NOVEL HYPERSPECTRAL ANOMALY DETECTION METHODS BASED ON UNSUPERVISED NEAREST REGULARIZED SUBSPACE

Anomaly detection has been of great interest in hyperspectral imagery analysis. Most conventional anomaly detectors merely take advantage of spectral and spatial information within neighboring pixels. In this paper, two methods of Unsupervised Nearest Regularized Subspace-based with Outlier Removal Anomaly Detector (UNRSORAD) and Local Summation UNRSORAD (LSUNRSORAD) are proposed, which are based on the concept that each pixel in background can be approximately represented by its spatial neighborhoods, while anomalies cannot. Using a dual window, an approximation of each testing pixel is a representation of surrounding data via a linear combination. The existence of outliers in the dual window will affect detection accuracy. Proposed detectors remove outlier pixels that are significantly different from majority of pixels. In order to make full use of various local spatial distributions information with the neighboring pixels of the pixels under test, we take the local summation dual-window sliding strategy. The residual image is constituted by subtracting the predicted background from the original hyperspectral imagery, and anomalies can be detected in the residual image. Experimental results show that the proposed methods have greatly improved the detection accuracy compared with other traditional detection method.


INTRODUCTION
Hyperspectral imagery contains a wealth of spectral information and is widely used in the field of target detection because of the unique advantages.Since it is difficult for researchers to obtain enough prior knowledge to characterize the statistical information of target categories, the detection without a priori spectral information of target, which is called anomaly detection, has been of significant interest (Niu and Wang 2016).Anomaly detection models the background and using the differences between pixels and the background to detect anomalous pixels which is a small quantity of pixels in the hyperspectral image whose spectral characteristics differ significantly from those of a large proportion of pixels in the hyperspectral data cube (Vafadar and Ghassemian 2017).
Anomaly detection is an important application in the field of hyperspectral remote sensing, which can be widely applied to detect location of crop stress in precision farming, to find scarce minerals in geology, to analyse oil and environmental pollution, and to detect landmines for public safety (Li, Zhang et al. 2015, Taghipour, Ghassemian et al. 2016, Zhao, Du et al. 2016).
Many anomaly detection algorithms have been proposed.The classical detection algorithm called the Reed-Xiaoli (RX) (Reed and Yu 1990) detector, is developed by Reed and Xiaoli in 1990, which has been considered as the benchmark for In recent years, the method based on signal sparse representation has been applied to hyperspectral image target detection.This method aims to use background and target signals as few as possible to concisely represent the entire image information in an overcomplete dictionary composed of background information and target information (Liu, Lin et al. 2010, Chen and Chang 2013, Xu, Wu et al. 2016) values.By comparison with five popular and classical methods, the proposed UNRSORAD and LSUNRSORAD method provide higher detection accuracies.The rest of the paper is organized as follows: In Section 2, the proposed methods and main concepts will be presented.Section 3 shows the experimental and illustrates the superiority of the proposed methods.Finally, Section 4 draws our conclusions.

PROPOSED METHODS
In this section we introduce the proposed UNRSORAD and LSUNRSORAD methods.Before explaining the proposed methods, we provide a short review of Unsupervised Nearest Regularized Subspace (UNRS) (Li and Du 2014) algorithm.It is an important technique used in our proposed approach.Then we explain the outliers removal strategy, then the dual-window sliding strategy.

Algorithm
Let a given hyperspectral image dataset X ∈ R d be expressed as: The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-3, 2018 ISPRS TC III Mid-term Symposium "Developments, Technologies and Applications in Remote Sensing", 7-10 May, Beijing, China where  = the total number of the image pixels   = each pixel in the image For each testing pixel y ∈   (of size  × 1), we assume an appriximation y′ ∈   (of size  × 1) calculated via the linear combination of the surrounding selected data can be expressed as: where   = the surrounding background pixels α = weight vector represents the surrounding background pixels collected inside the outer windows while outside the inner window(Fig.1), in which  is the number of chosen surrounding background pixels between dual-window, which can be calculated by: where   = the size of outer windows   = the size of inner windows Then, the original space is 'shifted' via centering on the focal point y , and then it can be calculated: where   =   −  ∈  ×1 = symmetric matrix called  ∈  × , and   ∈  × is a symmetric matrix called Gram matrix, denoted by .In order to estimate the weight vector α, we consider using a Lagrange multiplier to solve the function under the sum-to-one constraint condition (Li and Du 2014).Finally, the value of α can be expressed as: where i, j = the rows and columns of the matrix index Note that matrix  −1 ∈  ×s .We apply this technique by adding the regularized term to the objective function.Then, it can be modified as: where  = a constant Thus, we convert the above problem to an equivalent problem solved by using Lagrange multiplier method.Finally, we can obtain the value of α which minimizes the new cost function: where  = a identify matrix

