Energy-based models for inverse imaging problems

TL;DR

This paper provides a comprehensive overview of energy-based models (EBMs) in inverse imaging, covering theoretical foundations, learning techniques, sampling algorithms, and numerical results demonstrating their effectiveness.

Contribution

It offers a rigorous theoretical framework for EBMs in Bayesian inverse problems and discusses practical algorithms and numerical validation for imaging applications.

Findings

EBMs are effective for inverse imaging problems.

Sampling algorithms like Langevin Monte Carlo are suitable for EBMs.

Numerical results validate the use of EBMs in real inverse imaging tasks.

Abstract

In this chapter we provide a thorough overview of the use of energy-based models (EBMs) in the context of inverse imaging problems. EBMs are probability distributions modeled via Gibbs densities $p(x) \propto \exp{-E(x)}$ with an appropriate energy functional $E$. Within this chapter we present a rigorous theoretical introduction to Bayesian inverse problems that includes results on well-posedness and stability in the finite-dimensional and infinite-dimensional setting. Afterwards we discuss the use of EBMs for Bayesian inverse problems and explain the most relevant techniques for learning EBMs from data. As a crucial part of Bayesian inverse problems, we cover several popular algorithms for sampling from EBMs, namely the Metropolis-Hastings algorithm, Gibbs sampling, Langevin Monte Carlo, and Hamiltonian Monte Carlo. Moreover, we present numerical results for the resolution of several inverse imaging problems obtained by leveraging an EBM that allows for the explicit verification of those properties that are needed for valid energy-based modeling.