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.

翻译：本文提出了一种新颖的贝叶斯框架，通过设计一种考虑底层非欧几里得几何结构的蒙特卡洛采样算法来解决泊松逆问题。为应对泊松似然函数带来的挑战（如非利普希茨梯度和正性约束），我们推导出一个利用精确及渐近精确数据增强的贝叶斯模型。特别地，增强模型引入基于Burg熵的Bregman散度导出的两组分裂变量。值得注意的是，所得增强后验分布的条件分布具有自然共轭特性，并保持了潜在变量与分裂变量的内在几何结构。这使得所有条件分布（除包含正则化势函数的一项外）均可通过吉布斯步骤进行显式高效采样。对于后者，我们采用适用于处理具有显式或易计算得分函数先验的Hessian黎曼朗之万蒙特卡洛（HRLMC）算法。通过在镜像流形上操作，该朗之万步骤确保采样满足正性约束，并更准确地反映底层问题结构。在去噪、去模糊及正电子发射断层扫描（PET）实验中获得的性能结果表明，与基于优化和采样的方法相比，该方法在重建质量方面具有竞争优势。