A General Framework for Group Sparsity in Hyperspectral Unmixing Using Endmember Bundles

Due to low spatial resolution, hyperspectral data often consists of mixtures of contributions from multiple materials. This limitation motivates the task of hyperspectral unmixing (HU), a fundamental problem in hyperspectral imaging. HU aims to identify the spectral signatures (\textit{endmembers}) of the materials present in an observed scene, along with their relative proportions (\textit{fractional abundance}) in each pixel. A major challenge lies in the class variability in materials, which hinders accurate representation by a single spectral signature, as assumed in the conventional linear mixing model. Moreover, To address this issue, we propose using group sparsity after representing each material with a set of spectral signatures, known as endmember bundles, where each group corresponds to a specific material. In particular, we develop a bundle-based framework that can enforce either inter-group sparsity or sparsity within and across groups (SWAG) on the abundance coefficients. Furthermore, our framework offers the flexibility to incorporate a variety of sparsity-promoting penalties, among which the transformed $\ell_1$ (TL1) penalty is a novel regularization in the HU literature. Extensive experiments conducted on both synthetic and real hyperspectral data demonstrate the effectiveness and superiority of the proposed approaches.