This paper presents a scalable approximate Bayesian method for image restoration using Total Variation (TV) priors, with the ability to offer uncertainty quantification. In contrast to most optimization methods based on maximum a posteriori estimation, we use the Expectation Propagation (EP) framework to approximate minimum mean squared error (MMSE) estimates and marginal (pixel-wise) variances, without resorting to Monte Carlo sampling. For the classical anisotropic TV-based prior, we also propose an iterative scheme to automatically adjust the regularization parameter via Expectation Maximization (EM). Using Gaussian approximating densities with diagonal covariance matrices, the resulting method allows highly parallelizable steps and can scale to large images for denoising, deconvolution, and compressive sensing (CS) problems. The simulation results illustrate that such EP methods can provide a posteriori estimates on par with those obtained via sampling methods but at a fraction of the computational cost. Moreover, EP does not exhibit strong underestimation of posteriori variances, in contrast to variational Bayes alternatives.

中文翻译：

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使用总变分先验和期望传播的快速可扩展图像恢复

本文提出了一种可扩展的近似贝叶斯方法，用于使用总变差 (TV) 先验进行图像恢复，并具有提供不确定性量化的能力。与大多数基于最大后验估计的优化方法相比，我们使用期望传播 (EP) 框架来近似最小均方误差 (MMSE) 估计和边际（像素级）方差，而不使用蒙特卡洛采样。对于经典的基于各向异性 TV 的先验，我们还提出了一种迭代方案，通过期望最大化 (EM) 自动调整正则化参数。使用具有对角协方差矩阵的高斯近似密度，得到的方法允许高度并行化的步骤，并且可以扩展到大图像以解决去噪、反卷积和压缩感知 (CS) 问题。模拟结果表明，这种 EP 方法可以提供与通过抽样方法获得的后验估计相同的后验估计，但计算成本只是其中的一小部分。此外，与变分贝叶斯替代方案相比，EP 并未表现出强烈低估后验方差。