# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 5 Unstructured Background Models
### Reed-Xiaoli (RX) Algorithm
Windows for \(\underline{\hat{\mu}}\) and \(\underline{\underline{\hat{\mathsf{f}}}}\) defined as hollow boxes bounded with guard window from the inside are used for estimation of \(\underline{\hat{\mu}}\) and \(\underline{\underline{\hat{\mathsf{f}}}}\). It was shown in [52; 53] that local Gaussianity on the image can be forced by subtracting the nonstationary mean. In [34], it is suggested that residual background is a zero-mean Gaussian and independent in spatial domain:  
\[ \begin{array}{ll}\underline{\underline{\underline{\mathsf{X}}}}|H_{0}= \underline{\underline{B}}&\in\mathcal{N}\left(\underline{0},\underline{ \underline{\hat{\mathsf{f}}}}\right)\\ \underline{\underline{\underline{\mathsf{X}}}}|H_{1}=\underline{\underline{ \mathsf{s}}}+\underline{\underline{B}}&\in\mathcal{N}\left(\underline{ \underline{\hat{\mathsf{s}}}},\underline{\underline{\hat{\mathsf{f}}}}\right) \end{array} . \tag{14}\]  
The difference in window sizes for \(\underline{\hat{\mu}}\) and \(\underline{\underline{\hat{\mathsf{f}}}}\) arises from the stationarity of these unknown parameters [51]; the mean should vary more frequently than the covariance matrix. In selecting the largest window size, a balance between two opposing requirements should be found. The window for \(\underline{\underline{\hat{\mathsf{f}}}}\) should include enough samples for stable determination of the inverse of the covariance matrix. [PERSON] and [PERSON] [54] argue that a reliable  
Figure 6: Geometrical principle of local RX detector with three specific sliding windows: guard window and windows for calculation of \(\underline{\underline{\hat{\mathsf{f}}}}\) and \(\underline{\underline{\hat{\mathsf{f}}}}\) with their respective widths \(w_{g}\), \(w_{\underline{\hat{\mu}}}\) and \(w_{\underline{\hat{\mathsf{f}}}}\). Detection statistic is calculated in a convolutional manner using the squared Mahalanobis distance. estimate of covariance matrix requires at least 10 times and preferably 100 times more samples than the number of spectral bands in a hyperspectral image. The opposing requirement states that the window size should be as small as possible to capture a homogenous background and reduce the computational load needed for inverse calculation of the covariance matrix. Quasi-local implementation of the RX detector includes the use of one global covariance matrix but performs local demeaning. This approach requires the calculation of only one covariance matrix using all available samples, which leads to a more stable inverse and lower computational complexity than the local implementation of the RX detector. Advantages of the global RX detector are simple implementation and fast computation, as there is only one covariance matrix to be inverted. If the assumptions about the local multivariate normal model (14) and \(\underline{\underline{\Gamma}}_{B}=\underline{\underline{\Gamma}}_{\rm s}= \underline{\underline{\Gamma}}\) are fulfilled, then the squared Mahalanobis distance follows a non-central chi-squared distribution \(\chi^{2}_{K}(\beta)\) with K degrees of freedom and non-centrality parameter \(\beta\)[38]:  
\[D_{RX}(\underline{x}|\mathcal{B})\,\sim\,\left\{ \begin{array}{l}\chi^{2}_{K }(0),\,\,under\,\,H_{0}\\ \chi^{2}_{K}(d^{2}_{M}),\,\,under\,\,\,H_{1}\end{array} \right., \tag{15}\]  
thereby enabling the CFAR property. The limitation of the local RX detector is that the covariance matrix is determined from a small number of high-dimensional and highly correlated samples. That can result in rank deficiency of covariance matrix or often leads to instability of its inverse. Matrix inversion needs to be calculated for every pixel, which makes the local RX detector computationally intensive. The local RX detector suffers from an increased number of false alarms because some pixels can be anomalies in the local background but not on the entire image (e.g., isolated tree).