  We propose a regularization scheme for image reconstruction that leverages
the power of deep learning while hinging on classic sparsity-promoting models.
Many deep-learning-based models are hard to interpret and cumbersome to analyze
theoretically. In contrast, our scheme is interpretable because it corresponds
to the minimization of a series of convex problems. For each problem in the
series, a mask is generated based on the previous solution to refine the
regularization strength spatially. In this way, the model becomes progressively
attentive to the image structure. For the underlying update operator, we prove
the existence of a fixed point. As a special case, we investigate a mask
generator for which the fixed-point iterations converge to a critical point of
an explicit energy functional. In our experiments, we match the performance of
state-of-the-art learned variational models for the solution of inverse
problems. Additionally, we offer a promising balance between interpretability,
theoretical guarantees, reliability, and performance.
