  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.
