# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 5 Unstructured Background Models
### Reed-Xiaoli (RX) Algorithm
The main limitation of RX detectors is that in hyperspectral imaging in remote sensing applications, the assumption of Gaussianity is often violated due to the presence of multiple materials in the scene or in the local background, which adversely impacts detection performance. Improved Variants of the RX Detector  
Many authors have tried to tackle the outlined shortcomings of the RX detector and improve its detection performance. First of all, it should be remarked that there is a confounding aspect to the naming of the new anomaly detectors. In our opinion, the RX detector should only refer to the local algorithm defined and explained above, with the exceptions of the global and quasi-local spatial implementations. The anomaly detection methods that use the squared Mahalanobis distance as detection statistics are often named RX variants [15; 18; 26]. To mitigate the computational cost of the (local) RX detector, a variety of parallel implementations for multicore platforms (CPU and GPU) have been developed [55; 56; 57; 58]. [PERSON] et al. [57] proposed optimized versions of the RX detectors based on the Cholesky decomposition of the correlation matrices with parallel implementations on high-performance computing platforms: multicore and GPU. [PERSON] [10] stated that the normal distribution model accurately fits the body of the data but not its tails. That is particularly important in anomaly detection, as the distribution tails have the most influence on the false alarms. Therefore, the family of multivariate elliptically contoured \(t\) distribution, which can model heavier distribution tails, is suggested in lieu of the multivariate normal distribution [59; 60; 61]. The quasi-local (QL) RX detector is a compromise between the local and global RX detector and should not be confused with the quasi-local implementation of the RX detector. The original QL idea was applied to achieve the QL covariance matrix [62; 63; 64], but in later publications, this approach is referred to as the QL RX detector [65; 66]. The basic idea of the QL RX detector is to decompose the global covariance matrix using the eigenvalue decomposition:  
\[\underline{\underline{\Gamma}}=\underline{\underline{E}},\underline{ \underline{\Lambda}},\underline{\underline{E}}^{T}. \tag{16}\]The eigenvectors in the \(\underline{E}\) are forwarded to the detector, but the eigenvalues in the \(\underline{\Delta}\) are replaced by the larger of the local and the global variance [62]:  
\[\lambda_{QL}^{i}=max\Big{(}\lambda_{LOCAL}^{i},\lambda_{GLOBAL}^{i}\Big{)}. \tag{17}\]  
That enables lower detection scores in the image areas with higher variance, and it thus may lead to a lower probability of false alarms. The main benefits of the QL approach are a more stable estimation of the covariance matrix and a much less expensive way of computing its inverse. Due to the high correlation of the spectral bands, the high dimensionality of the hyperspectral data, and a limited number of samples for estimation of the local covariance matrices, they often suffer from ill-conditioning. A widely accepted method for improving the estimation of the inverse covariance matrix issue is shrinkage [67; 68; 69; 70]. The most straightforward method for the shrinkage of the covariance is the addition of a scaled identity matrix, which is often applied in the ridge regression technique [71]. The ridge regularized (RR) squared Mahalanobis distance (SMD) detector is then formulated as [39]:  
\[D_{RR\;SMD}(\underline{x}|\mathcal{B})=d_{M\;RR}^{2}\Big{(}\underline{x}, \underline{\hat{\mu}}\Big{)}=\Big{(}\underline{x}-\underline{\hat{\mu}}\Big{)} ^{T}\cdot\big{(}\underline{\underline{\underline{\underline{\  
In [79], a superpixel-based dual window RX (SPDW RX) detector is presented to address the same issue. The SPDW RX uses superpixel segmentation to adaptively determine the dual windows, instead of the two fixed-size windows of the local RX detector. Firstly, the hyperspectral image is divided into superpixels. Then for every superpixel, a minimum bounding rectangle is defined.