We consider the task of localizing targets of interest in a hyperspectral (HS) image based on their spectral signature(s), by posing the problem as two distinct convex demixing task(s). With applications ranging from remote sensing to surveillance, this task of target detection leverages the fact that each material/object possesses its own characteristic spectral response, depending upon its composition. However, since $\textit{signatures}$ of different materials are often correlated, matched filtering-based approaches may not be apply here. To this end, we model a HS image as a superposition of a low-rank component and a dictionary sparse component, wherein the dictionary consists of the $\textit{a priori}$ known characteristic spectral responses of the target we wish to localize, and develop techniques for two different sparsity structures, resulting from different model assumptions. We also present the corresponding recovery guarantees, leveraging our recent theoretical results from a companion paper. Finally, we analyze the performance of the proposed approach via experimental evaluations on real HS datasets for a classification task, and compare its performance with related techniques.

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 corroborate our theoretical results by presenting empirical evaluations in terms of phase transitions in rank and sparsity, in comparison to related techniques. Investigation of a specific application in hyperspectral imaging is included in an accompanying paper.