In this study, we propose a novel algorithm for compressive imaging using digital micromirror device (DMD) modulated focal plane array (FPA) data. In this setting, DMD modulates the scene in the image domain by blocking some of the pixels at a higher resolution level. For reconstruction, a regularized optimization problem is solved, whereas reconstruction time is crucial for a practical compressive sensing application. Here, we propose an augmented Lagrangian based method that employs sparsity in both Fourier and gradient spaces, where we apply a separate proximal mapping for each snapshot. We demonstrate the performance of the algorithm via comparison to the state-of-the-art. The proposed method yields higher pSNR at shorter convergence times, especially at higher compression ratios.