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 ifnd 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 diferences between the original data and the assumed background, the residual of their corresponding low-rank coeficient matrices are computed and utilized as a part of the detection output together with the column-wise ℓ2 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.
limitations in practical applications.
To overcome the insuficient 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 perspatial 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 diferent 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 repreabnormal 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 lowthe blocked adaptive computationally eficient 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 mathe background. However, this hypothesis has obvious trix or back to the statistical estimation, a better combination of the assumed background component and anomaly component may let the detection more reasonable.
As is well known, subspace clustering (SC) is gradually developed for unsupervised HSIs interpretation [16].
It can learn the similarity between pixels through selfdictionary learning. Inspired by the sketched-SC and representation theory, we propose a double low-rank regularization model for hyperspectral anomaly detecwhere and are the regularization parameters, is the tion. The proposed method is solved via the alternatself-representation coeficient matrix, ing direction method of multipliers (ADMMs) method background coeficient matrix, and [17] and the anomalies are finally detected by computing part indicating the anomalies.
is the assumed
a background-related residual matrix and an anomalyrelated distance matrix.
optimization problem is given by: Let X = {x}=1 ∈ R× denotes the data collections, where is the number of bands and is the total pixel number of the image. Then, the low-rank representation model aims to decompose the data into a lowest-rank matrix L ∈ R× and a sparse matrix E ∈ R× . The mL,iEn ||L||⋆ + ||E||0 .. X = L + E, (1) where is the regularization parameter, || · || ⋆ and || · || 0 represents the nuclear norm and the ℓ0 norm, respectively.
For a given dictionary , the low-rank matrix L can be rewritten as a linear combination of the and its corresponding coeficient matrix . And the NP-hard problem eq. (1) can be represented as: m,iEn ||||⋆ + ||E||2,1 .. X = + E, (2) where ∈ R× has dictionary samples, ∈ R× , and || · || 2,1 is the ℓ2,1 norm of the matrix. ℓ2,1 norm can be regarded as the ℓ1 norm of the ℓ2 norm of matrix columns. + + 2 ||X −
Assume that the data can be self-represented: data X˜ = X , we also have: where is the coeficient matrix. Then for a sketched X = X,
X = X˜, where ∈ R× is defined as a random projection matrix to compress X while preserving its main information, and ∈ R× .
By means of the sketched data X˜, the proposed double low-rank regularization (DLRR) model based on sketched
m,in,E ||||⋆ + ||||⋆ + ||E||2,1 .. X = X˜, X = + E, (3)
Considering that the diference of and may reflect the abnormal information of the image, then the residual of these two matrices can be utilized as the reference of detection output when = . And the final detection result is formulated by the sum of the column-wise ℓ2 norm of the low-rank coeficient matrices residual and the sparse coeficient matrix: (x) = ||:, − :,||2 + ||:,||2. (4)
To solve the proposed DLRR model, the ADMM method is employed and the detailed optimization process is described as follows.
First, and are introduced as the auxiliary variables for the coeficient matrix
and , respectively: ,,,,E ||||⋆ + ||||⋆ + ||E||2,1 min ..
X = X˜, X = + E, = , = .
Then, the augmented Lagrangian function of (5) can be constructed: ,,,,E,1,2,3,4 ||||⋆ + ||||⋆ + ||E||2,1 min
X˜ + 1/ ||2 + 2 ||X − − + 2/ || 2 2 || − + 3/ ||2 + 2 || − + 4/ || , 2 is the penalty parameter. where 1 ∈ R× , 2 ∈ R× , 3 ∈ 4 ∈ R× are the Lagrangian multipliers, and > R× , and 0
Then the equation (6) can be divided into five optimization problems and be updated one by one with iterative procedures. The updating rules of these variables are: 1) step with fixed and 3: 2) step with fixed and 4: min ||||⋆ + 2 || − + 3/ || .
2
(7)
min ||||⋆ + 2 || − (+1) + 4/ || . (8)
2 6) The Lagrangian multipliers and the penalty pa- Figure 1: The San Diego dataset. (a) Image scene. (b) Groundmin 2 ||X− X˜+1/ ||2 + 2 ||− +3/ || . 5) step with fixed , E, 2 and 4: min 2 ||X− − +2/ ||2 + 2 ||− +4/ || .
2 1 = 1 + (X − 2 + (X − − ), 3 = 3 + ( − ), 4 = 4 + ( − ), = min{1.1, }.
X˜), 4/ ).
The solutions of (7) and (8) are calculated by = Θ (1/ )( + 3/ ) and = Θ (/ )( + 4/ ), respectively, where Θ is the singular value thresholding (SVT) operator. Then E is updated by (/ )(X − + 2/ ) where is a ℓ2,1-min thresholding operator [18]. and are respectively solved by finding the partial derivative and setting it to zero. Their optimized solutions are = ( + X˜⊤ X˜)− 1( X˜⊤X + X˜⊤1/ + − 3/ ) and = ( + ⊤)− 1(⊤X − ⊤E + ⊤2/ + −
The initial settings of this optimization process are: 0 = 0 = 0 = 0, E0 = 0, 0 = 1 = 0, 3 = 4 = 0, 0 = 0.01, = 106. And the convergence conditions are || − , || − || < , || −
X˜|| < , || −
+ || < || < , or the iteration times exceeds the predefined upper limit. Empirically, the predefined value of the error tolerance is = 10− 6 and the maximum iteration time is 100.
In this section, the performance of the proposed DLRR method is assessed on two real-world HSI scenes: the San Diego dataset and the Urban dataset. Four classical anomaly detection methods, which are RX, BACON, Kernel-RX (KRX) [19], and the low probability anomaly detector (LPAD) [20], respectively, are applied for comparable analysis. The regularization parameters and are set as 5 and 10, respectively. The background dictionaries are collected by the mean vector of the K-means clusters, and the dictionary number is set as 400. 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 dataset has 224 original spectral bands in total, (a) (a) (b) (b) 2 (10) (11) (12) (13) (14) (15) (16) parable algorithms are conducted in the aforementioned San Diego and Urban datasets, and the detection performances evaluated by the ROC curves and the normalized background-anomaly statistical range are shown in Figure 3 and Figure 4, respectively. It can be seen that in the San Diego dataset, the proposed DLRR model has the smallest false alarm rate when the detection probability reaches 1. And for the Urban dataset, our method has the largest separation range between the backgrounds and the anomalies. For further evaluation, the area under ROC curve (AUC) values are computed and the results are shown in Table 1. The results show that the proposed DLRR method has the largest AUC values compared with other methods in both two datasets. [19] H. Kwon, N. M. Nasrabadi, Kernel RX-algorithm: A nonlinear anomaly detector for hyperspectral imagery, IEEE Trans. on Geosci. Remote Sens. 43 (2005) 388–397. [20] Z. Li, L. Wang, S. Zheng, Applied low dimension linear manifold in hyperspectral imagery anomaly detection, in: Proc. SPIE, volume 9142, Art, 2014, p. 91421P.