Outliers Removal Strategy
The dual-window makes use of two windows, called inner window and outer window to capture the characteristics of targets and background respectively (Liu and Chang 2008).In order to make full use of spatial information, many anomaly detection algorithms use the dual-window, such as LRX, CRD, UNRS and so on, while the dual-window can still not rule out the influence of anomalous pixels between outer window and inner window.In order to improve the detection accuracy, we adopt outliers removal strategy (Vafadar and Ghassemian 2017) to remove the outlier pixels in the double window that are obviously different from others as shown in Fig. 2

. The light
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-3, 2018 ISPRS TC III Mid-term Symposium "Developments, Technologies and Applications in Remote Sensing", 7-10 May, Beijing, China green square is anomalous pixel that is significantly different from other dual-window within pixels, and the green squares are background pixels.The UNRS algorithm aims to represent testing pixels(the blue square in Fig. 2) linearly with the background pixels(the green squares in Fig. 2), while the existence of anomalous pixels will affect the accuracy of the linear representation of the test pixels.The approximation of testing pixels sample(the light blue square in Fig. 2) is generated via a linear combination from the background pixels after removing the anomalous pixels, so what is the outlier on earth?It requires to find a suitable maximum and minimum threshold based on statistical theorem.Pixels with intensity values greater than maximum threshold or smaller than minimum threshold are considered the outliers.Similar to reference (Vafadar and Ghassemian 2017), we calculate mean and standard deviation of dual-window within pixels intensities values and construct threshold values simply by: where  = the mean of the background pixels    = the standard deviation of the background pixels   τ max and τ min represent the maximum and minimum of the background pixels intensities, respectively.Pixels with intensity value greater than τ max or less than τ min are removed.Therefore,   can be replaced by   ′ with the size of  ×  ′ in which  ′ is the number of background pixels after outliers removal, and   ′ is used to predict the testing pixel  ′ .Once the representation process is finished, we can obtain the residual image by: 11) where  ′ = the new weight vector after removing the outliers  As can be seen from Fig. 6 and Table 1,

The
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-3, 2018 ISPRS TC III Mid-term Symposium "Developments, Technologies and Applications in Remote Sensing", 7-10 May, Beijing, China This contribution has been peer-reviewed.https://doi.org/10.5194/isprs-archives-XLII-3-539-2018| © Authors 2018.CC BY 4.0 License.performance evaluation of hyperspectral anomaly detectors.It is a second-order matched filtering algorithm that calculates Mahalanobis distance of testing pixel and the background to complete the anomaly detection.When the entire image is considered for background modeling, it is also called Global RX (GRX).If the RX detector estimates the background using local statistics, it is also called Local RX (LRX).In real hyperspectral imagery, the background information is very complicated and cannot be described merely using multivariate normal distribution, and it may be difficult to use the estimated covariance and mean vector to represent the background statistics because of the existence of noise and anomalies.Under the circumstances, some improved algorithms such as Weighted-RX(W-RXD)(Guo, Zhang et al. 2014) and Linear Filter-Based RXD(LF-RXD)(Guo, Zhang et al. 2014) aim at improving the probability of anomaly detection by improving the estimation of the background information.Some kernel-based detection algorithms, such as the classical nonlinear kernel RX(Kwon and Nasrabadi 2005) detection algorithm, have also achieved better detection performance than the conventional algorithms in anomaly detection.

𝑟 1
Fig. 2 outliers removal and liner representation process

Fig. 5
Fig.4The San Diego airport hyperspectral data with its test region and the target map

Fig
Fig. 6 ROC curves of different methods

. However, this method merely takes advantage of spectral information and rarely gives the consideration to spatial information. It's difficult to obtain satisfactory performance when applied to the
dual-window sliding strategy.The residual image is constituted by subtracting the predicted background from the original imagery, and anomalies can be detected in the residual image.The detection results are assessed using Receiver Operating Characteristic (ROC)(Hanley and Mcneil 1982, Crichton 2002)curves and Area Under Curve (AUC)(Hanley and Mcneil 1982)