We investigate the learning of a convex patch-based synthesis model for the reconstruction of images. In essence, we propose to learn a dictionary via bilevel optimization for denoising. Using implicit differentiation, we find a closed-form formula of the derivative of the minimizer of an objective with respect to the atoms of the dictionary. We also propose a novel way to handle the mean of each patch of the predictions, which improves our model when it is applied to other inverse problems. For minimizing the objective involving the learnt dictionary, we propose an early stopping criterion to further improve the performance of the model for denoising. Finally, we assess our model in a compressed sensing MRI inverse problem and show that, despite being trained on denoising only, our model yields good reconstruction performances.