A sharp sufficient condition of block signal recovery via $l_2\mathrm{/}{l}_1$-minimization

This work gains a sharp sufficient condition on the block restricted isometry property for the recovery of sparse signal. Under the certain assumption, the signal with block structure can be stably recovered in the present of noisy case and the block sparse signal can be exactly reconstructed in the noise-free case. Besides, an example is proposed to exhibit the condition is sharp. As byproduct, when $t=1$, the result improves the bound of block restricted isometry constant ${\delta }_{s\mathrm{\mid }\mathcal{I}}$ in Lin and Li (Acta Math. Sin. Engl. Ser. 29(7): 1401-1412, 2013).