Learning of Patch-Based Smooth-Plus-Sparse Models for Image Reconstruction

We aim at the solution of inverse problems in imaging, by combining a penalized sparse representation of image patches with an unconstrained smooth one. This allows for a straightforward interpretation of the reconstruction. We formulate the optimization as a bilevel problem. The inner problem deploys classical algorithms while the outer problem optimizes the dictionary and the regularizer parameters through supervised learning. The process is carried out via implicit differentiation and gradient-based optimization. We evaluate our method for denoising, superresolution, and compressed-sensing magnetic-resonance imaging. We compare it to other classical models as well as deep-learning-based methods and show that it always outperforms the former and also the latter in some instances.