Bayesian imaging with data-driven priors

Abstract

This thesis addresses the challenging domain of imaging inverse problems, with particular emphasis on low-photon scenarios common in astronomy, biology, and medicine where reliable reconstruction is crucial. We leverage the Bayesian framework to develop efficient sampling algorithms for posterior inference, aiming to enhance reconstruction quality and provide principled uncertainty quantification. Key aspects include integrating data-driven models within Bayesian methods and establishing theoretical convergence guarantees.

As our first contribution, we introduce an accelerated proximal Markov chain Monte Carlo (MCMC) methodology for Bayesian inference in convex imaging problems. This strategy, formulated as a stochastic relaxed proximal-point iteration, has two interpretations. For smooth or Moreau-Yosida regularized models, it corresponds to an implicit midpoint discretization of overdamped Langevin diffusion, achieving provably accelerated convergence (O(√κ) for κ-strongly log-concave targets), improving over prior work. For non-smooth models, it acts as a Leimkuhler-Matthews discretization targeting a Moreau-Yosida approximation, significantly reducing bias compared to conventional Euler-based methods. Our non-asymptotic analysis also identifies the optimal time step to maximize convergence speed.

In our second contribution, we address low-photon Poisson imaging by introducing a novel plug-and-play (PnP) Langevin sampling methodology. Standard PnP Langevin algorithms are not well-suited for Poisson data due to high uncertainty, exploding gradients, and non-negativity constraints. We propose two approaches: (i) an accelerated PnP Langevin method incorporating boundary reflections and a likelihood approximation, and (ii) a mirror sampling algorithm utilizing Riemannian geometry to handle constraints and poor likelihood regularity directly, without approximations.

The proposed methodologies are evaluated through experiments including image deconvolution under Gaussian and Poisson noise with assumption-driven and data-driven priors, and compared against state-of-the-art methods for low-photon Poisson problems. Empirically, these sampling-based approaches significantly improve reconstruction quality and provide valuable uncertainty estimates in challenging low-photon regimes, establishing a strong basis for further development.

Building upon this work, future directions include embedding the proposed samplers within more complex Bayesian schemes (e.g., empirical Bayes and model selection), adapting the first contribution to non-convex problems, and integrating diffusion model priors with the second. Applying these techniques to broader low-photon imaging challenges remains a key objective.