An ordered-subsets proximal preconditioned gradient algorithm for edge-preserving PET image reconstruction

Purpose: In iterative positron emission tomography (PET) image reconstruction, the statistical variability of the PET data precorrected for random coincidences or acquired in sufficiently high count rates can be properly approximated by a Gaussian distribution, which can lead to a penalized weighted least-squares (PWLS) cost function. In this study, the authors propose a proximal preconditioned gradient algorithm accelerated with ordered subsets (PPG-OS) for the optimization of the PWLS cost function and develop a framework to incorporate boundary side information into edge-preserving total variation (TV) and Huber regularizations. Methods: The PPG-OS algorithm is proposed to address two issues encountered in the optimization of PWLS function with edge-preserving regularizers. First, the second derivative of this function (Hessian matrix) is shift-variant and ill-conditioned due to the weighting matrix (which includes emission data, attenuation, and normalization correction factors) and the regularizer. As a result, the paraboloidal surrogate functions (used in the optimization transfer techniques) end up with high curvatures and gradient-based algorithms take smaller step-sizes toward the solution, leading to a slow convergence. In addition, preconditioners used to improve the condition number of the problem, and thus to speed up the convergence, would poorly act on the resulting ill-conditioned Hessian matrix. Second, the PWLS function with a nondifferentiable penalty such as TV is not amenable to optimization using gradient-based algorithms. To deal with these issues and also to enhance edge-preservation of the TV and Huber regularizers by incorporating adaptively or anatomically derived boundary side information, the authors followed a proximal splitting method. Thereby, the optimization of the PWLS function is split into a gradient descent step (upgraded by preconditioning, step size optimization, and ordered subsets) and a proximal mapping associated with boundary weighted TV and Huber regularizers. The proximal mapping is then iteratively solved by dual formulation of the regularizers. Results: The convergence performance of the proposed algorithm was studied with three different diagonal preconditioners and compared with the state-of-the-art separable paraboloidal surrogates accelerated with ordered-subsets (SPS-OS) algorithm. In simulation studies using a realistic numerical phantom, it was shown that the proposed algorithm depicts a considerably improved convergence rate over the SPS-OS algorithm. Furthermore, the results of bias-variance and signal-to-noise evaluations showed that the proposed algorithm with anatomical edge information depicts an improved performance over conventional regularization. Finally, the proposed PPG-OS algorithm is used for image reconstruction of a clinical study with adaptively derived boundary edge information, demonstrating the potential of the algorithm for fast and edge-preserving PET image reconstruction. Conclusions: The proposed PPG-OS algorithm shows an improved convergence rate with the ability of incorporating additional boundary information in regularized PET image reconstruction.