Exploiting Two-Dimensional Group Sparsity in 1-Bit Compressive Sensing

We propose a new approach, {\it two-dimensional fused binary compressive sensing} (2DFBCS) to recover 2D sparse piece-wise signals from 1-bit measurements, exploiting 2D group sparsity for 1-bit compressive sensing recovery. The proposed method is a modified 2D version of the previous {\it binary iterative hard thresholding} (2DBIHT) algorithm, where the objective function includes a 2D one-sided $\ell_1$ (or $\ell_2$) penalty function encouraging agreement with the observed data, an indicator function of $K$-sparsity, and a total variation (TV) or modified TV (MTV) constraint. The subgradient of the 2D one-sided $\ell_1$ (or $\ell_2$) penalty and the projection onto the $K$-sparsity and TV or MTV constraint can be computed efficiently, allowing the appliaction of algorithms of the {\it forward-backward splitting} (a.k.a. {\it iterative shrinkage-thresholding}) family. Experiments on the recovery of 2D sparse piece-wise smooth signals show that the proposed approach is able to take advantage of the piece-wise smoothness of the original signal, achieving more accurate recovery than 2DBIHT. More specifically, 2DFBCS with the MTV and the $\ell_2$ penalty performs best amongst the algorithms tested.