Deep Gaussian Process Priors for Bayesian Image Reconstruction

TL;DR

The authors address uncertainty quantification in Bayesian image reconstruction by introducing deep Gaussian process priors that induce non-stationary, multi-scale behavior through a hierarchical SPDE formulation. They develop a scalable computational pipeline combining Matérn SPDE representations, finite element discretisation, and rational approximation of fractional operators, together with dimension-robust, determinant-free MCMC methods to sample from the posterior. The framework is validated on image upsampling, edge detection, and Radon-transform inversion, showing that non-stationary deep GPs can provide improved reconstructions over stationary priors, and that fractional regularity values can be meaningfully explored. The approach offers a flexible, scalable tool for Bayesian inverse problems in imaging, with potential extensions to adaptive meshing and anisotropic covariances.

Abstract

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.

Figures (22)

• Figure 1: One-dimensional GP regression example with two correlation length values.
• Figure 2: The left two plots show a realisation of a deep Gaussian process constructed using two layers, where the spatially varying length scale $\kappa^2 \in [50, 100^2]$. The right two plots show GP realisations corresponding to $\kappa^2=50$ and $\kappa^2=100^2$, respectively.
• Figure 3: Ground truth functions for reconstruction.
• Figure 4: Correlation lengths used in deep GP reconstructions of "square/circle" image with varying $\alpha$.
• Figure 5: Correlation lengths used in deep GP reconstructions of "corner/slope" image with varying $\alpha$.