(eng): Standard decompression of JPEG images produces artifacts along edges and a disturbing checkerboard pattern. To reduce these artifacts, decompression can be formulated as an image reconstruction problem within Bayesian maximum a posteriori probability framework. In this type of problem, the prior information about an image is typically given by the l1 norm of its sparse domain representation. In this paper, we show how the solution of this problem can be achieved very efficiently using the alternating direction method of multipliers if the sparsity domain forms a tight frame. The proposed algorithm restores images without disturbing JPEG artifacts in several\niterations, typically considerably less than competing algorithms. The quality of reconstruction both visually and in terms of SNR primarily depends on the tight frame used.\n