Fractional calculus is an important branch of mathematical analysis and played a fundamental role in different fields including signal processing and image processing. In this paper, we proposed a sparse super-resolution (SR) technique by using the Grünwald-Letnikov (G–L) fractional differential operator, which aims to reconstruct high-resolution images by recovering pixel information from low-resolution images. We suggested a modified fractional derivative mask based on G–L for image enhancement. The proposed method suppresses block artifacts, staircase edges, and false edges near the edges. The proposed method is very flexible and preserves detailed features. The obtained experiments demonstrated that fractional operator can nonlinearly preserve the low-frequency contour information in the smooth region and also nonlinearly improve well the high-frequency edge and texture of the image. In sparse SR images, the dictionary is constructed by texture information of the images by using integer derivatives. In this work, we computed dictionary matrices by applying the fractional masks. In fact, the extracted features based on fractional derivatives are utilized in the dictionary training procedure and sparse coding. The experimental results and analysis on natural images indicated that the proposed method achieved much better results than other algorithms in terms of both quantitative measures and visual perception.

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