**1. Introduction**

Attributed to the high spectral resolution, hyperspectral images are now capable of uncovering many subtle signal sources that cannot be known by prior knowledge or be visually inspected by image analysts [1,2]. Signal sources appear as anomalies in the data, such as unexpected presence, low probability of occurrence, small sample population whose signature is spectrally distinct from spectral signatures of its surrounding data samples. As a result, anomaly detection has received considerable interest in hyperspectral imaging in the last twenty years [3–6].

The RX detector developed by Reed and Yu [3] is acknowledged to be the most widely used anomaly detector. The classic RX algorithm is based on the global sample covariance matrix K, and is referred to as K-RXD. Since then, many RX-like anomaly detectors have been proposed [7–13]. Of particular interest are RXD using global sample correlation matrix R (R-RXD) [7,8], and RXD based on local background covariance matrix (L-RXD) [9]. The L-RXD uses not only spectral information but also spatial information to bring benefit for detection performance [10]. However, it may fail to obtain the best detection performance due to the penuriousness and unicity of local background distribution in every local window. A local summation anomaly detection (LSAD) is proposed in [13] by combining multiple local neighboring distributions of the pixel under test to ge<sup>t</sup> better performance. LSAD can be

considered as a local summation RXD (LS-RXD) using subspace feature projection for the stable local covariance estimation.

The hyperspectral remote sensing has developed rapidly in recent years, but as the satellite relocation cycle becomes shorter, some new problems come out. For instance, the massive data has brought some challenges to the data transmission and storage. Moreover, for the anomaly detection problem, the anomalies such as moving targets may show up for a short time and disappear quickly. In this case, timely detection is necessary. However, data transmission is quite time-consuming, to achieve timely detection, developing the recursive anomaly detection algorithms is important and necessary. Recently, several real-time anomaly detection methods [14–19] have been proposed. Specifically, real-time causal process of K-RXD and R-RXD detector (called as RT-CK-RXD, RT-CR-RXD) were developed in [14]. The real-time R-RXD and constrained energy minimization (CEM) are optimized and integrated in a dual-mode parallel Field-Programmable Gate Array (FPGA) based hardware platform in [16]. Unlike the RT-CR-RXD, in the FPGA-based implementations, each pixel under test is located in the middle region of the background, which can improve the performance of target detection. The computational performance of real-time causal linewise progressive anomaly detection (RCLPAD) based on Cholesky decomposition along with linear system solving were developed in [17]. An advanced anomaly detector using causal sliding array windows to capture local autocorrelation matrix statistics in the sense of causality was developed (CSA-RXD) [18], by virtue of causal sliding windows, a causal sample correlation matrix can be derived for causal anomaly detection. Recursive update equations are also derived and thus speed up real-time processing. A real-time L-RXD using the local casual square window is proposed in [19]. However, the method proposed in [19] still needs to calculate the inverse of a matrix to detect each pixel. Compared with sliding array window, setting a sliding square window usually contains much more spectral-spatial integration information. This paper addresses this issue and further develops the recursive processing for LS-RXD based on sliding square window. The contribution of this work is based on two points: a recursive version of LS-RXD according to a causal relation from the *Woodbury* identity, which reduce the runtime; and a background suppression algorithm integrated with the recursive procedure, which improves the detection accuracy.

The rest of the paper is organized as follows. In Section 2, several related RX anomaly detectors are briefly covered. Section 3 provides the design of recursive sliding window detector. Section 4 demonstrates the experiments of the proposed algorithm compared with some traditional anomaly detection algorithms. Finally, Section 5 draws our conclusions.

#### **2. Related Anomaly Detectors**

In this section, we provide a short overview of K-RXD, L-RXD and LS-RXD.

Assume that {*<sup>r</sup>i*}*Ni*=<sup>1</sup> is a set of data sample vectors, and *ri* = (*ri*1,*ri*2, ..,*riL*)*<sup>T</sup>* is the *i*th data sample vector, where *L* is the total number of spectral bands.
