Nonlocal low-rank regularization based approach (NLR) shows the state-of-the-art performance in compressive sensing (CS) recovery which exploits both structured sparsity of similar patches.However, it cannot efficiently preserve the edges because it only exploits the nonlocal regularization and ignores the relationship between pixels. Meanwhile, Logdet function that is used in NLR cannot well approximate the rank, because it is a fixed function and the optimization results obtained by this function essentially deviate from the real solution. A local and nonlocal regularization based CS approach is proposed toward exploiting the local sparse-gradient property of image and low-rank property of similar patches. Schatten-p norm is used as a better non-convex surrogate for the rank function. In addition, the alternating direction method of multipliers method (ADMM) is utilized to solve the resulting nonconvex optimization problem. Experimental results demonstrate that the proposed method outperforms existing state-of-the-art CS algorithms for image recovery.