Deep Gaussian Process Priors for Bayesian Image Reconstruction

2024-12-13

arXiv.org

AI Summary

To address the challenge of uncertainty quantification in image reconstruction arising from non-stationarity and multi-scale structures, this paper proposes a Bayesian prior modeling framework based on deep Gaussian processes (Deep GPs). Methodologically, it integrates rational approximations of fractional stochastic partial differential equations (SPDEs) with determinant-free Markov chain Monte Carlo (MCMC) to enable efficient high-dimensional posterior sampling for Deep GPs. A key contribution is the introduction of an interpretable, tunable regularity parameter that explicitly governs prior smoothness, yielding a controllable non-stationary prior paradigm. Evaluated on super-resolution, edge detection, and CT reconstruction tasks, the method achieves superior reconstruction accuracy and uncertainty calibration compared to conventional stationary GP priors. Furthermore, it enables systematic analysis and interpretation of the regularity parameter, facilitating principled model selection and physical interpretability.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Addresses non-stationary image reconstruction uncertainty quantification
• Overcomes high-dimensional sampling challenges in deep Gaussian processes
• Improves Bayesian image reconstruction via fractional SPDE techniques

Innovation

• Deep Gaussian processes for non-stationary image priors
• Fractional SPDEs for efficient sampling
• Determinant-free MCMC for posterior sampling