Deep Gaussian Process Priors for Bayesian Image Reconstruction

TL;DR

This paper develops efficient computational methods for deep Gaussian process priors in Bayesian image reconstruction, enabling better uncertainty quantification and non-stationary modeling in high-dimensional inverse problems.

Contribution

It introduces combined rational approximation and determinant-free MCMC techniques for sampling fractional SPDEs in deep GPs, improving computational efficiency.

Findings

Deep GPs outperform stationary GPs in image reconstruction quality.

The proposed methods enable effective sampling from complex high-dimensional posteriors.

Flexible regularity parameters allow tailored non-stationary priors for different imaging tasks.

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.