A Hybrid Bayesian-PDE Constrained Optimization Framework for High-Dimensional Image Reconstruction

Abstract

High-dimensional image reconstruction problems in fields such as medical imaging, astrophysics, and remote sensing are typically ill-posed inverse problems affected by noise, under sampling, and imperfections in the physical forward model. Traditional methods for resolving these conflicts suffer from an inherent trade-off: pure physics-based PDE-constrained models impose physical consistency but are deterministic and do not represent uncertainty, and fully Bayesian models provide principled uncertainty quantification but tend to become computationally intractable in very high-dimensional spaces. In response to these challenges, we propose the Hybrid Bayesian - PDE Constrained Optimization Framework. The Hybrid Bayesian - PDE Constrained Optimization Framework leverages the physical fidelity of PDE-based forward models with the expressive capability of Bayesian inference to model uncertainty. The reconstruction problem is cast as an optimization problem whereby a variational or hierarchical Bayesian prior is combined with a PDE-constrained data fidelity term, and the optimization objective is solved by an efficient stochastic variational optimization scheme. Experiments using a representative CT example and MRI datasets demonstrated how the hybrid methods provided (i) better reconstructions, preserving fine structures for substantially undersampled data and robust performance in noise when compared to the pure physics model along with providing (ii) clinically meaningful pixel-wise uncertainty maps. These results support the view that the proposed hybrid method provides a principled, computationally efficient, reliable approach to the challenge of solving large-scale inversion problems, while addressing the fundamental limitations of both deterministic, physics-based methods and probabilistic Bayesian inversion.