In image reconstruction, an accurate quantification of uncertainty is of great importance for informed decision making. Here, the Bayesian approach to inverse problems can be used: the image is represented through a random function that incorporates prior information which is then updated through Bayes' formula. However, finding a prior is difficult, as images often exhibit non-stationary effects and multi-scale behaviour. Thus, usual Gaussian process priors are not suitable. Deep Gaussian processes, on the other hand, encode non-stationary behaviour in a natural way through their hierarchical structure. To apply Bayes' formula, one commonly employs a Markov chain Monte Carlo (MCMC) method. In the case of deep Gaussian processes, sampling is especially challenging in high dimensions: the associated covariance matrices are large, dense, and changing from sample to sample. A popular strategy towards decreasing computational complexity is to view Gaussian processes as the solutions to a fractional stochastic partial differential equation (SPDE). In this work, we investigate efficient computational strategies to solve the fractional SPDEs occurring in deep Gaussian process sampling, as well as MCMC algorithms to sample from the posterior. Namely, we combine rational approximation and a determinant-free sampling approach to achieve sampling via the fractional SPDE. We test our techniques in standard Bayesian image reconstruction problems: upsampling, edge detection, and computed tomography. In these examples, we show that choosing a non-stationary prior such as the deep GP over a stationary GP can improve the reconstruction. Moreover, our approach enables us to compare results for a range of fractional and non-fractional regularity parameter values.

翻译：在图像重建中，准确量化不确定性对于知情决策至关重要。本文采用贝叶斯方法处理逆问题：通过随机函数表示图像，该函数融合了先验信息，并随后通过贝叶斯公式进行更新。然而，先验的选取颇具挑战，因为图像常呈现非平稳效应与多尺度特性。因此，常规的高斯过程先验并不适用。相比之下，深度高斯过程凭借其层次化结构，能够自然地编码非平稳行为。为应用贝叶斯公式，通常采用马尔可夫链蒙特卡洛方法。对于深度高斯过程，在高维情形下采样尤为困难：其关联的协方差矩阵规模庞大、结构稠密，且随样本变化而变化。降低计算复杂度的一种常用策略是将高斯过程视为分数阶随机偏微分方程的解。本研究探讨了用于深度高斯过程采样的分数阶SPDE高效计算策略，以及从后验分布采样的MCMC算法。具体而言，我们结合有理逼近与无行列式采样方法，通过分数阶SPDE实现采样。我们在标准贝叶斯图像重建问题中测试了所提技术：上采样、边缘检测与计算机断层扫描。在这些示例中，我们证明选择非平稳先验（如深度GP）相较于平稳GP能够改善重建效果。此外，我们的方法使得我们能够比较一系列分数阶与非分数阶正则化参数值下的结果。