IEEE Geoscience and Remote Sensing Magazine - September 2021 - 78

SINCE MAKING SENSIBLE
PRIOR ASSUMPTIONS
ABOUT SPECTRAL
DISTRIBUTION IS DIFFICULT,
THE CHALLENGE OF
FINDING A PERFECT
DISTANCE MEASURE IS
STILL OPEN.
it more acceptable than other measures. He also explored
the importance of the Euclidean minimum distance, which
is distinct from the SAM. However, the performances of
the SAM and the Euclidean
minimum distance are equivalently
related. The major advantage
of the SAM is that it
produces the distance in the
form of an angle, which can
be projected to any other distance
measure [150]. Further
modifications are carried
out to create different hybrid
measures. An approach similar
to SCA-based measurement
has been proposed; it
is a derivative of the SAM,
where mean is subtracted from the spectra [151]. Because
the KL distance is nonsymmetric, its symmetric version is
used to find the dissimilarity between two discrete random
variables [146]. Later, an add-on to the KL measure was performed
[22], where clustering was done prior to establishing
the distance measures. This approach was found to be
quite efficient in terms of time complexity.
The MD approach to BS relies on a prior assumption
that 1) the classes follow a multivariate Gaussian distribution,
2) all classes have a similar covariance, and 3) the pixel
densities are large enough for covariance representation.
The results obtained by the MD are similar to those of BD
[152]. The EMD mainly uses a histogram distance function
that minimizes the cost while transforming the distributions.
A thresholded EMD is combined with sparse nonnegative
matrix factorization (NMF) for BS, which is found
to be more efficient than multivariate PCA (MVPCA), the
ED, the Kullback-Leibler divergence (KLD), and their variants
[44]. The NSSA measured in an n-dimensional space
()n 2$ is capable of calculating the high-dimensional
solid spectral angle, rather than 2D spectra as in the SAM,
and thus has been deemed effective [153], [154]. A novel
dual (spectral-spatial) clustering approach based on minimizing
the band correlation is proposed in [50]. It uses the
SAM and a pairwise hyperspectral angle descriptor (incorporating
neighborhood pixel information) as two distance
measures, resulting in a time-consuming process. A list of
distance matrices commonly used in the BS context is presented
in " Common Band Selection Distance Matrices. "
Since making sensible prior assumptions about spectral
distribution is difficult, the challenge of finding a perfect
distance measure is still open, and the problem needs further
exploration.
Generally, stand-alone distance metrics are more useful
for SBS and SSBS approaches, due to the availability of GT
information. This helps to select bands by creating discrimination
through intra-/interclass separability measures. However,
for UBS approaches, the assumption of class-specific
spectra can be obtained through endmember extraction,
78
spectral clustering, and segmentation. The superpixel-based
UBS strategy is an example [99]. Although direct distance
measures are useful, the UBS strategy cannot depend solely
on such criteria, as errors may occur during endmember extraction,
clustering, and so on. Curve fitting-based, thresholded
BS [155] suffers from such problems, and the resulting
bands have a high correlation. Hence, for UBS strategies,
intra/interband measures are emphasized more than intra-/
interclass performance (Figure 3) to reduce redundancy in
the BS process.
Such an intraband relationship can be obtained by using
correlation measures [156]. In statistics, the correlation
measure defines the relationship between random variables.
The correlation between two random variables Ii
Ij can be represented as in
c =ij,
where c ,ij
VarI VarI
CovI I
of two random variables Ii
and Ij
() ( )
(, )
ij
ij
,
(1)
and Cov are the correlation and the covariance
and I ,j
defines the variance of any variable Ii
are two arbitrary band images represented
respectively, and Var
or .Ij For HSIs,
the correlation can be defined according to (1), where
Ii
as vectors.
Another similar correlation measure is the joint correlation
coefficient [157]. This calculates the angle between two
arbitrary (mean-subtracted) band images Ii
and :Ij
R == icos
ij
where IIi
T
,, ,
II
II
T
ij
i
j
ij
j
is the inner product of vectors,
clidean norm, and i ([ij ,
(2)
$
is the Eu0
r- ] radians) defines the angle
between band i and band j. The correlation information
can help to find the intraband relationship, which further
reduces redundancy in the selection process [158].
The shared-nearest-neighbor (SNN) correlation analysis
[159] method computes the k-nearest-neighborhood bands
and assigns weights to each one using correlation coefficients,
which are further ordered and ranked. A similar approach
using ranking was proposed in [81] and is discussed
in the " Grouping and Ranking Strategies " section.
Specifically, as the adjacent bands are highly correlated,
an adjacency correlation coefficient is incorporated to
partition them, and further BS is carried out [85]. However,
instead of considering the images directly, features
can be derived from the images and used to define
the similarity between bands. Such an attempt has been
made to determine the correlation among band images
by using the structural similarity (SSIM) measure [73].
Figure 5 presents a false gray-scale representation of the
Indian Pines data set's feature-based (SSIM) intraband
correlation matrix. These methods generally reveal the
similarity of HSI bands, which may further help to cluster
the band images.
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