| Accurate statistical models for hyperspectral image (HSI) data distribution are useful for many applications. These models provide the foundation for development and evaluation of reliable algorithms for detection, classification, clustering, and estimation. It is well known that real world HSI data exhibit heavy tail due to nonhomogeneity. While the multivariate normal distribution is often used for multidimensional modeling, this distribution exhibits some limitations in capturing the heavy tail behavior of HSI data. In this thesis, a family of multivariate elliptically contoured distributions (ECDs) is investigated as an extension to well-known multivariate normal distribution, which gives more freedom in accurately capturing the decay rate and maintains most of appealing properties of multivariate normal distribution. The procedure to obtain the valid theoretical probability density function (PDF) of an ECD and methods to generate synthetic elliptically contoured random vector data are presented in detail. In order to test the symmetry assumption of real data, several graphical plots including scatter plots, t plots and beta plots are proposed in terms of some invariant statistics under orthogonal transformations. These symmetry tests are applied to both symmetric and non-symmetric synthetic data as well as to real data from the AVIRIS sensor. Correlation coefficients which are numerical measures of detecting deviation from symmetry are also calculated as an auxiliary metric as part of these graphical tools. A weighted mixture of multivariate t distributions is proposed to model the main body and heavy tail Mahalanobis distribution of real data using an Exceedance Metric in a logarithmic scale. Expectation-Maximization (EM) and Stochastic Expectation-Maximization (SEM) methods for clustering multimodal data into several unimodal clusters are also included for completeness. |
