Variational and Bayesian methods are two widely used set of approaches to solve image denoising problems. In a Bayesian setting, these approaches correspond, respectively, to using maximum a posteriori estimators and posterior mean estimators for reconstructing images. In this paper, we propose novel theoretical connections between Hamilton–Jacobi partial differential equations (HJ PDEs) and a broad class of posterior mean estimators with quadratic data fidelity term and log-concave prior. Where solutions to some first-order HJ PDEs with initial data describe maximum a posteriori estimators, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. We use these connections to establish representation formulas and various properties of posterior mean estimators. In particular, we use these connections to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of smooth functions and derive representation formulas for these functions.

变分和贝叶斯方法是解决图像去噪问题的两种广泛使用的方法。在贝叶斯环境中，这些方法分别对应于使用最大后验估计器和后均值估计器来重建图像。在本文中，我们提出了汉密尔顿-雅各比偏微分方程（HJ PDE）与具有二次数据保真度项和对数凹入先验的广泛后验均值估计器之间的新颖理论联系。其中一些具有初始数据的一阶HJ PDE的解描述了最大后验估计量，这里我们展示了一些具有初始数据的粘性HJ PDE的解描述了一类后验均值估计子。我们使用这些联系来建立表示公式和后验均值的各种性质。特别是，