A multi-grid approach to ML reconstruction in PET: A fast alternative to EM-based techniques

Maximum likelihood reconstruction in PET is computationally challenging. Iterative methods including OSEM[8], [14] and the more recent block iterative techniques[1] require many time-consuming projection and back-projection steps in order to obtain a solution. While operational ML reconstruction methods are generally available now, the processing times and convergence characteristics of these methods are not entirely satisfactory and not sufficient to allow ready exploration of reconstructions over a range of resolution bandwidths, as would be needed for data-dependent bandwidth selection [13]. The problems with convergence are a particular issue for low count reconstructions, such as individual time frames of a PET dynamic study, where adequate temporal resolution requires under-smoothing. This work considers a direct Newton-Raphson approach to computation of ML type reconstruction. The novel aspect of the approach is construction of a quadratic image-domain approximation to the ML objective function. This approximation has the potential to dramatically reduce the need for projection and back-projection. The quadratic approximation is solved subject to positivity constraints. A matrix-splitting algorithm is proposed for solution of this problem - c.f. [1], [9]. One, perhaps novel, feature is the use of over-lapping grid blocks. The structure of the quadratic program involved leads to certain numerical efficiencies. A linear analysis finds that the use of multiple but overlapping grids has potential to speed up convergence. The approach has the potential for application as a post-processing scheme for traditional filtered back-brojection reconstructions. Illustrations of this are provided. On the order of 10-15 iterations are required to obtain convergence for raw/unsmoothed ML reconstructions. This is a dramatic improvement in performance relative to operational OSEM approaches.