IEEE Geoscience and Remote Sensing Magazine - December 2017 - 52

In other words, the mixture problem can be approached
in a macroscopic fashion, which means that only a few
macroscopic components and their associated abundances
should be derived. However, intimate mixtures happen
at microscopic scales, thus complicating the analysis with
nonlinear mixing effects [133]. In addition to spectral mixing effects, there are many other interfering factors that can
significantly affect the analysis of remotely sensed hyperspectral data. For instance,
atmospheric interferers are a
potential source of errors in
THE PIXEL PURITY INDEX
spectral unmixing. Multiple
IS PERHAPS THE MOST
scattering effects can also lead
POPULAR ENDMEMBER
to model inaccuracies.
In linear spectral unmixEXTRACTION ALGORITHM
ing, the macroscopically pure
BECAUSE OF ITS
components are assumed to
AVAILABILITY IN
be homogeneously distributSOFTWARE PACKAGES.
ed in separate patches within
the field of view. In nonlinear spectral unmixing, the
microscopically pure components are intimately mixed. A
challenge is how to derive the nonlinear function, because
nonlinear spectral unmixing requires detailed a priori
knowledge about the materials. Responding to this limitation, a vast majority of techniques have -focused on linear
spectral unmixing, where the goal is to find a set of macroscopically pure spectral components (called endmembers)
that can be used to unmix all the other pixels in the data.
Unmixing thus amounts to finding the fractional coverage
(abundance) of each endmember in each pixel of the scene,
which can be approached as a geometrical problem [134]. In
the following section, we focus on the most relevant parts of
the linear spectral unmixing chain. We also summarize the
main efforts in nonlinear spectral unmixing.
ESTIMATION OF THE NUMBER OF ENDMEMBERS
Determining the number of pure spectral endmembers in
HSIs is a challenging problem. One of the most commonly
used approaches to this problem is the virtual dimensionality (VD) method [135], which follows the pigeon-hole
principle. If we represent a signal source as a pigeon and
a spectral band as a hole, we can use a spectral band to accommodate one source. Thus, if a signal source is present in
our remotely sensed hyperspectral data set, we should be
able to detect this particular source in the relevant spectral
band. This can be accomplished by calculating the eigenvalues of both the data-correlation and covariance matrices. A
source is present if their difference is positive.
Another popular approach is hyperspectral signal identification with minimum error (HySime) [136]. The idea of
HySime is to find the first k eigenvectors that contain the
most data information, i.e., to find k such that the meansquare error (MSE) between the original data and their
projection onto the eigenvector subspace is minimized.
Subspace k is ranked in terms of data variance, but noise
52

variance is not unitary in different directions, and the contribution from signals may be smaller than from noise.
HySime addresses this issue by using subspace projection
techniques, thus contributing an additional feature with
regard to VD: the modeling of noise before the estimation.
The eigenvalue likelihood maximization (ELM) method
[137], in turn, implements a modification of the VD concept based on the following observations: 1) the eigenvalues corresponding to the noise are identical in the covariance and the correlation matrices, and 2) the eigenvalues
corresponding to the signal (the endmembers) are larger in
the correlation matrix than in the covariance matrix. The
ELM takes advantage of this fact and provides a fully automatic method that does not need an input parameter (as
does VD) or estimation of the noise (as does HySime).
Finally, the normal compositional model (NCM) [138]
addresses the possibility that, in real images, there may not
be any pure pixels. To address this issue, NCM assumes
that the HSI pixels are linear combinations of an unknown
number of random endmembers (the opposite of the deterministic approach). This model provides more flexibility
with respect to the observed pixels and the endmembers,
which are allowed to be a greater distance from the observed pixels.
ENDMEMBER EXTRACTION
The identification of endmembers is a challenging problem, for which many different strategies have been proposed [134]. To categorize algorithms, we consider three
different scenarios.
1) The data contain at least one pure pixel per endmember,
i.e., there is at least one spectral vector in each vertex of
the data simplex (pure pixel assumption).
2) The data do not contain pure pixels but contain enough
spectral vectors on each facet. In this case, we may fit a
minimum volume simplex to the data.
3) The data are highly mixed, with no spectral vectors near
the facets. In this case, minimum volume algorithms
fail, and we need to resort to a statistical framework. We
also consider algorithms that include spatial information in addition to spectral information for this purpose.
PURE PIXEL ASSUMPTION
Pure pixel methods assume a classic spectral unmixing
chain with three stages: DR, endmember selection, and
abundance estimation. Here, the endmembers are directly
derived from the original hyperspectral scene. The pixel
purity index (PPI) [139] is perhaps the most popular endmember extraction algorithm because of its availability
in software packages. PPI has many parameters involved
and is not an iterative algorithm. Manual intervention is
required to select a final set of endmembers, which makes
it unattractive for automatization purposes.
An alternative is the N-FINDR [140], which assumes the
presence of pure pixels in the original hyperspectral scene
and further maximizes the volume that can be formed
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DECEMBER 2017

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2017