This work addresses image recovery problem in the presence of salt-and-pepper noise and image blur. The salt-and-pepper noise reviewed as the impulsive noise, in this paper, is modeled as a sparse signal because of its impulsiveness. To accurately reconstruct the clean image and the blur kernel, the framelet domains are exploited to sparsely represent the image and the blur kernel. From the reformulations conducted, a joint estimation is devised to simultaneously perform the image recovery, the salt-and-pepper noise suppression and the blur kernel estimation under a optimization framework. To solve the optimization problem, an efficient solver based on accelerated proximal gradient (APG) is developed to obtain the joint estimation solution. Numerical studies demonstrate the superior performance of the joint estimation algorithm compared with the state-of-the-art approaches in terms of both objective and subjective evaluation standards.