Blind image deblurring is one of the challenging problems in image processing and computer vision. The main purpose of blind image deblurring is to estimate the correct blur kernel and restore the latent image with edge-preservation, details-protection, and ringing suppression. In order to achieve ideal results, an innovative compound L0-regularized model is proposed to estimate the blur kernel by regularizing the sparsity property of natural images and two characteristics of blur kernel, such as continuity and sparsity. In the alternating direction framework, the split Bregman algorithm and half-quadratic splitting rule are alternatively employed to optimize the proposed kernel estimation model. Finally, a nonblind restoration method with ringing suppression is developed to obtain the ultimate latent image. Extensive experiments demonstrate the efficiency and viability of the proposed method compared with some state-of-the-art blind deblurring methods.