IEEE Geoscience and Remote Sensing Magazine - December 2021 - 45

MULTIOBJECTIVE-BASED CLUSTERING
Multiobjective-based clustering methods generally address
more than one optimization problem and simultaneously
search for optimal solutions based on certain intelligent
computing techniques. Compared with the singleobjective-based
clustering methods, multiobjective-based
clustering approaches are more popular and generally perform
better, as they consider numerous factors at the same
time, e.g., spectral and spatial information, local and global
information. A representative example is the automatic
fuzzy clustering based on the multiobjective differential
evolution (AFCMDE) algorithm [32]. It extends the MoDEFC
model to a multiobjective version for an improved
ability to learn the complexity of remote sensing images,
with two objective functions included, i.e., the partition error
and the Xie-Beni index, as in (9):
minff fYY Y12
c MN
ff2
1=- =
i =1 j =1
i j
m
, ji
2
2
MN
n== c,. (9c)
21h
/ U Yj
j =1
ij
m
,
i
MN
/ U ,
j =1
ij
m
Uij
,
Yji
-n
/ Yjl
l =1
-n ---^
^
/
m
/ m
Specifically, AFCMDE consists of two layers, i.e., optimization
and clustering. In the optimization layer, a feasible number
of clusters is obtained by minimizing these two objective
functions. In the clustering layer, a nondominated sorting
method is utilized to update the population and search the
Pareto front to obtain the final clustering result. Through
multiobjective optimization, AFCMDE outperforms MoDEFC.
Then, based on AFCMDE, a multiobjective memetic FCM
algorithm (AFCMOMA) was presented to further improve
the optimization capability of the model [94]. The approach
introduces the memetic algorithm to balance the local and
global search ability and adds a new population-updating
strategy to obtain more high-quality individual samples. As a
result, the clustering accuracy is further improved.
In addition, a novel social recognition-based multiobjective
gravitational algorithm (SMGSA) was developed for
HSIs to learn the similarity relationships among pixels [95].
The SMGSA algorithm searches individual pixels among the
elite ones obtain by the gravitational force and the general
ones learned from the social recognition model, based on
the whole population, to generate an outstanding exploitation
ability. Furthermore, in [38], a novel multiobjective
PSO (MOPSO) method was proposed for HSIs to simultaneously
solve three problems, i.e., clustering the statistical
parameter estimation, searching for the best discriminative
bands, and estimating the number of clusters, using three
different optimization criteria. Moreover, based on the advanced
sparse subspace clustering (SSC) model, a multiobjective
SSC (MOSSC) method was put forward for HSIs. It
DECEMBER 2021 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
21h
^^=6 ,,hh ^ h@
c MN
/ /UY n ,, (9b)
/ /UYji
i =1 j =1
ij
m
,
!
-
-
MNminik ik
nn
n
2
2
2
2
(9a)
treats the sparse constraint term and the data fidelity term as
two objective functions to avoid the manual determination
of the regularization parameter, as in the SSC model [96].
Commonly, with the help of advanced intelligent optimization
algorithms to search for the global optimal solution
to the clustering model, intelligent computing-based
clustering methods may perform better than traditional
clustering approaches. However, such techniques still have
several disadvantages that limit their practical applications
to some degree. For example, the principle of such methods
is relatively complex, with a high application threshold.
In addition, such techniques are generally based on other
clustering models, and their performance is limited by the
inherent defects of the foundation clustering models, such
as FCM and GMM.
GRAPH-BASED CLUSTERING METHODS
Graph-based clustering is one of the recently developed advanced
hyperspectral clustering approaches that is evolved
from graph theory. Such methods generally model the relationships
among hyperspectral pixels with an adjacency
matrix
W RMN MN
#
u !
, also known as a similarity graph,
whose element represents the similarity between a corresponding
pair of pixels or the penalty factor when separating
the corresponding two pixels into different subgraphs.
The adjacency matrix is the basis of graph clustering. The
quality of the matrix directly affects the final clustering
accuracy. In practice, it is generally constructed by the
f -ball strategy [97], the KNN strategy [98], and the full connection
strategy [99]. Then, by applying a certain graph cut
algorithm to minimize the total cutting cost of the adjacency
matrix, the final clustering result can be obtained, as
shown in Figure 7.
Specifically, graph cut is a very important part of graph
theory. It aims to segment a graph into several disjoint and
distinctive subgraphs by maximizing the intrasubgraph
similarity and minimizing the intersubgraph similarity,
with each subgraph denoting a specific class. With decades
of development, many graph cut algorithms have been developed,
including minimum cut [100], radio cut [101], normalized
cut [102], average cut [103], minimum-maximum
cut [104], and so on. Among them, the normalized cut algorithm
is the most widely used. According to the differences
among the constructed graphs, graph-based clustering
methods can be coarsely divided into three main kinds: 1)
complete graph-based clustering, 2) bipartite graph-based
clustering, and 3) abbreviated graph-based clustering.
COMPLETE GRAPH-BASED CLUSTERING
Complete graph-based clustering methods group HSIs based
on an adjacency matrix Wu
that consists of all pixels; the matrix
contains the similarity between any pair of pixels, at a size
of
MN MN#
. A typical example is spectral clustering (SC)
[102], [105], which generally employs the normalized cut
algorithm to conduct graph cutting, with a spectral analysis
model formulated as the following optimization problem:
45

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