Hyperspectral Anomaly Detection Based on Low-Rank
Structure Exploration
Shizhen Chang1 , Pedram Ghamisi1,2
1
Institute of Advanced Research in Artificial Intelligence (IARAI), Landstraรer Hauptstraรe 5, 1030 Vienna, Austria.
2
Helmholtz-Zentrum Dresden-Rossendorf, Helmholtz Institute Freiberg for Resource Technology, Machine Learning Group, Chemnitzer Str. 40,
D-09599 Freiberg, Germany
Abstract
As one of the typical research area in unsupervised hyperspectral image learning, anomaly detection needs to accomplish the
abnormal pixels separation process without prior spectral knowledge. Recently, the representation-based detectors which can
find the spectral similarity between pixels under no statistical distribution assumption have attracted extensive attention and
been frequently used. To this end, low-rank regularization methods can approximately decompose the hyperspectral data into
a low-rank background part and a sparse anomaly part. Based on the theory of representation and self-representation, this
paper proposed a double low-rank regularization (DLRR) model for hyperspectral anomaly detection. To further explore the
reconstructed structure differences between the original data and the assumed background, the residual of their corresponding
low-rank coefficient matrices are computed and utilized as a part of the detection output together with the column-wise โ2
norm of the sparse matrix. Experiments carried out on two real-world hyperspectral datasets show promising performances
compared with other state-of-the-art detectors.
Keywords
Hyperspectral imagery, anomaly detection, low-rank representation, sketched-subspace clustering
1. Introduction limitations in practical applications.
To overcome the insufficient accuracy happened in
With continuous and redundant spectral bands, hyper- the distribution-based models, the representation-based
spectral images (HSIs) carry a wealth of spectral and detectors have been proposed and shown intended per-
spatial information of land-covers [1, 2]. This promotes formances. Representative methods are the collaborative
military and civilian applications utilizing the spectral representation detector (CRD) [11], the background joint
characteristics of different materials. And lots of research sparse representation detector (BJSRD) [12], etc. A dual
works have been conducted, such as feature extraction concentrate window is utilized to extract the possible
[3], noise reduction [4], unmixing [5], classification [6], background information as the confidence dictionaries
and detection [7], etc. As a special branch of HSIs re- of the background at each test pixel. And the detection
searches, anomaly detection aims to extract potential result is approximately derived by calculating the repre-
abnormal pixels without any prior knowledge [8]. There- sentation residual of the pixel. Nowadays, the low-rank
fore, suitable methods need to be designed. representation is widely used in hyperspectral anomaly
Traditionally, classic anomaly detectors have been de- detection which takes advantage of the repeatability of
veloped mostly based on the assumption of data sta- the background spectrum and decomposes the original
tistical distributions. The benchmark RX detector, the data matrix. Considering that the anomalies are usually
cluster-based anomaly detection (CBAD) [7] algorithm, rare and sparse, Chen et. al. [13] first utilized the low-
the blocked adaptive computationally efficient outlier rank decomposition model for anomaly detection. Later,
nominators (BACON) [9] and the random selection-based many types of research have been carried out based on
anomaly detector (RSAD) [10] assume that the data fol- the low-rankness of the background subspace and the
lows the Gaussian or Gaussian mixture distributions, sparsity of the anomaly subspace [14, 15]. However, after
then they implement the detection task according to the doing the low-rank decomposition, the detection decision
Mahalanobis distance between the pixel-under-test and of these methods either focus on analyzing the sparse ma-
the background. However, this hypothesis has obvious trix or back to the statistical estimation, a better combina-
tion of the assumed background component and anomaly
CDCEO 2021: 1st Workshop on Complex Data Challenges in Earth component may let the detection more reasonable.
Observation, November 1, 2021, Virtual Event, QLD, Australia.
As is well known, subspace clustering (SC) is gradu-
" szchang@whu.edu.cn (S. Chang); pedram.ghamisi@iarai.ac.at
(P. Ghamisi) ally developed for unsupervised HSIs interpretation [16].
0000-0001-2345-6789 (S. Chang); 0000-0003-1203-741X It can learn the similarity between pixels through self-
(P. Ghamisi) dictionary learning. Inspired by the sketched-SC and
ยฉ 2021 Copyright for this paper by its authors. Use permitted under Creative
Commons License Attribution 4.0 International (CC BY 4.0). representation theory, we propose a double low-rank
CEUR
Workshop
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http://ceur-ws.org
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CEUR Workshop Proceedings (CEUR-WS.org)
regularization model for hyperspectral anomaly detec- where ๐ฝ and ๐ are the regularization parameters, ๐ด is the
tion. The proposed method is solved via the alternat- self-representation coefficient matrix, ๐ is the assumed
ing direction method of multipliers (ADMMs) method background coefficient matrix, and E denotes the sparse
[17] and the anomalies are finally detected by computing part indicating the anomalies.
