization parameter setting the relative weight between the data
term and the volume term. Most of the methods adopting
the above formulation implement an a nonlinear block
Gauss-Seidel iterative scheme minimizing successively with
respect to M and to A. This is the case of iterative constrained
endmembers (ICE) algorithm [50] and of the minimum volume transform-nonnegative matrix factorization (MVC-NMF)
[51], whose main differences are related with the way
they define the regularizer V (M). For variations of these
ideas recently introduced, see [3]. The sparsity-promoting
ICE (SPICE) [52] is an extension of the ICE algorithm that
incorporates sparsity-promoting priors aiming at finding
the number of endmembers.
Problem (3) is non-convex. Thus the solutions provided
by greedy solvers are strongly dependent on the initialization. This handicap was circumvented in the simplex identification via variable splitting and augmented Lagrangian (SISAL)
[53], the minimum volume enclosing simplex (MVES) [54] by
reformulating (3) with respect to M -1 instead of M.
D. stAtisticAl AlgoRithMs
The MV simplex shown in the right hand side example of
Fig. 8 is smaller than the true one. This situation corresponds to a highly mixed data set where there are no spectral vectors near the facets. For these classes of problems,
the MV algorithms fail and we usually resort to the statistical framework, formulating HU as a statistical inference
problem, usually adopting the Bayesian paradigm.
The Bayesian approaches often have the following flavor
(see, e.g., [55] [3] and references therein): The posterior distribution of the parameters of interest is computed from the
linear observation model (1) within a hierarchical Bayesian model, where conjugate prior distributions are chosen
for some unknown parameters to account for physical constraints. The hyperparameters involved in the definition of
the parameter priors are then assigned non-informative priors. Due to the complexity in obtaining close-form expression for the posterior density, the parameters of interest,
namely the mixing matrix and the fractional abundances,
are, often, estimated from samples of the posterior density generated with Markov chain Monte Carlo (MCMC)
techniques.
A clear illustration of the potential of the Bayesian
approach to cope with highly mixed data sets is provided
by the DECA [56] algorithm; it models the abundance
fractions as mixtures of Dirichlet densities. A cyclic minimization algorithm is developed where: 1) the number
of Dirichlet modes is inferred based on the minimum
june 2013

ieee Geoscience and remote sensinG maGazine

description length (MDL) principle; 2) a generalized expectation maximization (GEM) algorithm is derived to infer
the model parameters.
Finally, we note that most of the matrix factorization
methods referred to in sections III-B and III-C may be also
be formulated as Bayesian inference problems, with the
advantage of attaching meaning to the model parameters
and providing a suitable framework to deal with them.
e. unMixing exAMple
In this section, we illustrate part
of the concepts presented before
nonlInEaR UnmIxInG Is
by unmixing the publicly availa moRE complEx
able TERRAIN HSI9 acquired
pRoblEm tHan lInEaR
by the HYDICE sensor [57] (see
UnmIxInG. FUlly
HYDICE parameters in Table 1).
UnsUpERvIsED
The low SNR bands due to water
nonlInEaR UnmIxInG
absorption were removed yieldmEtHoDs HavE only
ing a data set with 166 bands.
The TERRAIN HSI, shown in
vERy REcEntly bEEn
the top left column of Fig. 9, was
ExploRED.
calibrated to reflectance, has size
500 # 307, and is mainly composed of soil, trees, grass, a lake,
and shadows, disposed on a flat surface. The signal subspace
was identified with the HySime [58] algorithm and the original data projected onto this subspace. The identified subspace dimension was 20. We have, however, discarded those
orthogonal directions corresponding to SNR 1 10 to avoid
instability of the endmember identification (see [3] for more
details). After this procedure, we ended up with a subspace
of dimension 6.
The plots on the top right column of Fig. 9 show the
identified endmember signatures with the VCA algorithm
[44]. The corresponding pixels are referenced in the original
image. They represent three types of soil, trees, grass, and
a spectrum obtained in the lake, which we termed shade
due to its low amplitude. The figure in the middle of the left
column shows a scatterogram of the data set projected on
the subspace defined by the first two subspace eigen directions determined by HySime. The endmembers identified
by VCA and N-FINDR area also represented. The solution
provided by the two algorithms are identical and, due to the
high spatial resolution of the sensor, correspond to nearly
pure pixels. Notice that there are endmembers placed in
all the "extremes" of the scatterogram, which is coherent
with the pure pixel hypothesis. The remaining parts of Fig. 9
shows the estimated abundance fractions for soil 1, trees,
and grass.
F. nonlineAR unMixing
A complete physics based approach to nonlinear HU would
involve the inversion of the RTT, which is an extremely
complex ill-posed problem, relying on scene parameters
9

Data set available at http://www.agc.army.mil/Missions/Hypercube.aspx.

http://www.agc.army.mil/Missions/Hypercube.aspx

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2013