In this talk we revisit the problem of Positron Emission Tomography (PET) reconstruction with non-smooth and convex priors. As the data fidelity term in PET is the Poisson likelihood, there are not many algorithms that can solve this problem. A very popular choice for solving it is the Primal-Dual Hybrid Gradient (PDHG) method proposed by Chambolle and Pock. While this algorithm works well for small to medium size problems in PET, in the case when the data size becomes large, computational issues arise, which is due to the fact that the system matrix for clinical PET scanners is very large and cannot be stored in the memory of most computers. Instead, an expensive algorithm to compute matrix-vector products has to be employed.
In this work we circumvent this issue by marrying the PDHG method with a randomized dual decomposition strategy (somewhat reminiscent of ART, Kaczmarz, and OSEM, which operate in the primal). PDHG arises as a special (and suboptimal) case of our more general method. We prove that our algorithm converges for a wide range of random samplings (e.g., uniform sampling, importance sampling, parallel samplings, ...), enabling us to implement various decomposition rules which can be chosen to fit the particular computing environment in use. We prove rigorous iteration complexity bounds for standard and accelerated variants of our method. Numerical examples show that the new method is much faster than PDHG.
This is joint work with Antonin Chambolle (Ecole Polytechnique), Matthias Ehrhardt (Cambridge), and Carola-Bibiane Schoenlieb (Cambridge).
15:30 - 16:00