In this paper, we develop a decomposition model to restore blurred images with missing pixels. Our assumption is that the underlying image is the superposition of cartoon and texture components. We use the total variation norm and its dual norm to regularize the cartoon and texture, respectively. We recommend an efficient numerical algorithm based on the splitting versions of augmented Lagrangian method to solve the problem. Theoretically, the existence of a minimizer to the energy function and the convergence of the algorithm are guaranteed. In contrast to recently developed methods for deblurring images, the proposed algorithm not only gives the restored image, but also gives a decomposition of cartoon and texture parts. These two parts can be further used in segmentation and inpainting problems. Numerical comparisons between this algorithm and some state-of-the-art methods are also reported.