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

We propose a new approach, two-dimensional binary fused compressive sensing (2DBFCS) to recover 2D sparse piece-wise signals from 1-bit measurements, exploiting group sparsity in 2D 1-bit compressive sensing. The proposed method is a modified 2D version of the previous binary iterative hard thresholding (2DBIHT) algorithm, where, the objective function consists of a 2D one-sided l1 (or l2) function and an indicator function of K-sparsity and an indicator function of total variation (TV) or modified TV (MTV) constraint (the MTV favors both sparsity and piece/wise smoothness while the TV promotes the whole smoothness). The subgradient of 2D one-sided l1 (or l2) barrier and the projection onto the Ksparsity and TV or MTV constraint set are easy to compute, such that the forward-backward splitting can be applied in 2DBFCS efficiently. Experiments on the recovery of 2D sparse piece-wise smooth signals show that the proposed 2DBFCS with the TV or MTV is able to take advantage of the piece-wise smoothness of the original signal, achieving more accurate recovery than 2DBIHT. Especially, the 2DBFCS with the MTV and the l2 barrier performs best amongst the algorithms.