Variational Analysis of Adversarial Regularization for Solving Inverse Problems

Inverse problems form the backbone of modern signal/image processing and computational imaging, where signal reconstruction from corrupted measurements follows an optimization problem. The objective function is the sum of a data-fidelity term and a regularization functional that enforces desired properties in the reconstruction. The adversarial regularization (AR) framework is an unsupervised, data-driven approach for solving inverse problems, where the regularization function is learnt adversarially as a critique between the ground-truth distribution and the distribution of unregularized reconstructions. Thereafter, the solution to the regularized inverse problem follows an iterative technique. In this paper, we analyze the AR framework from a variational perspective, and, using Euler-Lagrange conditions, obtain the optimal regularization function in closed-form. The overall objective function is smooth and readily amenable to gradient descent minimization. We introduce momentum into the iterates as a natural extension to accelerate convergence. Since the optimal solutions are obtained in closed-form, our approach to solving inverse problems does not require prior training whilst being data-driven. We demonstrate the proposed technique on image deconvolution and show that the reconstruction performance of the proposed techniques measured in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM) are identical to the learnt counterparts.