This talk presents a new and highly efficient MCMC methodology to perform Bayesian inference in low-photon imaging problems, with particular attention to situations involving observation noise processes that deviate significantly from Gaussian noise, such as binomial, geometric and low-intensity Poisson noise. These problems are challenging for many reasons. From an inferential viewpoint, low photon numbers lead to severe identifiability issues, poor stability and high uncertainty about the solution. Moreover, low-photon models often exhibit poor regularity properties that make efficient Bayesian computation difficult; e.g., hard non-negativity constraints, non-smooth priors, and log-likelihood terms with exploding gradients. More precisely, the lack of suitable regularity properties hinders the use of state-of-the-art Monte Carlo methods based on numerical approximations of the Langevin stochastic differential equation (SDE) or other similar dynamics, as both the continuous-time process and its numerical approximations behave poorly. We address this difficulty by proposing an MCMC methodology based on a reflected and regularised Langevin SDE, which is shown to be well-posed and exponentially ergodic under mild and easily verifiable conditions. This then allows us to derive four reflected proximal Langevin MCMC algorithms to perform Bayesian computation in low-photon imaging problems. The proposed approach is illustrated with a range of experiments related to image deblurring, denoising, and inpainting under binomial, geometric and Poisson noise.

In this talk, we will summarize the recent advacements made for non-Gaussian process priors for statistical inversion. This will be primarily focused on $\alpha$-stable distributions which provide a natural generalization of a family of distributions, such as the normal and Cauchy. We discuss recently proposed priors which include various Cauchy priors, hierarchical and neural-network based $\alpha$-stable priors. The focus will be computational where we demonstrate their gains on a range of examples for fully Bayesian and MAP-based estimation. We also provide some theoretical insights which include error bounds.

Gaussian process priors are often used in practice due to their fast computational properties. The smoothness of the resulting estimates, however, is not well suited for modelling functions with sharp changes. We propose a new prior that has same kind of good edge-preserving properties than total variation or Mumford-Shah but correspond to a well-defined infinite dimensional random variable. This is done by introducing a new random variable $T$ that takes values in the space of ‘trees’, and which is chosen so that the realisations have jumps only on a small set.

In this talk we present CUQIpy (pronounced ”cookie pie”) - a new computational modelling environment in Python that uses uncertainty quantification (UQ) to access and quantify the uncertainties in solutions to inverse problems. The overall goal of the software package is to allow both expert and non-expert (without deep knowledge of statistics and UQ) users to perform UQ related analysis of their inverse problem while focusing on the modelling aspects. To achieve this goal the package utilizes state-of-the-art tools and methods in statistics and scientific computing specifically tuned to the ill-posed and often large-scale nature of inverse problems to make UQ feasible. We showcase the software on problems relevant to imaging science such as computed tomography and partial differential equation-based inverse problems. CUQIpy is developed as part of the CUQI project at the Technical University of Denmark and is available at https://github.com/CUQI-DTU/CUQIpy.