Maximum a Posteriori Training of Diffusion Models for Image Restoration

Inverse problems involve reconstructing clean images from degraded observations. Maximum a Posteriori (MAP) estimation reconstructs the most probable source image from noisy measurements. When combined with Plug-and-Play (PnP) priors defined by an image denoising algorithm, MAP estimation yields high-quality reconstructions. In contrast, Diffusion Models (DMs) address inverse problems by sampling from the posterior distribution using score functions trained on images perturbed by Gaussian noise. Prior work reformulated diffusion sampling as Deep Equilibrium (DEQ) models but did not fine-tune DMs for inverse problems. This work introduces MaximUm a PostEriori Training (MUPET), a framework that leverages PnP gradient descent to enable DEQ fine-tuning of DMs on inverse problems. By refining a generative prior at the fixed-point of MAP estimation, MUPET enhances image restoration via posterior sampling while maintaining quality when sampling from the prior.