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, super-resolution, 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.

翻译：我们旨在通过结合图像块的惩罚稀疏表示与无约束平滑表示来解决成像中的逆问题。这种方法为重建过程提供了直观的解释。我们将优化问题表述为双层规划：内层问题采用经典算法，外层问题通过监督学习优化字典和正则化参数。该过程通过隐式微分和基于梯度的优化实现。我们在去噪、超分辨率和压缩感知磁共振成像任务中评估了所提出的方法，并与经典模型及基于深度学习的方法进行了比较。结果表明，该方法始终优于经典模型，并在某些情况下也优于基于深度学习的方法。