IEEE Geoscience and Remote Sensing Magazine - June 2019 - 121

and maximum SNR PCA (MSNRPCA). The MVPCA sorts
all of the bands by defining a variance-based band-power
ratio, denoted by R var ^m h, as
m

R var ^m h =

/ i =1v i
E

(1)

N
i =1

where E = R v i is the sum of variances of all the band images, v i is the variance of the ith band, and m is the number
of selected bands. The advantage of MSNRPCA over MVPCA is that noise is whitened before the PCA to eliminate
noise impact. Neither MVPCA nor MSNRPCA took spectral
correlation into account, and, accordingly, the divergencebased band-decorrelation scheme [22] based on the Kullback-Leibler distance was implemented to remove either
redundant or insignificant bands. If the divergence between
two bands was below a predefined threshold, the band with
low priority was removed. Later, mutual information (MI)
was used to gauge band dissimilarity [23]. In [24], the covariance-based method prioritized all of the spectral bands to
minimize their impact on target detection using a matched
filter and adaptive coherence estimator.
LOW-CORRELATION CRITERIA
The selected bands are required to have low mutual correlations. In [25], band selection was implemented into
a framework of target detection, where a band image was
treated as a desired target signature and other bands were
unknown signature vectors; the objective was to minimize
the correlations of selected bands. The constrained band
selection (CBS) method adopted constrained energy minimization (CEM) to linearly constrain a band image while
minimizing the band correlation (BC) or dependence provided by other band images. The resulting CEM-CBS was
formulated as
" w Ti Qw i , subject to b Ti w i = 1,
min
wi

(2)

where b i is a column vector of the ith band image, w i is the
column vector specifying a finite-impulse response filter designed for b i, and Q is the BC matrix. The solutions to the
CEM-CBS problem were extended using four different criteria, i.e., band-correlation minimization (BCM), band-correlation constraint, band-dependence constraint, and band-dependence minimization. Unfortunately, CEM-CBS required
tremendous computational time because of the enormous
dimension of vectors converted from band images. Therefore,
[25] developed the linearly constrained MV (LCMV)-CBS to
constrain a band image into an image matrix without vector conversion, thereby dramatically reducing computational
complexity. Later, the constrained multiple-band selection
(CMBS) expanded the CEM-CBS and implemented the
LCMV to constrain and select multiple bands simultaneously
[26]. In [27], the optimal band index criterion based on fractal dimension was put forward to quantify the correlation of
spectral and spatial information among all bands for band
JUNE 2019

IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE

selection. Based on spectral redundancy and band decorrelation, the local curve-fitting technique [28] was employed to
select optimal bands by ranking the absolute difference between the average reflectance and smoothed average reflectance generated from curve fitting.
LARGE-DISSIMILARITY CRITERIA
The selected bands are expected to be dissimilar to each
other. Inspired by the fast density peak-based clustering
(FDPC) algorithm in [29], a ranking-based exemplar component analysis algorithm was proposed to automatically
find cluster centers from all the bands. The algorithm required only the measuring of the distance between all pairwise bands and did not need to parametrize a probability
density function. A simple criterion was adopted to find the
independent density peaks. Each band i was ranked by ES i,
which is the product of its local density t i and the distance
d i to other bands with higher densities,
ES i = t i d i .

(3)

A band with an anomalously large ranking score, i.e., ES,
was recognized as an exemplar or representative band. Later, enhanced FDPC (E-FDPC) [30] was presented to solve
two issues related to the scale problem of t i and d i as well
as the cutoff threshold. The E-FDPC algorithm computed the ranking score of each band by weighting the local
density and the intracluster distance, and an exponentialbased learning rule was introduced to properly determine
the cutoff threshold. In [31], to benefit the red-green-bluebased hyperspectral visualization, the CBS for visualization
^ CBSVx h adopted the second and third orders of normalized
information measured to select the triplet of bands containing minimal redundancy and maximal informative content.
SUPERVISED RANKING-BASED METHODS
Different from unsupervised ranking-based methods,
supervised methods involve the prior knowledge of HSI
data when constructing a band-prioritization criterion.
The designed criteria may correlate closely with certain
applications, i.e., classification, spectral unmixing, and
so on.
1) Classification-aimed criteria: The criteria were designed to
ensure excellent classification performance using selected bands. In [23], minimum misclassification canonical
analysis (MMCA) was proposed to rank bands according
to their classification abilities. Derived from Fisher's discriminant function, the MMCA aimed to minimize the
misclassification error of selected bands by solving the
generalized eigenvalue problem,
S b v = mS w v,

(4)

where S b and S w are the between-class and within-class scatter matrices, respectively, and m and v are the corresponding eigenvalue and eigenvectors, respectively. Similarly, a
121

IEEE Geoscience and Remote Sensing Magazine - June 2019

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - June 2019