IEEE Geoscience and Remote Sensing Magazine - December 2021 - 47

better performance [112]. In [113], a rank 2 NMF-based hierarchical
clustering (H2NMF) algorithm was developed
for HSIs. It first treats all pixels as a cluster and then splits
one cluster into two disjoint clusters using a rank 2 NMF
model until obtaining stable results.
In addition, in [114], an orthogonal graph-regularized
NMF (OGNMF) method was introduced. It combines the
orthogonal graph constraints with the NMF model to learn
the local structure information of HSIs and achieves a relatively
good clustering effect. In addition, a robust manifold
factorization-based clustering (RMFC) algorithm was proposed
for HSIs [22]. It employs a low-rank matrix factorization
framework to simultaneously deal with the dimension
reduction (DR) task and the clustering task, with manifold
regularization to enhance the robustness of the clustering
model. With the help of the out-of-sample extension trick,
it can be extended to large HSIs.
BIPARTITE GRAPH-BASED CLUSTERING
Bipartite graph-based clustering is an extended version of
complete graph-based clustering, and it has been successfully
applied to HSIs to obtain good effect. In contrast to
the completed graph, the bipartite graph models the relationships
between two different sets, i.e., the anchor set
and the pixel set, to obtain a structured similarity matrix
W R MN nMNn
t ^^hh
!
++#
at a larger size, as in (11):
.
W
t =cm
0 A
AT 0
(11)
Here, A is generally constructed based on Gaussian kernel
distances, with the KNN strategy utilized, as in (12):
A =)
ij
exp
,
where Y RDn
t !
#
_-- t
d !! h
YY MM^
ij ik jj ki
2i ,
otherwise
YY YY
^tt
hor
, (12)
is the anchor matrix derived from the HSI
matrix and d is the kernel parameter.
A representative bipartite graph-based clustering method
is spectral-spatial coclustering based on a bipartite graph
(SSCC-BG) [115]. It extracts anchors from the cluster centroids
of k-means and then constructs a bipartite graph
between centroids and pixels. SSCC-BG obtains good clustering
results for HSIs by fusing spectral information and
spatial information into the graph. In addition, in [116], a
sequential SC (S-SC) method was developed to efficiently
cluster HSIs. It employs the minibatch k-means to determine
the anchors and conduct cluster assignments. Then,
based on the bipartite graph, S-SC utilizes the sequential
SVD for the product of the rows and columns of
AA ,
T
instead
of directly decomposing it, which effectively reduces
the computational complexity and improves the efficiency
of the model.
Furthermore, in [117], a novel bipartite graph partitionbased
coclustering with joint sparsity (BGP-CJS) was put
forward for HSIs. The technique builds a more informative
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
bipartite graph with a learned A from the joint-sparsity-constrained
optimization problem. Then, an efficient spectral
graph-based normalized cut method is proposed to simultaneously
cluster the rows and columns of the similarity matrix.
Consequently, the BGP-CJS further improves the clustering
accuracy.
ABBREVIATED GRAPH-BASED CLUSTERING
To overcome the large computational complexity of complete
graph-based clustering, efficient abbreviated graphbased
clustering has been developed, which selects only a
few important and representative points, i.e., key points, to
construct a similarity graph at a much smaller size of
nn ,#
where n is the number of key points. In practice, the abbreviated
graph is generally induced by the anchor graph.
Hence, anchor graph-based clustering methods can be the
most representative abbreviated graph-based clustering
models. In many recent studies, the anchor graph is utilized
to evaluate the similarity among pixels, instead of the complete-graph
[118], [119].
Such methods generally include two main steps, i.e.,
anchor selection and relation matrix construction. The representative
pixels or cluster centroids are considered as anchors,
which are commonly obtained via random selection
or preclustering. The relation matrix A RMN n
!
# models the
relationships among anchors and pixels, and it is generally
constructed based on certain similarity measurements,
such as the Gaussian kernel distance. In [120], A is taken as
a variable and obtained by learning, as in (13), which leads
to a more accurate :A
MN n
AA1=1, ij $0
min // YY AAij ij
t
-+c
2
2
i=1 =j 1
where c is the regularization parameter and 1 Rn 1
#
!
is a vector whose elements are all ones. With the obtained
WA
,AT1
u = KA
, the adjacency matrix Wu
where K! R nn#
jj
can be constructed as
is a diagonal matrix with
ij
the diagonal element being K =/i= A . Such methods
MN
1
are generally much more efficient, and they are more scalable
to large HSIs, given their small computing demands.
However, because only a few key points are utilized to approximate
the structure information of HSIs, the underlying
adjacency among pixels cannot be accurately mined.
Consequently, the clustering accuracy of such methods is
generally discounted.
A typical example of the abbreviated graph-based clustering
method is the fast SC with an anchor graph (FSCAG)
algorithm [43]. To ensure the clustering efficiency, FSCAG
randomly selects the anchors from the original hyperspectral
pixels. Then, the relation matrix A is learned from (13),
with a spatial constraint based on the mean filtered results
of HSIs, inspired by FCM-S1, to incorporate spatial information
into the anchor graph. Last, through spectral analysis
of the adjacency matrix Wu
induced by ,A as in (10),
the final clustering result is yielded. In addition, in [121],
47
2
F
,
(13)
IEEE Geoscience and Remote Sensing Magazine - December 2021

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