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In this paper, we present a Bayesian-based superresolution algorithm that uses approximations of symmetric alpha-stable (SαS) Markov random fields as prior. The approximated SαS prior is used to perform maximum a posteriori (MAP) estimation for the high-resolution (HR) image reconstruction process. Compared with other state-of-the-art prior models, the proposed prior can better capture the heavy tails of the distribution of the HR image. Thus, the edges of the reconstructed HR image are preserved better in our method. As the corresponding energy function is nonconvex, the graduated nonconvexity method is used to solve the MAP estimation. Experiments confirm the better fit achieved by the proposed model to the actual data distribution and the consequent improvement in terms of visual quality over previously proposed super-resolution algorithms.