# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 5 Unstructured Background Models
### Kernel-Based Models
#### 5.3.3 Support Vector Data Description (SVDD)  
Most anomaly detectors on hyperspectral images try to model or estimate the PDF of the background. However, one could instead try to directly estimate the size and shape of the background support region for a given dataset. That is the basic point of the SVDD based anomaly detector [31; 103], which is basically a single-class support vector machine (SVM) classifier [104; 105; 106]. It avoids a-priori assumptions about the underlying background PDF and directly estimates the region of support for the background. SVMs are large-margin techniques that achieve good generalization of high-dimensional non-Gaussian data by directly estimating a maximum separability decision boundary [31]. SVMs showed great potential in classifying hyperspectral images [107; 108], and in [31], are extended for the anomaly detection problem. The SVDD benefits for anomaly detection are listed in [31]: (1) it is nonparametric (data-driven), (2) requires a few training samples for background characterization, (3) avoids overfitting and provides good generalization, and (4) can model nontrivial multimodal distributions by applying the kernel trick. In a geometrical sense, the SVDD finds a minimum enclosing hypersphere that includes the background data in either original data space or in the high-dimensional feature space. In the former case, the linear SVDD is applied, while in the latter case, the non-linear kernel-based SVDD is used. As the derivation of both SVDD algorithms is essentially the same, and the only change is in the non-linear mapping \(\boldsymbol{\Phi}(\cdot)\), only the algorithm for the non-linear SVDD is presented. The linear SVDD can be derived by omitting the non-linear mapping function. The need for mapping in the higher dimensional space arose as the hypersphere in the original data space does not provide a tight representation of the complex distributions found in the background. A minimum enclosing hypersphere in higher dimensional feature space corresponds to a much more complex boundary in the input data space. The smallest enclosing hypersphere in the feature space \(S=\left\{\underline{\Phi}(\underline{x}):\left\|\underline{\Phi}(\underline{ x})-\underline{c}\right\|^{2}<R^{2}\right\}\) that contains the set of mapped training data \(T=\left\{\underline{\Phi}(\underline{x}_{i}),\ i=1,\ \ldots,\ M\right\}\), where \(\underline{c}=\sum\
olimits_{i}\alpha_{i}\cdot\underline{\Phi}(\underline{x} _{i})\) represents the center of the hypersphere that corresponds to the center of gravity of the support vectors given the optimal weights \(\alpha_{i}\). The optimal weights are scalars (Lagrange multipliers) that need to satisfy sum-to-one and non-negativity constraints. The center \(\underline{c}\) and the radius \(R\) of the minimum enclosing hypersphere are determined by optimizing the Lagrangian, the optimal solution of which must satisfy Karush-Kuhn-Tucker (KKT) conditions [109]. The decision statistic for the non-linear SVDD of the test pixel \(\underline{x}\) can be now formulated as [31]:  
\[ \begin{array}{l} SVDD_{\Phi}(\underline{x}|\mathcal{B})=\ \left\| \underline{\Phi}(\underline{x})-\underline{c}\right\|^{2}\\ =\left\|\underline{\Phi}(\underline{x})-\sum\limits_{i}^{N}\alpha_{i}\cdot \underline{\Phi}\left(\underline{y}_{i}\right)\right\|\\ \end{array}  \tag{29}\]  
\[=\ \underline{\Phi}(\underline{x})^{T}\cdot\ \underline{\Phi}(\underline{x})-2\cdot\sum \limits_{i=1}^{N}\alpha_{i}\cdot\ \underline{\Phi}(\underline{x})^{T}\cdot\underline{\Phi}\left(\underline{y}_{ i}\right)+\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\alpha_{i}\alpha_{j}\cdot\ \underline{\Phi}\left(\underline{y}_{i}\right)^{T}\cdot\underline{\Phi}\left( \underline{y}_{j}\right) \begin{array}{l}>\\ <\\ \end{array} \]  
where \(\underline{y}\) denote the training data, \(N\) is the number of examples in each training set, \(R\) is the hypersphere radius, and \(\alpha_{i}\) are the Lagrange multipliers.