## 1 Introduction  
Hyperspectral images (HSI) [1] have attracted widespread attention in many applications [2], such as geological exploration, environmental monitoring, and target detection [3] because they possess both spatial information and rich spectral information, which makes them appropriate for material identification and object detection. Among them, the utilization of HSIs for anomaly detection [4], which is a type of target detection without prior knowledge of the target [5, 6], has practical significance for both civilian and military purposes. The anomaly [7] refers to minority pixels with significantly different spectral characteristics from the majority pixels, which are referred to the background of the whole HSI. Desirable anomaly detection methods require appropriate background modeling [8] without prior knowledge to suppress the background and make the anomaly prominent [9]. Therefore, an accurate and effective model of the complex background [10] is critical in hyperspectral anomaly detectors [11], which can be generally categorized into two major types: statistical theory-based methods [12, 13, 14] and representation-based methods [15, 16, 17]. Statistical theory-based methods typically hypothesize that the background obeys a certain distribution and use statistical methods to estimate the likelihood that a pixel is attached to the background. As a representative of statistical theory-based methods, the Reed-Xiaoli (RX) detector [12] describes the background via a statistical multivariate Gaussian distribution model. However, this may not be correct for real HSI, leading to a higher false alarm rate. A series of improved algorithms have been proposed to improve the efficiency of the RX detector [18, 19, 20]. For example, the covariance matrix of the background is standardized in a regularized-RX detector [18], which can avoid the pathological state of matrix inversion. A nonlinear RX algorithm named kernel RX (KRX) [19] utilizes the kernel theory to project linearly inseparable HSI data to a high-dimensional feature space, where the differences between anomaly and background are strengthened. However, statistical theory-based methods suffer in situations with many kinds of background objects with irregular distributions. Given that these objects together do not necessarily follow some statistical distribution, it is difficult to correctly model many different background objects. The rationale for the representation-based methods is that each background pixel in the HSI can be well expressed as a linear combination of a background dictionary by a function model, whereas the abnormal pixel cannot be similarly expressed. The abnormal degree of the pixel is evaluated by the representation residual; the larger the representation residual of a pixel, the higher the probability that the pixel will be attached to the anomaly. The representation-based methods mainly include sparse representation [21; 22] and collaborative representation [23; 24; 25] according to different regularization constraints and different background dictionary constructions. Sparse representation holds that background pixels can be linearly and sparsely represented by a few atoms in an overcomplete dictionary, whereas anomaly pixels cannot. Therefore, sparse model optimization and complete dictionary construction are two key research aspects of sparse representation hyperspectral anomaly detectors. In [22], a low-rank and sparse representation (LRASR) detector applied a low-rank representation to model the background, while a sparsity-induced regularization term was employed to constrain the representation coefficient matrix. K-means was used to select the background dictionary atoms from the HSI data cube. Although sparse representation methods can detect anomalous targets to some extent, it is difficult to construct an over-complete background dictionary without abnormal pixel pollution when prior information is absent.