Improved modeling of background distributions in an end-to-end spectral imaging system model

Previously, an analytical end-to-end spectral imaging system model has been developed. The model is constructed around the propagation of spectral statistics from the scene, through the sensor, and processing transformations to lead to prediction of a performance metric. In this analytical framework the description of the class statistics has been by their spectral mean vector and spectral covariance matrix (first and second order statistics). This representation is only strictly accurate when the underlying classes are Gaussian in nature. While some background classes fall into this category, many have been observed to be nonGaussian in nature. As a work-around for this limitation, we have often formed subclasses in the background, which when combined form a "composite" background class that can be multimodal. However, we have observed in estimates of empirical data distributions that unimodal backgrounds often have longer tails than those predicted by the Gaussian distribution. Recently, it has been demonstrated that a family of distributions, known as the elliptically contoured multivariate t-distributions, can provide an accurate depiction of empirically observed backgrounds. These distributions are parameterized by their multivariate mean vector and covariance matrix, but also by a degree of freedom parameter, M. By varying M, excellent fits to empirical distributions have been observed. Another key feature of these distributions is that the number of degrees of freedom has been shown to be invariant to linear transformations. Since the analytical model operates by performing a sequence of linear transformations on the statistics, the input value of M is preserved and can be used at any stage of the model to represent the class statistics. This paper describes an implementation of the elliptically contoured t-distributions to represent background classes in the end-to-end system model. The functional form and examples of the t-distributions are shown. Results are presented comparing predictions of target detection performance using backgrounds modeled by multiclass Gaussian distributions with the new elliptical-t distributions.