Linear Inverse Problems Using a Generative Compound Gaussian Prior (2406.10767v1)

Abstract: Since most inverse problems arising in scientific and engineering applications are ill-posed, prior information about the solution space is incorporated, typically through regularization, to establish a well-posed problem with a unique solution. Often, this prior information is an assumed statistical distribution of the desired inverse problem solution. Recently, due to the unprecedented success of generative adversarial networks (GANs), the generative network from a GAN has been implemented as the prior information in imaging inverse problems. In this paper, we devise a novel iterative algorithm to solve inverse problems in imaging where a dual-structured prior is imposed by combining a GAN prior with the compound Gaussian (CG) class of distributions. A rigorous computational theory for the convergence of the proposed iterative algorithm, which is based upon the alternating direction method of multipliers, is established. Furthermore, elaborate empirical results for the proposed iterative algorithm are presented. By jointly exploiting the powerful CG and GAN classes of image priors, we find, in compressive sensing and tomographic imaging problems, our proposed algorithm outperforms and provides improved generalizability over competitive prior art approaches while avoiding performance saturation issues in previous GAN prior-based methods.