Sub-pixel detection in hyperspectral imaging with elliptically contoured $t$-distributed background

Detection of a target with known spectral signature when this target may occupy only a fraction of the pixel is an important issue in hyperspectral imaging. We recently derived the generalized likelihood ratio test (GLRT) for such sub-pixel targets, either for the so-called replacement model where the presence of a target induces a decrease of the background power, due to the sum of abundances equal to one, or for a mixed model which alleviates some of the limitations of the replacement model. In both cases, the background was assumed to be Gaussian distributed. The aim of this short communication is to extend these detectors to the broader class of elliptically contoured distributions, more precisely matrix-variate $t$-distributions with unknown mean and covariance matrix. We show that the generalized likelihood ratio tests in the $t$-distributed case coincide with their Gaussian counterparts, which confers the latter an increased generality for application. The performance as well as the robustness of these detectors are evaluated through numerical simulations.

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