Bregman geometry-aware split Gibbs sampling for Bayesian Poisson inverse problems
Elhadji Cisse Faye, Mame Diarra Fall, Nicolas Dobigeon, Eric Barat
TL;DR
This work tackles photon-limited Poisson inverse problems by introducing a geometry-aware Bayesian framework that preserves positivity through Burg-entropy–based Bregman divergences. It combines exact data augmentation with a double AXDA splitting, enabling a tractable posterior that decouples data fidelity from priors and yields efficient Gibbs updates; the remaining non-Gaussian prior step is handled with Hessian Riemannian Langevin Monte Carlo on a mirror manifold. The proposed HRLwSGS algorithm supports exact or easily computable prior score functions, enabling the use of data-driven priors such as RED and BSD, and providing quantified uncertainty. Empirical results on Poisson denoising, deconvolution, and PET reconstruction show competitive reconstruction quality and robust uncertainty quantification, highlighting the method’s practical impact for photon-limited imaging modalities.
Abstract
This paper proposes a novel Bayesian framework for solving Poisson inverse problems by devising a Monte Carlo sampling algorithm which accounts for the underlying non-Euclidean geometry. To address the challenges posed by the Poisson likelihood -- such as non-Lipschitz gradients and positivity constraints -- we derive a Bayesian model which leverages exact and asymptotically exact data augmentations. In particular, the augmented model incorporates two sets of splitting variables both derived through a Bregman divergence based on the Burg entropy. Interestingly the resulting augmented posterior distribution is characterized by conditional distributions which benefit from natural conjugacy properties and preserve the intrinsic geometry of the latent and splitting variables. This allows for efficient sampling via Gibbs steps, which can be performed explicitly for all conditionals, except the one incorporating the regularization potential. For this latter, we resort to a Hessian Riemannian Langevin Monte Carlo (HRLMC) algorithm which is well suited to handle priors with explicit or easily computable score functions. By operating on a mirror manifold, this Langevin step ensures that the sampling satisfies the positivity constraints and more accurately reflects the underlying problem structure. Performance results obtained on denoising, deblurring, and positron emission tomography (PET) experiments demonstrate that the method achieves competitive performance in terms of reconstruction quality compared to optimization- and sampling-based approaches.