a background-related residual matrix and an anomaly- Considering that the difference of ๐ด and ๐ may reflect
related distance matrix. the abnormal information of the image, then the residual
of these two matrices can be utilized as the reference of
detection output when ๐ก = ๐. And the final detection
2. Proposed method result is formulated by the sum of the column-wise โ2
norm of the low-rank coefficient matrices residual and
2.1. Related works the sparse coefficient matrix:
Let X = {x๐ }๐ ๐=1 โ R
๐ฟร๐
denotes the data collections,
where ๐ฟ is the number of bands and ๐ is the total pixel ๐ท๐ท๐ฟ๐
๐
(x๐ ) = ||๐ด:,๐ โ ๐:,๐ ||2 + ||๐ธ:,๐ ||2 . (4)
number of the image. Then, the low-rank representation
model aims to decompose the data into a lowest-rank 2.3. Problem optimization
matrix L โ R๐ฟร๐ and a sparse matrix E โ R๐ฟร๐ . The
To solve the proposed DLRR model, the ADMM method
optimization problem is given by:
is employed and the detailed optimization process is de-
min ||L||โ + ๐ผ||E||0 ๐ .๐ก. X = L + E, (1) scribed as follows.
L,E First, ๐ต and ๐ป are introduced as the auxiliary variables
for the coefficient matrix ๐ด and ๐, respectively:
where ๐ผ is the regularization parameter, || ยท ||โ and || ยท ||0
represents the nuclear norm and the โ0 norm, respec- min ||๐ต||โ + ๐ฝ||๐ป||โ + ๐||E||2,1
tively. ๐ต,๐ด,๐ป,๐,E
For a given dictionary ๐ท, the low-rank matrix L can ๐ .๐ก. X = Xฬ๐ด, X = ๐ท๐ + E, (5)
be rewritten as a linear combination of the ๐ท and its ๐ด = ๐ต, ๐ = ๐ป.
corresponding coefficient matrix ๐. And the NP-hard
problem eq. (1) can be represented as: Then, the augmented Lagrangian function of (5) can
be constructed:
min ||๐||โ + ๐ผ||E||2,1 ๐ .๐ก. X = ๐ท๐ + E, (2)
๐,E
min ||๐ต||โ + ๐ฝ||๐ป||โ + ๐||E||2,1
๐ต,๐ด,๐ป,๐,E,๐1 ,๐2 ,๐3 ,๐4
where ๐ท โ R๐ฟร๐ has ๐ dictionary samples, ๐ โ R๐ร๐ , ๐ ๐
+ ||X โ Xฬ๐ด + ๐1 /๐||2๐น + ||X โ ๐ท๐ โ ๐ธ + ๐2 /๐||2๐น
and || ยท ||2,1 is the โ2,1 norm of the matrix. โ2,1 norm 2 2
can be regarded as the โ1 norm of the โ2 norm of matrix ๐ ๐
+ ||๐ด โ ๐ต + ๐3 /๐||๐น + ||๐ โ ๐ป + ๐4 /๐||2๐น ,
2
columns. 2 2
(6)
2.2. Problem formulation where ๐1 โ R๐ฟร๐ , ๐2 โ R๐ฟร๐ , ๐3 โ R๐กร๐ , and
Assume that the data can be self-represented: ๐4 โ R๐ร๐ are the Lagrangian multipliers, and ๐ > 0
is the penalty parameter.
X = X๐ถ, Then the equation (6) can be divided into five optimiza-
tion problems and be updated one by one with iterative
where ๐ถ is the coefficient matrix. Then for a sketched procedures. The updating rules of these variables are:
data Xฬ = X๐ , we also have:
1) ๐ต step with fixed ๐ด and ๐3 :
X = Xฬ๐ด,
๐
min ||๐ต||โ + ||๐ด โ ๐ต + ๐3 /๐||2๐น . (7)
where ๐ โ R๐ ร๐ก is defined as a random projection ma- ๐ต 2
trix to compress X while preserving its main information, 2) ๐ป step with fixed ๐ and ๐4 :
and ๐ด โ R๐กร๐ .
By means of the sketched data Xฬ, the proposed double ๐
min ๐ฝ||๐ป||โ + ||๐ โ ๐ป (๐+1) + ๐4 /๐||2๐น . (8)
low-rank regularization (DLRR) model based on sketched- ๐ป 2
SC can be formulated as:
3) E step with fixed ๐ and ๐2 :
min ||๐ด||โ + ๐ฝ||๐||โ + ๐||E||2,1 ๐
๐ด,๐,E
(3) min ๐||E||2,1 + ||Xโ๐ท๐ โ๐ธ +๐2 /๐||2๐น . (9)
E 2
๐ .๐ก. X = Xฬ๐ด, X = ๐ท๐ + E,
4) ๐ด step with fixed ๐ต, ๐1 and ๐3 :
๐ ๐
min ||XโXฬ๐ด+๐1 /๐||2๐น + ||๐ดโ๐ต+๐3 /๐||2๐น .
