IEEE Geoscience and Remote Sensing Magazine - March 2022 - 70

where the radius R and center c of the smallest closed hypersphere
can be obtained.
The distance in the higher feature space between the
testing pixel and c is considered the abnormal level of the
testing pixel. It should be noted that a Gaussian radial basis
function (RBF) is usually chosen as the kernel function in
which a parameter, sigma, needs to be precisely set. To find
the optimal parameter, [37] proposed an adaptive support
vector method that estimates sigma by using a geometric
interpretation of the SVDD based on the Gaussian RBF, and
[38] can find optimal kernel parameters via a gradient descent
algorithm to improve SVDD performance. Another
SVDD method based on a sparse kernel has also been proposed;
it constructs the minimum hyperspheres in multiple
feature subspaces and uses a sparse kernel to learn weights
of multiple hyperspheres to form a joint hypersphere [39].
This technique uses ensemble learning and has some advantages
in complex and changeable hyperspectral images.
CLUSTERING-BASED METHODS
As a classical unsupervised classification method, clustering
has been widely used in various fields (including image
and nonimage processing). In hyperspectral anomaly
detection without prior knowledge, anomaly-background
separation can be considered an unsupervised binary classification
problem. Therefore, it is logical to use a clustering
approach, which considers outliers to be points deviating
from the center of a cluster. Original data are clustered, and
the Mahalanobis distance is calculated for each cluster to
estimate the anomaly degree of the testing pixel. An anomaly
detection algorithm based on k-means strictly delimits
each class [40], while one based on fuzzy clustering can use
several possible member functions [41]. Recently, a densebased
clustering method was proposed in [109] that is easy
to operate and has been applied in band selection and classification.
Based on it, a new anomaly detection approach
was designed in [42], and it mainly takes advantage of the
small proportion of abnormal pixels in an image. It divides
hyperspectral images into multiple local windows and calculates
the density of each pixel in a window according to
a fast density peak clustering algorithm. The density score
is the abnormal level of the testing pixels. The lower the
density is, the more likely a pixel belongs to an abnormal
target. This approach is more effective for small targets; its
accuracy for large targets depends on whether the parameters
can be appropriately set.
GRAPH-BASED METHODS
Graph theory is a branch of mathematics that reflects the
internal structure of data. In hyperspectral images, pixels
can be considered the set of vertices of a graph, and any two
pixels can form an edge of a graph. Hyperspectral anomaly
detection based on graph theory calculates the similarity
between any two pixels. If the similarity is greater than a
given threshold, the pixels are connected with an edge and
become inseparable. Pixels of the large component of the
70
graph are considered background pixels and form a background
set. The distance between the testing pixel and the
nearest background point is taken as the abnormal level of
the testing pixel [43]. However, because appropriate parameters
are difficult to estimate, this method may perform no
better than the RX algorithm in actual applications.
In [44], a graph is constructed; it calculates the degree
of each pixel (the number of edges connected to it) and
the weight of each edge, followed by the weighted vertex
volume as the abnormal level of the testing pixels according
to these two variables. The authors of [45] introduced
the concept of vertex weight, which is the reconstructing
residual of each pixel by surrounding pixels in a manifold
space, and proposed the notion of edge weight, which is
the affinity between any two pixels. The affinity matrix is
constructed from these two weight matrices. Finally, the
probability of each pixel belonging to abnormal targets is
obtained by solving a quadratic programming problem. In
[46], the authors introduced spatial neighborhood information
for anomaly detection based on a Laplace graph. In
[47], a graph is used as the initial detection to separate potential
anomalies from the background, and a multisparse
representation is employed to obtain spectral features that
can enhance anomaly-background separability; finally, a
fusion strategy is used to realize anomaly detection. Recently,
an anomaly detection algorithm similar to the graph
theory-based one has been studied. In [48], anomaly detection
can be realized by constructing a kernel isolation
forest from root to leaf, which uses samples randomly selected
from the image kernel feature space. In the kernel
isolation forest, background pixels have longer paths than
anomalous pixels because outliers are more easily isolated.
Therefore, the path length can be used to judge abnormal
levels of testing pixels.
RECONSTRUCTION-BASED METHODS
In recent years, with the development of compressed sensing
theory and machine learning methods, anomaly detection
based on reconstruction has become a research
hotspot. The basic idea is to reconstruct all pixels using a
certain dictionary in a reasonable model, and the residual
is the abnormal level of pixels. The reconstruction model
can be expressed as
xD ,. a
(3)
where x is a pixel signal, D is a dictionary, and a is a
representation coefficient vector. Depending on the regularization
constraints on ,a hyperspectral anomaly detection
based on reconstruction is mainly divided into sparse
representation (SR), collaborative representation (CR), and
low-rank representation (LRR).
SPARSE REPRESENTATION-BASED METHODS
In an SR model, it is assumed that a normal pixel x can
be represented by several atoms from an overcomplete
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2022
IEEE Geoscience and Remote Sensing Magazine - March 2022

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