IEEE Geoscience and Remote Sensing Magazine - June 2019 - 126

graph clustering utilized Bhattacharyya distance to construct a band adjacency graph, and the Markov clustering
algorithm was adopted to cluster the band-adjacent graph
into different clusters. The band that has minimal total distances with other bands in each cluster was considered as
the representative band.
In [106], the structure-aware measurement exploited the
structural information of HSI data by means of spatial-spectral consistency analysis to construct an undirected weighted graph, and the dominant-set extraction was adopted for
clustering on graph representation and select dominant vertices as representative bands. In [107], HSI data were transformed into a complex network graph where band selection
was carried on by analyzing the network's topological feature corresponding to each band, and the bands that are the
most qualified for discriminating the network's demarcation units were selected.
SPARSITY-BASED METHODS
According to the sparsity theory, each band can be sparsely
represented using only a few nonzero coefficients associated with atoms in a suitable basis or dictionary. The sparsitybased band selection methods make use of sparse representation or regression to reveal certain underlying structures
within HSI data. The representative bands can be found by
solving an optimization problem with sparsity constraints.
Current sparsity-constrained methods can be categorized
into sparse nonnegative matrix factorization (SNMF)based, sparse representation-based, and sparse regressionbased approaches.
SPARSE NONNEGATIVE MATRIX
FACTORIZATION-BASED METHODS
The SNMF decomposes an original HSI data matrix into
the product of a set of bases (i.e., the basis matrix) and
encodings (i.e., the coefficient matrix), where the basis
is nonnegative and the encodings are both negative and
sparse. The nonnegative constraint in bases and encodings
brings about the parts-based feature of SNMF because only
additive combinations are allowed. In [108], the SNMF was
introduced to solve the HSI band selection problem, where
the representative bands were chosen by clustering the
sparse coefficients. Specifically, it aims to factorize the HSI
band matrix B into the product of an unknown dictionary
matrix W ! R D # r and an unknown sparse coefficient matrix H ! R r # N by optimizing the objective function
1 B - WH 2,
min fr ^ W, H h = 2
F
W, H
s.t., W, H $ 0, and h i 0 % r, 1 # i # N,

(12)

where the subscript r in fr denotes the desired low rank r.
The largest entry in each column of H indicates the cluster or subspace of each band it belongs to. The constraint
h i 0 % r explains that each column vector h i is sparse
and the number of nonzero entries is far smaller than
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the dimensionality r. Unfortunately, the adopted ED distance measurement in (12) does not perfectly represent
the approximate error between X and its approximation
WH. This is because the Gaussian distribution assumption behind the ED measurement contradicts the nature
of HSI data. Therefore, the Earth's mover distance (EMD)
was adopted to measure the approximation error and improve the band selection from SNMF [109]. The EMD is a
cross-bin distance that mainly resolves the histogram- or
image-matching problems and aims to minimize the cost
that must be incurred when transforming one feature distribution (i.e., the histogram) into the other. Let X l represent the source histogram, which corresponds to the spectral responses of pixels in the lth band, and let the column
vector Xu l be the target normalized histogram, which corresponds to another band. The EMD distance between two
histograms b l and bu l is formulated as a linear programming
problem with the objective of minimizing the total cost for
transforming the source b l to the target bu l as
EMD ^b l, bu l h = / fl ^i, j h D ^i, j h,
i, j

Z
]] fl ^i, j h $ 0, / fl ^i, j h =1
i, j
s.t., [
,
] / fl ^i, j h = X(i, l), / fl ^i, j h = X (j, l)
j
\ i

(13)

where fl ^i, j h and d^i, j h are the flow amount and flow cost
between the ith pixel of the source histogram b l and the jth
pixel of the target histogram bu l, respectively, and the flow
cost measures the L1 ground distance.
SPARSE REPRESENTATION-BASED METHODS
Unlike SNMF-based methods, sparse representation-based
methods learn or manually define a dictionary in advance.
They often select the informative bands according to sparse
coefficients. In [110], using the sparse representation of
HSI bands with a K-SVD-learned dictionary, the sparse
representation-based band selection algorithm ranked the
bands according to sparse coefficients using majority voting, and the bands with high occurrence in the histograms
of sparse coefficients were selected. Band selection [111]
was formulated into a multitask sparse learning problem,
and an immune clonal strategy was adopted to search for
the best band combinations based on sparse coefficients.
The collaborative sparse representation model can force all
bands to share the same active atoms if possible. In [112],
the collaborative sparse model was adopted to refine the
preselected bands and minimize the number of selected
bands. When the dictionary in a sparse representation
model is set to be equal to the HSI data themselves, all
of the band vectors are assumed to be sampled from several independent subspaces. The improved sparse subspace
clustering (ISSC) model [113] was formulated into the optimization problem
argmin Z q, s.t., B = BZ and diag ^Z h = 0,
Z

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