# Plug-and-Play Deep Energy Model for Inverse problems

## TL;DR

This paper introduces an energy-based Plug-and-Play framework for image recovery that learns a CNN-based prior with convergence guarantees, enabling more complex priors and improved MRI reconstruction performance.

## Contribution

It presents the first energy-based PnP formulation using a CNN score function with convergence guarantees, allowing for more expressive priors and better image reconstruction.

## Key findings

- Improved MRI reconstruction performance over traditional PnP methods.
- Energy-based formulation ensures convergence even without contraction constraints.
- CNN score function models complex image priors effectively.

## Abstract

We introduce a novel energy formulation for Plug- and-Play (PnP) image recovery. Traditional PnP methods that use a convolutional neural network (CNN) do not have an energy based formulation. The primary focus of this work is to introduce an energy-based PnP formulation, which relies on a CNN that learns the log of the image prior from training data. The score function is evaluated as the gradient of the energy model, which resembles a UNET with shared encoder and decoder weights. The proposed score function is thus constrained to a conservative vector field, which is the key difference with classical PnP models. The energy-based formulation offers algorithms with convergence guarantees, even when the learned score model is not a contraction. The relaxation of the contraction constraint allows the proposed model to learn more complex priors, thus offering improved performance over traditional PnP schemes. Our experiments in magnetic resonance image reconstruction demonstrates the improved performance offered by the proposed energy model over traditional PnP methods.