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A highly compressed image is usually not only of low resolution, but also suffers from compression artifacts (blocking artifact is treated as an example in this paper). Directly performing image super-resolution (SR) to a highly compressed image would also simultaneously magnify the blocking artifacts, resulting in an unpleasing visual experience. In this paper, we propose a novel learning-based framework to achieve joint single-image SR and deblocking for a highly-compressed image. We argue that individually performing deblocking and SR (i.e., deblocking followed by SR, or SR followed by deblocking) on a highly compressed image usually cannot achieve a satisfactory visual quality. In our method, we propose to learn image sparse representations for modeling the relationship between low- and high-resolution image patches in terms of the learned dictionaries for image patches with and without blocking …