In this paper, we propose a novel coupled dictionary training method for single image super-resolution based on patchwise sparse recovery, where the learned couple dictionaries relate the low- and high-resolution image patch spaces via sparse representation. The learning process enforces that the sparse representation of a low-resolution image patch in terms of the low-resolution dictionary can well reconstruct its underlying high-resolution image patch with the dictionary in the highresolution image patch space. We model the learning problem as a bilevel optimization problem, where the optimization includes an 1-norm minimization problem in its constraints. Implicit differentiation is employed to calculate the desired gradient for stochastic gradient descent. We demonstrate that our coupled dictionary learning method can outperform the existing joint dictionary training method both quantitatively and qualitatively. Furthermore, for real applications, we speed up the algorithm approximately 10 times by learning a neural network model for fast sparse inference and selectively processing only those visually salient regions. Extensive experimental comparisons with stateof- the-art super-resolution algorithms validate the effectiveness of our proposed approach.