Deep End-to-End Posterior ENergy (DEEPEN) for image recovery

Current end-to-end (E2E) and plug-and-play (PnP) image reconstruction algorithms approximate the maximum a posteriori (MAP) estimate but cannot offer sampling from the posterior distribution, like diffusion models. By contrast, it is challenging for diffusion models to be trained in an E2E fashion. This paper introduces a Deep End-to-End Posterior ENergy (DEEPEN) framework, which enables MAP estimation as well as sampling. We learn the parameters of the posterior, which is the sum of the data consistency error and the negative log-prior distribution, using maximum likelihood optimization in an E2E fashion. The proposed approach does not require algorithm unrolling, and hence has a smaller computational and memory footprint than current E2E methods, while it does not require contraction constraints typically needed by current PnP methods. Our results demonstrate that DEEPEN offers improved performance than current E2E and PnP models in the MAP setting, while it also offers faster sampling compared to diffusion models. In addition, the learned energy-based model is observed to be more robust to changes in image acquisition settings.