IEEE Geoscience and Remote Sensing Magazine - March 2022 - 72

each dictionary atom is estimated using SR, and the
atoms with higher frequency are selected as the final
background dictionary atoms. Highly correlated bands
not only cause complex calculation but also redundant
information. To address this, a multiple-dictionary SR
model via discriminative feature learning is proposed in
[54], and it can simultaneously learn spectral features
and robust dictionaries.
COLLABORATIVE REPRESENTATION-BASED METHODS
SR takes advantage of the competition between atoms in
an overcomplete dictionary to achieve the best representation
with a small number of samples. In contrast, CR
assumes that collaboration between dictionary atoms is
more important than competition in the case of small
samples [113], and it allows all atoms in a dictionary to
participate in linear representation to obtain better performance.
Based on this, hyperspectral anomaly detection
harnessing CR [55] has been proposed; it considers
that each normal pixel can be well represented by all atoms
in the background dictionary, while abnormal pixels
cannot, and the representation residual is regarded as the
anomaly degree of the testing pixels. The objective function
of CR is
argmin xDaam
a -+ ,
2
2
where
:
2
is the l2
2
2
(6)
norm. The constraint allows each
term of a to have a small value; i.e., all atoms from D can
participate in the testing pixel representation. It should be
noted that the l2
norm leads to a closed solution, which
makes the CR model simpler; a diagram is given in Figure
5(b). To adaptively adjust the contribution of the dictionary
atoms, a Tikhonov regularization matrix is added
according to the similarity between the testing pixel and
dictionary atoms.
CR has rapidly attracted a lot of attention because it
is simple and efficient. However, its dictionary construction
is also implemented via a dual window, which does
not take account for the pollution of abnormal pixels.
An improved outlier removal anomaly detector based
on CR is proposed in [56] that can remove outliers that
are significantly different from other background clutter
in a dual window by using a Gaussian distribution
model. In [13], spatial PCA is used to extract major pixel
information and global and local backgrounds to form
a dictionary, which reduces the pollution of abnormal
pixels. In [57], density peak clustering is employed to
calculate the density of each pixel in a local background
selected through a dual window, and only the pixels with
high density participate in the construction of the background
dictionary. A morphology-based CR detector
(CRD) is proposed in [58], and it extracts morphological
features for better performance. In [59], local collaboration
and global low-rank attributes are combined to detect
anomalies.
72
LOW-RANK REPRESENTATION-BASED METHODS
Both SR and CR anomaly location methods are pixel-bypixel
detection. LRR describes the global structure of data
[114]. In a hyperspectral image, high spatial and spectral
correlation cause global low-rank characteristics. For anomaly
detection, the background is low rank due to spatial
homogeneity and spectral correlation, and anomalies are
sparsity due to a low probability of occurrence and small
proportion. Hence, based on LRR, 2D hyperspectral data
are divided into two parts, the background and anomalies,
which can be expressed as
mi .,ns.t21,aam+= +EX DE
)
a.E
(7)
where Da represents the background matrix, which can
be expressed by a background dictionary D with coefficient
matrix ;a E indicates an anomaly matrix; a )
the nuclear norm of a , which is a convex optimization
of the rank function; E 21,
is
is the sum of the l2
norm of
each column in anomaly matrix E, which encourages as
many columns as possible to be zeros, i.e., the sparsity of
E; and m is the regulation parameter, which can balance
low-rank and sparse terms. A model diagram is presented
in Figure 5(c).
It should be noted that if the background is assumed
to be composed of a single subspace, i.e., D is an identity
matrix, the LRR model is degenerated into a low-rank and
sparse decomposition (LRaSMD) one, also known as robust
PCA (RPCA) [60]. In [61], a hyperspectral anomaly detection
method based on LRaSMD is proposed, and the Go
Decomposition algorithm is adopted to solve the objective
function. Based on LRaSMD, anomaly detection using the
Mahalanobis distance is proposed in [62]. It uses a robust
background obtained after LRaSMD to estimate the covariance
matrix and mean, and it takes the Mahalanobis
distance between the testing pixels and background as the
abnormal level of the testing pixels. Experimental results
show that using the Mahalanobis distance to detect anomalies
is better than directly applying the anomaly matrix
after LRaSMD.
However, it may not be reasonable to assume that the
background belongs to the same subspace. Therefore,
multiple subspaces are considered for anomaly detection.
In [63], LRR with a learned dictionary (LRRaLD) is proposed,
adopting sparse dictionary learning to learn a background
one. However, this method does not consider the
problem of abnormal pollution during dictionary learning.
Similarly, [64] uses a clustering method to obtain an
initial dictionary and adopts SR to represent a randomly
selected background pixel to obtain the usage frequency
of dictionary atoms, taking those with high occurrences
as dictionary atoms for LRR. In [65], an abundance matrix
is employed to replace the original image, and the
clustering method is used to construct a dictionary and
input it into an LRR model to detect anomalies, which effectively
suppresses the background and greatly improves
IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE MARCH 2022
IEEE Geoscience and Remote Sensing Magazine - March 2022

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - March 2022
