# Systematic Review of Anomaly Detection in Hyperspectral Remote Sensing Applications
## 6 Structured Background Models
### Representation-based Models
An approach that implements the spatial similarity between pixels in local regions is displayed in the work of [PERSON] et al. [147]. They suggest incorporating spatial constraints in the detection model to improve the detection performance of the LRR model. Blind source component separation by unmixing was presented by [PERSON] et al. [151] to identify anomalous components. The sparse representation model assumes that a hyperspectral signal (pixel) can be adequately represented by a sparse linear combination of dictionary atoms, i.e., a pixel can be reconstructed by only a few atoms [135]. The PUT is sparsely represented using the \(\ell_{0}\) or \(\ell_{1}\) norm regularization. The main goal of the sparse representation is to determine the sparse (weight) vector \(\underline{\underline{a}}_{SR}\) such that \(\left\|\underline{x}-\underline{\underline{D}}\underline{\cdot}\underline{ \underline{a}}_{SR}\right\|_{2}\) is minimized while \(\|\underline{\underline{a}}_{SR}\|_{l}\leq L_{0}\)[134]. Namely, vector \(\underline{\underline{a}}_{SR}\) can be determined by solving the optimization problem:  
\[\underline{\underline{a}}_{SR}=\operatorname*{argmin}_{\underline{\underline{ a}}_{SR}}\left\|\underline{x}-\underline{\underline{D}}\underline{\cdot}\underline{ \underline{a}}_{SR}\right\|_{2}\text{ such that }\|\underline{\underline{a}}_{SR}\|_{l}\leq L_{0},\ l\in\{0,1\} \tag{47}\]  
where \(L_{0}\) is the constant (regularization) parameter that balances the \(\underline{x}\) reconstruction error and the sparsity of \(\underline{\underline{a}}_{SR}\), and \(l\) represents the choice of \(\ell_{0}\) or \(\ell_{1}\) norm. The listed optimization problem (47) is NP-hard if the \(\ell_{0}\)-norm is applied; it can be solved by greedy-pursuit based algorithms such as the orthogonal matching pursuit [152] or a subspace pursuit [153]. If \(\ell_{1}\)-norm is used, the \(\underline{\underline{a}}_{SR}\) can be determined by convex relaxation algorithms such as in [154; 155; 156], or by pursuit methods such as basis pursuit [157] or basis pursuit denoising [158]. [PERSON] et al. [134] proposed the AD based on the joint sparse representation (JSR) framework. It uses a dual sliding window approach to estimate an active dictionary, and the detection statistic is determined by the length of the matched projection on the orthogonal complementary background subspace (estimated by the JSR). An adaptive weighted sparse representation and background estimation-based AD is presented in [145]. It uses the endmember extraction method to characterize their spectra and respective abundances. The sparse representation was adaptively weighted on both global and local domains, and the detection statistic was determined as the residual of PUT reconstructed by the background dictionary. Discriminative feature learning with multiple-dictionary was introduced in the work of [159]. The detection statistic is designed with a global multiple-view AD strategy that incorporates multiple use of global RX detectors that are mutually fused to produce the final result. An innovative sparsity score estimation was presented in [160] that implements atom usage probability, which helps in improving the discriminative power of the background dictionary. [PERSON] et al. [138] proposed the combined use of LRR and sparse representation based on the separation of the background and the anomalies in the observed data. LRR is used to model the background, and the sparsity-inducing regularization term is introduced to the representation coefficients, which enables the description of global and local structures of the hyperspectral dataset. The final detection statistic is determined by the response of the residual matrix. Collaborative representation techniques take the opposite approach of sparse representation methods. \"Collaborative representation means that all atoms 'collaborate' in the representation of a single pixel, and each atom has an equal chance to participate in the representation.\" [135].