Reconstruction Guarantee Analysis of Binary Measurement Matrices Based on Girth

Binary 0-1 measurement matrices, especially those from coding theory, were introduced to compressed sensing (CS) recently. Good measurement matrices with preferred properties, e.g., the restricted isometry property (RIP) and nullspace property (NSP), have no known general ways to be efficiently checked. Khajehnejad \emph{et al.} made use of \emph{girth} to certify the good performances of sparse binary measurement matrices. In this paper, we examine the performance of binary measurement matrices with uniform column weight and arbitrary girth under basis pursuit. Explicit sufficient conditions of exact reconstruction %only including $\gamma$ and $g$ are obtained, which improve the previous results derived from RIP for any girth $g$ and results from NSP when $g\mathrm{/}2$ is odd. Moreover, we derive explicit $l_1\mathrm{/}{l}_1$, $l_2\mathrm{/}{l}_1$ and $l_{\mathrm{\infty }}\mathrm{/}{l}_1$ sparse approximation guarantees. These results further show that large girth has positive impacts on the performance of binary measurement matrices under basis pursuit, and the binary parity-check matrices of good LDPC codes are important candidates of measurement matrices.