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The JPEG compression method is among the most successful compression schemes since it readily provides good compressed results at a rather high compression ratio. However, the decompressed result of the standard JPEG decompression scheme usually contains some visible artifacts, such as blocking artifacts and Gibbs artifacts (ringing), especially when the compression ratio is rather high. In this paper, a novel artifact reducing approach for the JPEG decompression is proposed via sparse and redundant representations over a learned dictionary. Indeed, an effective two-step algorithm is developed. The first step involves dictionary learning and the second step involves the total variation regularization for decompressed images. Numerical experiments are performed to demonstrate that the proposed method outperforms the total variation and weighted total variation decompression methods in the measure of peak of signal to noise ratio, and structural similarity.