We present a variant of the classic problem of anomaly detection in hyperspectral imagery. In this variant, the anomalous signatures are assumed to be additive and to exhibit spectra that are sparse – that is, only a few of the many hyperspectral channels are significantly nonzero.
When the background data are Gaussian, and there is no structure in the anomalous signatures, then the optimal detector is given by a Mahalanobis distance and exhibits contours that are ellipsoids. When the desired signature is known, then the solution is given by a matched filter that is specifically optimized for that signature; the contours are parallel planes whose orientation depends on both the desired signature and the covariance of the background. We address an in-between problem, one for which the detailed signature is not known, but a more generic description of the structure is available.
We propose that this solution might have application to the detection of gaseous plumes, when the chemistry of the gas is unknown. Such plumes have approximately additive effect on their backgrounds, and – especially in the thermal infrared "fingerprint region" – tend to have very sparse absorption and emission spectra.