# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 6 Structured Background Models
### Representation-based Models  
The rising paradigm of compressed sensing [131; 132; 133] propelled the popularity of representation-based anomaly and target detection [27; 40; 134; 135; 136; 137; 138; 139]. Consider having a set of \(n\ll N\) labeled samples (pixels) \(\underline{\underline{D}}=\{\underline{x}_{i}\}_{i=1}^{n}\in\mathbb{R}^{K \times n}\) where \(K\) refers to number of spectral bands. In the following expressions, input HS image \((\underline{X})\) is regarded as a \(K\times N\) matrix \((\underline{\underline{X}}\in\mathbb{R}^{K\times N})\) which is a transpose of the definition set in Section 4 (\(\underline{\underline{X}}\in\mathbb{R}^{N\times K}\)). A dataset with all available samples \(\underline{\underline{D}}\) is usually called a dictionary, which is constructed of atoms (labeled pixels). The labeled datasets are not available in anomaly detection problems, but they can be reconstructed from the data [140; 141; 142]. The idea of decomposing the hyperspectral image into a low-rank background matrix and a sparse anomaly matrix was implemented in the robust principal component analysis (RPCA) [138; 143; 144]. This model did not account for the presence of noise in the input dataset, which adversely impacted detection performance. An improvement over RPCA was achieved in [145; 146], where an additional noise factor was implemented in the low-rank and sparse matrix decomposition (LRaSMD) algorithm. The low-rank representation (LRR) [138; 147; 148] allows the background reconstruction using the multiple subspaces (unlike RPCA [149]) and therefore needs a dictionary to separate the anomalies from the background. LRR enables global background characterization, as it finds the lowest rank representation of all the hyperspectral pixels simultaneously. The adequacy of the LRR model for hyperspectral data modeling is nicely outlined in [142]: background pixels can be adequately represented as linear combinations of endmembers (described in a subspace), and anomalies are spatially sparse [146]. The LLR model for anomaly detection can be formulated as [142]:  
\[\min_{\underline{\underline{\xi}}\underline{\underline{\xi}}}\left\| \underline{\underline{\xi}}\right\|_{*}+\lambda\left\|\underline{\underline{ \underline{\xi}}}\right\|_{2,1}\text{ such that }\underline{\underline{\underline{\chi}}}= \underline{\underline{D}}\underline{\cdot}\underline{\underline{\xi}}+ \underline{\underline{\underline{\xi}}}, \tag{45}\]  
where the HS image \(\underline{\underline{\chi}}\) is decomposed into a background \(\underline{\underline{D}}\underline{\cdot}\underline{\cdot}\underline{ \underline{\xi}}\) and an anomaly component \(\underline{\underline{\xi}}=\left[\underline{\underline{\xi}}_{1},\underline {\underline{\xi}},\ldots,\underline{\underline{\xi}}_{N}\right]\). \(\underline{\underline{\underline{\xi}}}\) represents the dictionary, and \(\underline{\underline{\underline{\xi}}}\) contains the representation coefficients. \(\left\|\ \right\|_{*}\) is a nuclear norm, which is a good substitute for the rank function used in the original LRR model [138] because of the convex optimization problem it causes. A tradeoff parameter \(\lambda>0\) is used to balance the effects of the background and anomaly part. \(\left\|\ \right\|_{2,1}\) is the \(\ell_{2,1}\) norm defined as the sum of \(\ell_{2}\) norm of the columns of the matrix:  
\[\left\|\underline{\underline{\underline{\xi}}}\right\|_{2,1}=\sum_{i=1}^{N} \sqrt{\underline{\varepsilon}_{i}^{T}\underline{\varepsilon}_{i}} \tag{46}\]  
where \(\underline{\varepsilon}_{i}\) refers to a column of the matrix \(\underline{\underline{\underline{\xi}}}\). The role of the \(\ell_{2,1}\) norm is to encourage the columns of \(\underline{\underline{\underline{\xi}}}\) to be zero, indicating that anomalies are column-wise sparse or \"sample-specific\". [PERSON] and [PERSON] [150] show the hyperspectral AD based on the LRR and learned dictionary: LLRaLD AD. They propose using the basic detectors (such as global RX) on the sparse matrix for the detection statistic.