Mid Sweden University

Inverse problems refer to the task of reconstructing a clean signal from a degraded observation. In imaging, this pertains to restoration problems like denoising, super-resolution or in-painting. Because inverse problems are often ill-posed, regularization based on prior information is needed. Plug-and-play (pnp) approaches take a general approach to regularization and plug a deep denoiser into an iterative solver for inverse problems. However, considering the inverse problems at hand in training could improve reconstruction performance at test-time. Deep equilibrium models allow for the training of multi-task priors on the reconstruction error via an estimate of the iterative method’s fixed-point (FP). This paper investigates the intersection of pnp and DEQ models for the training of a regularizing gradient (RG) and derives an upper bound for the reconstruction loss of a gradient-descent (GD) procedure. Based on this upper bound, two procedures for the training of RGs are proposed and compared: One optimizes the upper bound directly, the other trains a deep equilibrium GD (DEQGD) procedure and uses the bound for regularization. The resulting regularized RG (RERG) produces consistently good reconstructions across different inverse problems, while the other RGs tend to have some inverse problems on which they provide inferior reconstructions.