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that approximate the solution of the SU problem as a direct
application of a single linear transformation [165].
GENERAL SPECTRAL TRANSFORMATIONS
Another group of approaches employ more flexible linear
transformations to better mitigate the effect of spectral
variability. These techniques consist of variations of FDA,
which is widely used for pattern classification. FDA aims to
find a transformation of the data to obtain a feature space
with the best separability between different classes [166].
In the context of SU, this amounts to minimizing the variance
of the signatures of each material while maximizing
the distance between the mean values of the different EM
classes [167]. Mathematically, this is formulated as
<
W argmin
W
m =
WS W
WS W
<
between
within
,
(12)
where Swithin is the weighted sum of the within-class covariance
matrices and Sbetween is the covariance matrix of the
mean EM spectra.
The first approaches directly applied FDA to SU by using
spectral libraries that were known a priori [167] or constructed
with pure pixels extracted from the observed hyperspectral
image [168]. Another work considered the augmentation
of the spectral library with pure pixels extracted from the
image to improve the discrimination among spectrally similar
vegetation species [169]. Later approaches studied other
variations, such as the iterative addition of more column vectors
to Wm
using a Gram-Schmidt orthonormalization procedure
to increase the dimensionality of the output space for
multispectral images with a small number of bands [170].
Another work proposed to make the spectral signatures of
different EMs orthogonal to one another and the spectral
signatures of the same EM all unitary and collinear to improve
the numerical conditioning of the SU problem [171].
FDA was successfully applied to improve the performance
of MESMA when unmixing urban surfaces (containing vegetation,
soil, water, and man-made materials) using imageextracted
spectral libraries [172]. In contrast to its improved
flexibility, FDA has as a downside: its dependence on good
estimation of the covariance matrices to be used in (12).
Thus, it may not perform well if the number of samples in
the libraries is not statistically representative [173].
UNMIXING METHODS THAT ESTIMATE
ENDMEMBERS FROM THE IMAGE
In recent years, a large number of works proposed to address
spectral variability in SU without relying on prior
knowledge about spectral libraries. Different strategies
have been proposed to this end, which we divide into four
groups. Local unmixing methods are computationally and
conceptually simple but require significant user supervision.
Parametric EM models provide more flexibility to
represent EM spectral variability but make the SU problem
harder to solve. EM-model-free methods address spectral
variability by using different modifications to the SU cost
240
function. Bayesian methods employ statistical representations
for the EMs, which leads to less user supervision at the
price of high computational complexity.
All these methods are able to estimate EMs and abundances
directly from the image. However, as seen in the " Unmixing
Methods That Use Spectral Libraries " section, the
reasoning underlying each group is quite different, which
leads to various levels of required user supervision, computational
complexity, and abundance estimation quality,
as illustrated in Figure 2. Moreover, prior knowledge in the
design of the algorithms is an important ingredient to guarantee
good performance, and it includes, e.g., the spectral
and spatial correlation of EM signatures and their statistical
properties. We review each family in the following.
LOCAL UNMIXING METHODS
LOCAL UNMIXING
Local unmixing techniques are characterized by the following:
◗ They are conceptually simple and physically motivated.
◗ They are computationally efficient.
◗ Using them requires a significant amount of user supervision.
◗ The selection of the local image regions has a significant impact on
the results.
◗ Local EM extraction can be difficult.
◗ Grouping the local estimates into global results is also challenging.
A conceptually simple and efficient method to deal with
spectral variability is to perform EM extraction and SU locally
for small, nonoverlapping regions of the hyperspectral
image. This approach, called local unmixing and detailed
in Figure 12, assumes the EM signatures to be constant in
each region of the image, benefiting from the knowledge
that spectral variability is often negligible in small areas.
The basic framework of local unmixing can be summarized
into the following steps:
1) Divide the observed image into a set of regions.
2) Estimate the number of spectral signatures and extract
the EMs in each region.
3) Perform SU with the local EM signatures.
4) Combine all the local SU results into global sets of EM
signatures and global abundance maps using, e.g., clustering
procedures.
Although local unmixing methods share similar overall
methodologies, there are important differences in the way
the hyperspectral image is partitioned (e.g., by using simple
square tiles or more advanced image segmentation) and
how the EMs are extracted from each region. This can have
a significant impact on the results.
The first approaches for local unmixing required
complete user supervision. For instance, the variable
MESMA algorithm proposed in [10] used manual image
segmentation to divide the image into local regions. SU
was then performed iteratively, updating the segmentation
maps and manually including additional EMs in
the process until a satisfying result was obtained. Later
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