# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 5 Unstructured Background Models  
Fundamental anomaly detection methods implement unstructured background models. They are often specified as data-driven, statistical, or probabilistic. These models do not assume any specific structure on the hyperspectral cubes based on a-priori knowledge, but integrate additive noise in the model.  
Let \(\underline{\mu}=[\mu_{1},\mu_{2},\ \cdots,\ \ \mu_{K}]\) be the mean vector of a hyperspectral cube with \(K\) bands, where \(\mu_{1}\) represents the average of all pixels in the first spectral band and \(\mu_{K}\) the average of the last spectral band. The mean vector \(\underline{\mu}\) is often called the background centroid or background prototype. In this respect, a squared Euclidean distance between PUT (\(\underline{x}\)) and background centroid can be used as a simple detection statistic:  
\[d_{E}^{2}(\underline{x},\underline{\mu})=\Big{(}\underline{x}-\underline{\mu }\Big{)}^{T}\cdot\Big{(}\underline{x}-\underline{\mu}\Big{)}. \tag{7}\]  
In hyperspectral images, due to their fine spectral resolution, spectral adjacent bands are usually highly correlated. Therefore in (7), a weighting matrix in the form of the inverse covariance matrix (\(\underline{\Gamma}^{-1}\)) could be added:  
\[d_{M}^{2}(\underline{x},\underline{\mu})=\Big{(}\underline{x}-\underline{\mu }\Big{)}^{T}\cdot\underline{\Gamma}^{-1}\cdot\Big{(}\underline{x}-\underline{ \mu}\Big{)}. \tag{8}\]Expression (8) represents the widely known squared Mahalanobis distance [44] between PUT and background centroid. It should be noticed that distance in (8) is proportional with negative log-likelihood of Gaussian or normal distribution:  
\[d_{M}\big{(}\underline{x},\underline{\mu}\big{)}\propto-\log p_{\mathcal{N}}( \underline{x}|\underline{\mu},\underline{\Gamma}), \tag{9}\]  
where  
\[p_{\mathcal{N}}(\underline{x}|\underline{\mu},\underline{\Gamma})=\frac{1}{(2 \pi)^{\frac{\xi}{8}}\cdot|\underline{\Gamma}|^{\frac{1}{2}}}exp\bigg{(}-\frac {1}{2}\Big{(}\underline{x}-\underline{\mu}\Big{)}^{\mathrm{T}}\cdot\underline{ \Gamma}^{-1}\cdot\Big{(}\underline{x}-\underline{\mu}\Big{)}\bigg{)}. \tag{10}\]  
Therefore, if a background can be adequately characterized with the multivariate normal distribution, then the squared Mahalanobis distance could be an appropriate detection statistic for anomaly detection. The multivariate normal model for anomaly detection can be written as:  
\[\begin{array}{ll}\underline{X}|H_{0}=\underline{B}&\in\ \mathcal{N}\left( \underline{\mu},\underline{\Gamma}\right)\\ \underline{X}|H_{1}=\underline{s}+\underline{B}&\in\ \mathcal{N}\left( \underline{s},\underline{\Gamma}\right)\end{array}, \tag{11}\]  
where \(\underline{B}\) denotes the spectral vector belonging to the background (which incorporates additive noise), \(\underline{s}\) is the spectral vector of the target, and \(\underline{\Gamma}\) is the background covariance matrix that is assumed to be the same for the background and target class. The most popular anomaly detector for hyperspectral remote sensed images is based on the multivariate normal model, and it is named the Reed-Xiaoli (RX) algorithm [34].