Split Bregman algorithms for sparse / joint-sparse and low-rank signal recovery: Application in compressive hyperspectral imaging

In this work we derive algorithms for solving two problems - the first one is the combined l₁-norm (sparsity) and nuclear norm (low rank) regularized least squares problem and the second one is the l_{2, 1}-norm (joint sparsity) and nuclear norm regularized least squares problem. There are no efficient general purpose solvers for these problems; our work plugs this gap by deriving Split Bregman based algorithms for solving the said problems. Both algorithms are applicable for recovering hyperspectral images from their compressive measurements obtained via the single pixel camera. We show that our proposed techniques significantly outperform previous methods in terms of recovery accuracy.