Comparison of ordered subset implementations for EM, preconditioned steepest ascent (PSA) and conjugate gradient (PCG) optimization tasks in PET image reconstruction

Objectives To investigate quantitative accuracy of images obtained via ordered subset (OS) implementations for maximum likelihood optimization in PET image reconstruction

Methods PET image reconstruction is commonly performed by maximization of the Poisson log-likelihood (LL) function; and may be achieved via the EM algorithm or numerical optimization methods. The EM algorithm can be formulated as fixed step-size special case of preconditioned steepest ascent (PSA), while PSA and preconditioned conjugate gradient (PCG) algorithms perform step-size optimization for each new direction. In the present work, the concept of ordered subsets (OS), commonly applied to achieve OS-EM, was also applied to result in OS-PSA and OS-PCG. In the context of simulated 3D brain imaging tasks, these algorithms were compared in terms of LL convergence and their noise vs. bias performance.

Results Without subsets, PSA and PCG algorithms can converge faster than the EM algorithm (though producing overlapping noise vs. bias trade-off curves). However, the OS-PCG algorithm is seen to converge significantly more poorly relative to both OS-EM and OS-PSA algorithms (asymptotic normalized LL error (NLLE) larger by factors of 1.76 and 2.48 respectively). This is attributed to the fact that while the PCG technique may have theoretical advantages compared to moving in the direction of steepest ascent, this effect disappears for OS data due to inherent inconsistencies between the data subsets. Furthermore, OS-EM is seen to outperform OS-PSA in terms of NLLE as well as noise vs. bias trade-off performance (10-15% reduced noise; matched bias).

Conclusions Usage of the PCG algorithm in the OS context is not recommended, while OS-PSA and especially OS-EM algorithms pose more favorable alternatives. In applications where the EM solution does not exist (e.g. non-linear 4D parametric imaging), usage of OS-PSA optimization is instead recommended