A Dictionary-Based Generalization of Robust PCA with Applications to Target Localization in Hyperspectral Imaging

We consider the decomposition of a data matrix assumed to be a superposition of a low-rank matrix and a component which is sparse in a known dictionary, using a convex demixing method. We consider two sparsity structures for the sparse factor of the dictionary sparse component, namely entry-wise and column-wise sparsity, and provide a unified analysis, encompassing both undercomplete and the overcomplete dictionary cases, to show that the constituent matrices can be successfully recovered under some relatively mild conditions on incoherence, sparsity, and rank. We leverage these results to localize targets of interest in a hyperspectral (HS) image based on their spectral signature(s) using the a priori known characteristic spectral responses of the target. We corroborate our theoretical results and analyze target localization performance of our approach via experimental evaluations and comparisons to related techniques.