  Modern imaging methods rely strongly on Bayesian inference techniques to
solve challenging imaging problems. Currently, the predominant Bayesian
computation approach is convex optimisation, which scales very efficiently to
high dimensional image models and delivers accurate point estimation results.
However, in order to perform more complex analyses, for example image
uncertainty quantification or model selection, it is necessary to use more
computationally intensive Bayesian computation techniques such as Markov chain
Monte Carlo methods. This paper presents a new and highly efficient Markov
chain Monte Carlo methodology to perform Bayesian computation for high
dimensional models that are log-concave and non-smooth, a class of models that
is central in imaging sciences. The methodology is based on a regularised
unadjusted Langevin algorithm that exploits tools from convex analysis, namely
Moreau-Yoshida envelopes and proximal operators, to construct Markov chains
with favourable convergence properties. In addition to scaling efficiently to
high dimensions, the method is straightforward to apply to models that are
currently solved by using proximal optimisation algorithms. We provide a
detailed theoretical analysis of the proposed methodology, including asymptotic
and non-asymptotic convergence results with easily verifiable conditions, and
explicit bounds on the convergence rates. The proposed methodology is
demonstrated with four experiments related to image deconvolution and
tomographic reconstruction with total-variation and $\ell_1$ priors, where we
conduct a range of challenging Bayesian analyses related to uncertainty
quantification, hypothesis testing, and model selection in the absence of
ground truth.
