• kernel density estimation
• hyperspectral images
• finite mixture model
• Bayesian learning
• anomaly detection
• variable bandwidth kernel density estimation

Detecting targets with unknown spectral signatures in hyperspectral imagery has been proven to be a topic of great interest in several applications. Because no knowledge about the targets of interest is assumed, this task is performed by searching the image for anomalous pixels, i.e. those pixels deviating from a statistical model of the background. According to the hyperspectral literature, there are two main approaches to Anomaly Detection (AD) thus leading to the definition of different ways for background modeling: global and local. Global AD algorithms are designed to locate small rare objects that are anomalous with respect to the global background, identified by a large portion of the image. On the other hand, in local AD strategies, pixels with significantly different spectral features from a local neighborhood just surrounding the observed pixel are detected as anomalies.
In this thesis work, a new scheme is proposed for detecting both global and local anomalies. Specifically, a simplified Likelihood Ratio Test (LRT) decision strategy is derived that involves thresholding the background log-likelihood and, thus, only needs the specification of the background Probability Density Function (PDF). Within this framework, the use of parametric, semi-parametric (in particular finite mixtures), and non-parametric models is investigated for the background PDF estimation. Although such approaches are well known and have been widely employed in multivariate data analysis, they have been seldom applied to estimate the hyperspectral background PDF, mostly due to the difficulty of reliably learning the model parameters without the need of operator intervention, which is highly desirable in practical AD tasks. In fact, this work represents the first attempt to jointly examine such methods in order to asses and discuss the most critical issues related to their employment for PDF estimation of hyperspectral background with specific reference to the detection of anomalous objects in a scene.
Specifically, semi- and non-parametric estimators have been successfully employed to estimate the image background PDF with the aim of detecting global anomalies in a scene by means of the use of ad hoc learning procedures. In particular, strategies developed within a Bayesian framework have been considered for automatically estimating the parameters of mixture models and one of the most well-known non-parametric techniques, i.e. the fixed kernel density estimator (FKDE). In this latter, the performance and the modeling ability depend on scale parameters, called bandwidths. It has been shown that the use of bandwidths that are fixed across the entire feature space, as done in the FKDE, is not effective when the sample data exhibit different local peculiarities across the entire data domain, which generally occurs in practical applications. Therefore, some possibilities are investigated to improve the image background PDF estimation of FKDE by allowing the bandwidths to vary over the estimation domain, thus adapting the amount of smoothing to the local density of the data so as to more reliably and accurately follow the background data structure of hyperspectral images of a scene.
The use of such variable bandwidth kernel density estimators (VKDE) is also proposed for estimating the background PDF within the considered AD scheme for detecting local anomalies. Such a choice is done with the aim to cope with the problem of non-Gaussian background for improving classical local AD algorithms involving parametric and non-parametric background models. The locally data-adaptive non-parametric model has been chosen since it encompasses the potential, typical of non-parametric PDF estimators, in modeling data regardless of specific distributional assumption together with the benefits deriving from the employment of bandwidths that vary across the data domain.
The ability of the proposed AD scheme resulting from the application of different background PDF models and learning methods is experimentally evaluated by employing real hyperspectral images containing objects that are anomalous with respect to the background.