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

## TL;DR

This paper introduces a convex demixing method for decomposing data matrices into low-rank and dictionary-sparse components, enabling effective target localization in hyperspectral images by leveraging spectral signatures.

## Contribution

It presents a unified theoretical framework for dictionary-based robust PCA accommodating both undercomplete and overcomplete dictionaries, with analysis of recovery conditions.

## Key findings

- Successful recovery of constituent matrices under mild conditions.
- Effective target localization in hyperspectral imaging using spectral signatures.
- Experimental validation demonstrating the approach's advantages.

## Abstract

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.