Sparse representation is one of the effective approaches to perform the feature extraction on the high-dimensional signals and has been widely used in the field of image processing. In this paper, a novel single image super-resolution (SR) reconstruction algorithm based on sparse coding is presented. Specifically, due to the fact that sparse coding has the potential to greatly facilitate the biological-neural system and to deal with a large amount of complex data, while consuming a finite amount of energy, a memristor crossbar array synthesized by a threshold-type memristor model is constructed for the implementation of a hardware-friendly sparse coding method, i.e., the Locally Competitive Algorithm (LCA). Meanwhile, the relevant online dictionary learning method is provided, along with the description of weight programming strategy. Furthermore, the specific description of the image SR via the LCA sparse coding is provided. For the verification purpose, a series of experiments are carried out to illustrate the validity and effectiveness of the entire scheme.