  Uncompressed clinical data from modern positron emission tomography (PET)
scanners are very large, exceeding 350 million data points (projection bins).
The last decades have seen tremendous advancements in mathematical imaging
tools many of which lead to non-smooth (i.e. non-differentiable) optimization
problems which are much harder to solve than smooth optimization problems. Most
of these tools have not been translated to clinical PET data, as the
state-of-the-art algorithms for non-smooth problems do not scale well to large
data. In this work, inspired by big data machine learning applications, we use
advanced randomized optimization algorithms to solve the PET reconstruction
problem for a very large class of non-smooth priors which includes for example
total variation, total generalized variation, directional total variation and
various different physical constraints. The proposed algorithm randomly uses
subsets of the data and only updates the variables associated with these. While
this idea often leads to divergent algorithms, we show that the proposed
algorithm does indeed converge for any proper subset selection. Numerically, we
show on real PET data (FDG and florbetapir) from a Siemens Biograph mMR that
about ten projections and backprojections are sufficient to solve the MAP
optimisation problem related to many popular non-smooth priors; thus showing
that the proposed algorithm is fast enough to bring these models into routine
clinical practice.