๐ด2 2
(10)
5) ๐ step with fixed ๐ป, E, ๐2 and ๐4 :
๐ ๐
min ||Xโ๐ท๐โ๐ธ+๐2 /๐||2๐น + ||๐โ๐ป+๐4 /๐||2๐น .
๐2 2 (a) (b)
(11)
6) The Lagrangian multipliers and the penalty pa- Figure 1: The San Diego dataset. (a) Image scene. (b) Ground-
rameter are updated as: truth.
๐1 = ๐1 + ๐(X โ Xฬ๐ด), (12)
๐2 + ๐(X โ ๐ท๐ โ ๐ธ), (13)
๐3 = ๐3 + ๐(๐ด โ ๐ต), (14)
๐4 = ๐4 + ๐(๐ โ ๐ป), (15)
๐ = min{1.1๐, ๐๐๐๐ฅ }. (16)
(a) (b)
The solutions of (7) and (8) are calculated by ๐ต =
ฮ(1/๐) (๐ด + ๐3 /๐) and ๐ป = ฮ(๐ฝ/๐) (๐ + ๐4 /๐), respec- Figure 2: The Urban dataset. (a) Image scene. (b) Ground-
truth.
tively, where ฮ is the singular value thresholding (SVT)
operator. Then E is updated by ๐ฎ(๐/๐) (X โ ๐ท๐ + ๐2 /๐)
where ๐ฎ is a โ2,1 -min thresholding operator [18]. ๐ด and
๐ are respectively solved by finding the partial deriva- and 189 bands are utilized for the detection task
tive and setting it to zero. Their optimized solutions are after eliminating the noisy bands. It records the
โค โค โค
๐ด = (๐ผ + Xฬ Xฬ) (Xฬ X + Xฬ ๐1 /๐ + ๐ต โ ๐3 /๐) and
โ1 area of the San Diego airport, CA, USA in 100 ร
๐ = (๐ผ + ๐ทโค ๐ท)โ1 (๐ทโค X โ ๐ทโค E + ๐ทโค ๐2 /๐ + ๐ป โ 100 pixels, three aircrafts including 58 pixels are
๐4 /๐). selected as the anomaly target. The visualized
The initial settings of this optimization process are: 2-D image scene and the ground-truth map of
๐ด0 = ๐0 = ๐ป0 = 0, E0 = 0, ๐0 = ๐1 = 0, ๐3 = this dataset are shown in Figure 1.
๐4 = 0, ๐0 = 0.01, ๐๐๐๐ฅ = 10 . And the convergence
6 2) Urban dataset: This dataset was collected by the
conditions are ||๐ โ Xฬ๐ด||๐น < ๐, ||๐ โ ๐ท๐ + ๐ธ||๐น < HYDICE airborne sensor, which has a spatial res-
๐, ||๐ด โ ๐ต||๐น < ๐, ||๐ โ ๐ป||๐น < ๐, or the iteration olution of 1 m and a spectral resolution of 10 nm.
times exceeds the predefined upper limit. Empirically, After removing low quality bands, 162 bands are
the predefined value of the error tolerance is ๐ = 10โ6 left for anomaly detection. This image scene con-
and the maximum iteration time is 100. tains 80 ร 100 pixels, and 17 small objects are
considered as anomaly targets. The 2-D visual-
ization map and the ground-truth of this dataset
3. Experiments are shown in Figure 2.
In this section, the performance of the proposed DLRR
method is assessed on two real-world HSI scenes: the 1 .0
1 .2 A n o m a ly (1 0 % ~ 9 0 % ) M in ~ M a x
N o r m a liz e d S ta tis tic a l R a n g e
P r o b a b ility o f d e te c tio n
B a c k g ro u n d (1 0 % ~ 9 0 % ) M e d ia n L in e
San Diego dataset and the Urban dataset. Four classi- 0 .8 1 .0
cal anomaly detection methods, which are RX, BACON, 0 .6 0 .8
Kernel-RX (KRX) [19], and the low probability anomaly
0 .6
0 .4 R X
B A C O N
0 .4
detector (LPAD) [20], respectively, are applied for com- 0 .2 K R X
L P A D
0 .2
parable analysis. The regularization parameters ๐ฝ and ๐
D L R R
0 .0
0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 0 .0
are set as 5 and 10, respectively. The background dictio- F a ls e A la r m R a te R X B A C O N K R X L P A D D L R R
naries are collected by the mean vector of the K-means (a) (b)
clusters, and the dictionary number ๐ is set as 400. Figure 3: The San Diego dataset. (a) ROC curves. (b) Normal-
ized background-anomaly statistical range.
1) San Diego dataset: This dataset was captured by
the AVIRIS sensor, which has a spatial resolution
of 3.5 m and a spectral resolution of 10 nm. This The proposed DLRR model together with other com-
dataset has 224 original spectral bands in total, parable algorithms are conducted in the aforementioned
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