Local MALA-within-Gibbs for Bayesian image deblurring with total variation prior

Rafael Flock, Shuigen Liu, Yiqiu Dong, Xin T. Tong
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Abstract

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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利用总变异先验对贝叶斯图像去模糊进行局部 MALA-within-Gibbs 处理
我们考虑用贝叶斯推理方法对图像去模糊进行总变异(TV)先验推理。由于后验难以分析,我们采用了马尔可夫链蒙特卡罗(MCMC)方法。然而,由于大多数 MCMC 方法在高维度下会显著恶化,因此不适合处理高分辨率成像问题。在本文中,我们展示了如何利用后验的稀疏条件结构来促进低维取样。为此,我们利用模糊算子和电视先验的局部结构,将图像分割成矩形块,并采用阻塞吉布斯采样器,其建议源自 Metropolis-Hastings 调整朗文算法 (MALA)。我们证明,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率。为了应用 MALA 建议,我们用平滑版本对电视进行了近似,并证明引入的近似误差是均匀分布且与维度无关的。由于后验是一个吉布斯密度,我们可以利用哈默斯利-克里福德定理来确定后验条件,这些后验条件只在局部取决于相邻的区块。我们概述了本地并行评估条件的计算策略,这些条件是吉布斯更新的目标密度。在两个数值实验中,我们验证了 MLwGalgorithm 与维度无关的特性,并证明其性能优于 MALA。
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