Cache-optimised 3D PET image reconstruction using ordered subsets in combination with highly rotation-symmetric voxel assemblies

The introduction of highly rotation-symmetric voxel assemblies with intrinsic polar symmetries, as realised for example with the PET Reconstruction Software Toolkit (PRESTO), has several advantages in terms of computational optimisation. First, due the applied symmetries a significant compression of the system response matrix can be achieved, thus allowing the pre-calculation and memory-resident storage of accurate matrix weights. Second, the optimised calculation of forward/backward projections with the Single Instruction Multiple Data (SIMD) technique strongly benefits from these symmetries. The symmetry-driven evaluation of projectors provides maximal memory through-put due to optimal cache usage with a minimal amount of cache misses (CM). Among these optimisations, further speed-up is expected when using Ordered-Subset-Expectation-Maximisation (OSEM) instead of Maximum-Likelihood-Expectation-Maximisation (MLEM). However, the evaluation of single subsets within OSEM is in conflict with the symmetry-driven evaluation of projectors in presence of polar symmetries. Usually, the exclusive addressing of disjoint subsets for any iteration would evoke a significant amount of cache misses and concurrent reduction of memory through-put, which would effectively outweigh the benefit of OSEM. Therefore, in this paper we present a method which can strongly reduce the amount of occurring CM by re-sorting the order of voxels for any group of symmetric voxels in memory. The memory access patterns for symmetric voxels are analysed for all subsets to find the optimal re-packing scheme depending on the desired number of subsets. In this way, the minimisation of computational burden can be achieved for any number of subsets, thus giving strong benefit for OSEM also in presence of polar symmetries.