基于全变分先验的贝叶斯图像去模糊的局部 MALA-within-Gibbs 方法
Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior
AI总结：
本文提出局部 MALA-within-Gibbs（MLwG）采样算法，通过利用后验的稀疏条件结构和分块 Gibbs 采样解决高分辨率图像去模糊中的高维采样难题，并证明其具有维度无关的接受率与收敛速率。
摘要：
我们考虑具有全变分（TV）先验的图像去模糊的贝叶斯推断。由于后验在解析上不可处理，我们采用马尔可夫链蒙特卡罗（MCMC）方法。然而，由于大多数 MCMC 方法在高维情况下性能显著恶化，它们不适合处理高分辨率成像问题。在本文中，我们展示了如何通过利用后验的稀疏条件结构来促进低维采样。为此，我们利用模糊算子和 TV 先验的局部结构，将图像划分为矩形块，并采用分块 Gibbs 采样器，其提议来自 Metropolis-Hastings 调整的 Langevin 算法（MALA）。我们证明了这种 MALA-within-Gibbs（MLwG）采样算法具有与维度无关的块接受率和与维度无关的收敛速率。为了应用 MALA 提议，我们通过平滑版本近似 TV，并证明引入的近似误差是均匀分布的且与维度无关。由于后验是 Gibbs 密度，我们可以使用 Hammersley-Clifford 定理来识别仅局部依赖于相邻块的后验条件分布。我们概述了在局部和并行地评估条件分布的计算策略，这些条件分布是 Gibbs 更新中的目标密度。在两个数值实验中，我们验证了 MLwG 算法的维度无关特性，并证明了其优于 MALA 的性能。
英文摘要：
We consider Bayesian inference for image deblurring with total variation (TV) prior. Since the posterior is analytically intractable, we resort to Markov chain Monte Carlo (MCMC) methods. However, since most MCMC methods significantly deteriorate in high dimensions, they are not suitable to handle high resolution imaging problems. In this paper, we show how low-dimensional sampling can still be facilitated by exploiting the sparse conditional structure of the posterior. To this end, we make use of the local structures of the blurring operator and the TV prior by partitioning the image into rectangular blocks and employing a blocked Gibbs sampler with proposals stemming from the Metropolis-Hastings adjusted Langevin Algorithm (MALA). We prove that this MALA-within-Gibbs (MLwG) sampling algorithm has dimension-independent block acceptance rates and dimension-independent convergence rate. In order to apply the MALA proposals, we approximate the TV by a smoothed version, and show that the introduced approximation error is evenly distributed and dimension-independent. Since the posterior is a Gibbs density, we can use the Hammersley-Clifford Theorem to identify the posterior conditionals which only depend locally on the neighboring blocks. We outline computational strategies to evaluate the conditionals, which are the target densities in the Gibbs updates, locally and in parallel. In two numerical experiments, we validate the dimension-independent properties of the MLwG algorithm and demonstrate its superior performance over MALA.
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