An ordered-subsets proximal preconditioned gradient algorithm for total variation regularized PET image reconstruction

Statistical variability of the PET data pre-corrected for random coincidences or acquired in sufficiently high count rates can be approximated by a Gaussian distribution, which results in a penalized weighted least-squares (PWLS) cost function. In this study, a proximal preconditioned gradient algorithm accelerated with ordered subsets (PPG-OS) is proposed for the optimization of the PWLS function, while addressing its two challenges encountered by previous algorithms such as separable paraboloidal surrogates accelerated with ordered-subsets (SPS-OS) and preconditioned conjugate gradient. First, the penalty and the weighting matrix of this function make its Hessian matrix ill-conditioned; thereby surrogate functions end up with high-curvatures and preconditioners would poorly approximate the Hessian matrix. The second challenge arises when using non-smooth penalty functions such as total variation (TV), which makes the PWLS function not amenable to optimization using gradient-based algorithms. To deal with these challenges, we used a proximal point method to surrogate the PWLS function with a proxy, which is then split into a preconditioned gradient descent and a proximal mapping associated with the TV penalty. A dual formulation was used to obtain the proximal mapping the TV penalty and also its smoothed version, i.e. Huber penalty. The proposed algorithm was studied for three different diagonal preconditioners and compared with the SPS-OS algorithm. Using simulation studies, it was found that the proposed algorithm achieves a considerably improved convergence rate over the state-of-the-art SPS-OS algorithm. Bias-variance performance of the algorithm was th evaluated for the preconditioners. Finally, the proposed PPG-OS algorithm was assessment using clinical PET data.