Spectral Unmixing With Negative and Superunity Abundances for Subpixel Anomaly Detection

We propose a low false alarm methodology to determine anomalies in hyperspectral data. The method is based on the assumptions that the linear mixing model is valid and that, due to the resolution of the image, most pixels are mixtures of common substances, of which pure pixels (not mixtures) are rare. In the first stage of the algorithm, the classes associated with the background, which are the dominant classes in the image, are found by clustering the image pixels. The resulting clusters may be considered as representatives of the background classes in the image. In order to determine the anomalous pixels, a threshold may be applied to the distance between the pixel spectrum and the cluster centers. However, pixels corresponding to anomalies and pure substances will both show high distances. If we consider that the background classes are themselves most likely mixtures of other materials, the pixels within the convex hull formed by the background classes will have positive fractions that are smaller than one. The pure substances, however, will be outside such a convex hull and will show negative or superunity fractions. Pixels with such mixing proportions are explained as linear combinations of the background classes and, therefore, as not true anomalies. Pixels corresponding to anomalies, however, when expressed as linear combinations of the background classes, show high residual error even with negative and superunity mixing proportions. We use the unmixing spectral linear model without the nonnegativity constraint to distinguish between false anomalies corresponding to pure substances and real man-made anomalies.