论文标题
在成像科学和汉密尔顿 - 雅各比的贝叶斯后平均估计器上
On Bayesian posterior mean estimators in imaging sciences and Hamilton-Jacobi Partial Differential Equations
论文作者
论文摘要
变分和贝叶斯方法是两种已广泛用于解决图像重建问题的方法。在本文中，我们提出了汉密尔顿-Jacobi（HJ）部分微分方程与具有高斯数据保真项和对数concove的大量贝叶斯方法和后平均估计器之间的原始连接。尽管具有初始数据的某些一阶HJ PDE的解决方案描述了贝叶斯环境中的最大后验估计器，但在这里，我们在这里表明，具有初始数据的某些粘性HJ PDE的解决方案描述了一类广泛的后平均估计器。这些连接使我们能够建立涉及后平均估计值的几个表示公式和最佳界限。特别是，我们使用这些连接到HJ PDES来表明某些贝叶斯后平均值估计器可以表示为两次连续可区分的函数的近端映射，并且我们得出了这些功能的表示公式。
Variational and Bayesian methods are two approaches that have been widely used to solve image reconstruction problems. In this paper, we propose original connections between Hamilton--Jacobi (HJ) partial differential equations and a broad class of Bayesian methods and posterior mean estimators with Gaussian data fidelity term and log-concave prior. Whereas solutions to certain first-order HJ PDEs with initial data describe maximum a posteriori estimators in a Bayesian setting, here we show that solutions to some viscous HJ PDEs with initial data describe a broad class of posterior mean estimators. These connections allow us to establish several representation formulas and optimal bounds involving the posterior mean estimate. In particular, we use these connections to HJ PDEs to show that some Bayesian posterior mean estimators can be expressed as proximal mappings of twice continuously differentiable functions, and furthermore we derive a representation formula for these functions.
