用能量模型解决图像逆问题，理论扎实且可验证。
Energy-based models for inverse imaging problems
- 基于能量函数构建概率模型，融合贝叶斯框架
- 在多个逆成像任务中实现稳定重建，结果可验证
- 适合需要可解释性与数学严谨性的图像恢复研究者
本章系统综述了能量基模型（EBMs）在图像逆问题中的应用。EBMs通过吉布斯密度 $p(x) /propto \\'exp{-E(x)}$ 建模概率分布，其中能量函数 $E$ 为关键设计。我们首先给出贝叶斯逆问题的严格理论，涵盖有限维与无限维情形下的适定性与稳定性结果。接着探讨如何利用数据学习EBM，并详细介绍从EBM采样的主流算法：梅特罗波利斯-哈斯蒂格、吉布斯采样、朗之万蒙特卡洛及哈密顿蒙特卡洛。最后，通过数值实验展示了基于EBM解决若干逆成像问题的效果，验证了能量建模所需的各项关键性质。
原文摘要 · Abstract (English)
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.
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