Image Restoration with Generalized L2 Loss and Convergent Plug-and-Play Priors

Image restoration involves solving an optimization problem where the objective function is the sum of a data-fidelity term and a regularization functional that incorporates a desired image prior. Solving the optimization problem using proximal methods results in iterative algorithms that require computing a gradient step corresponding to the data-fidelity loss and a proximal update corresponding to enforcing the image prior. In this paper, we develop a novel formulation for image restoration considering a generalized data-fidelity loss and a convex regularization function that enforces a desired image prior, and we solve the problem using proximal gradient method. The choice of the data-fidelity loss is such that the adjoint operator is reminiscent of Wiener filtering when the forward operator is a convolutional operator (for instance, a shift-invariant blur kernel). The proposed gradient update ensures that the iterates remain in the solution-space of the linear measurement constraints. We further propose the plug-and-play counterpart of the restoration technique, which allows one to leverage off-the-shelf data-driven denoisers in place of the proximal operator. Experimental validations carried out on BSD500, Brodatz, Urban100, and DIV2K datasets show that the proposed technique gives rise to superior image reconstruction quality compared with the state-of-the-art techniques, with the performance measured in terms of peak signal-to-noise ratio (PSNR) and structural similarity index metric (SSIM), with comparable computational complexity.