A Krasnoselskii-Mann Algorithm With an Improved EM Preconditioner for PET Image Reconstruction.

This paper presents a preconditioned Krasnoselskii-Mann (KM) algorithm with an improved EM preconditioner (IEM-PKMA) for higher-order total variation (HOTV) regularized positron emission tomography (PET) image reconstruction. The PET reconstruction problem can be formulated as a three-term convex optimization model consisting of the Kullback-Leibler (KL) fidelity term, a nonsmooth penalty term, and a nonnegative constraint term which is also nonsmooth. We develop an efficient KM algorithm for solving this optimization problem based on a fixed-point characterization of its solution, with a preconditioner and a momentum technique for accelerating convergence. By combining the EM precondtioner, a thresholding, and a good inexpensive estimate of the solution, we propose an improved EM preconditioner that can not only accelerate convergence but also avoid the reconstructed image being "stuck at zero." Numerical results in this paper show that the proposed IEM-PKMA outperforms existing state-of-the-art algorithms including, the optimization transfer descent algorithm and the preconditioned L-BFGS-B algorithm for the differentiable smoothed anisotropic total variation regularized model, the preconditioned alternating projection algorithm, and the alternating direction method of multipliers for the nondifferentiable HOTV regularized model. Encouraging initial experiments using clinical data are presented.