ABSTRACT

Among the most significant challenges with using Markov chain Monte Carlo (MCMC) methods for sampling from the posterior distributions of Bayesian inverse problems is the rate at which the sampling becomes computationally intractable, as a function of the number of estimated parameters. In image deblurring, there are many MCMC algorithms in the literature, but few attempt reconstructions for images larger than $512\times 512$ pixels (${10}^5$ parameters). In quantitative X-ray radiography, used to diagnose dynamic materials experiments, the images can be much larger, leading to problems with millions of parameters. We address this issue and construct a Gibbs sampler via a blocking scheme that leads to a sparse and highly structured posterior precision matrix. The Gibbs sampler naturally exploits the special matrix structure during sampling, making it ‘dimension-robust’, so that its mixing properties are nearly independent of the image size, and generating one sample is computationally feasible. The dimension-robustness enables the characterization of posteriors for large-scale image deblurring problems on modest computational platforms. We demonstrate applicability of this approach by deblurring radiographs of size $4096\times 4096$ pixels (${10}^7$ parameters) taken at the Cygnus Dual Beam X-ray Radiography Facility at the U.S. Department of Energy's Nevada National Security Site.

摘要

使用马尔可夫链蒙特卡罗 (MCMC) 方法从贝叶斯逆问题的后验分布中采样的最大挑战之一是采样变得难以计算的速率，作为估计参数数量的函数。在图像去模糊方面，文献中有很多 MCMC 算法，但很少有人尝试对大于$512\times 512$ 像素 （${10}^5$参数）。在用于诊断动态材料实验的定量 X 射线摄影中，图像可能更大，导致数百万个参数出现问题。我们解决了这个问题，并通过一种导致稀疏且高度结构化的后验精度矩阵的阻塞方案构建了一个 Gibbs 采样器。Gibbs 采样器在采样过程中自然地利用了特殊的矩阵结构，使其具有“维度稳健性”，因此其混合属性几乎与图像大小无关，并且生成一个样本在计算上是可行的。维度稳健性能够在适度的计算平台上表征大规模图像去模糊问题的后验。我们通过去模糊大小的射线照片来证明这种方法的适用性$4096\times 4096$ 像素 （${10}^7$ 参数）在美国能源部内华达州国家安全站点的 Cygnus 双光束 X 射线射线照相设施拍摄。