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A wavelet based compressed sensing Super Resolution algorithm is developed, in which the energy function optimization is approximated numerically via the Regularized Orthogonal Matching Pursuit. The proposed algorithm works well with a smaller quantity of training image patches and outputs images with satisfactory subjective quality. It is tested on classical images commonly adopted by Super Resolution researchers with both generic and specialized training sets for comparison with other popular commercial software and state-of-the-art methods. Experiments demonstrate that, the proposed algorithm is competitive among contemporary Super Resolution methods.