On the relation between Kaiser-Bessel blob and tube of response based modelling of the system matrix in iterative PET...

We investigate the question of how the blob approach is related to tube of response based modelling of the system matrix. In our model, the tube of response (TOR) is approximated as a cylinder with constant density (TOR-CD) and the cubic voxels are replaced by spheres. Here we investigate a modification of the TOR model that makes it effectively equivalent to the blob model, which models the intersection of lines of response (LORs) with radially variant basis functions ('blobs') replacing the cubic voxels. Implications of the achieved equivalence regarding the necessity of final resampling in blob-based reconstructions are considered. We extended TOR-CD to a variable density tube model (TOR-VD) that yields a weighting function (defining all system matrix elements) which is essentially identical to that of the blob model. The variable density of TOR-VD was modelled by a Gaussian and a Kaiser-Bessel function, respectively. The free parameters of both model functions were determined by fitting the corresponding weighting function to the weighting function of the blob model. TOR-CD and the best-fitting TOR-VD were compared to the blob model with a final resampling step (BLOB-RS) and without resampling (BLOB-NRS) in phantom studies. For three different contrast ratios and two different voxel sizes, resolution noise curves were generated. TOR-VD and BLOB-NRS lead to nearly identical images for all investigated contrast ratios and voxel sizes. Both models showed strong Gibbs artefacts at 4 mm voxel size, while at 2 mm voxel size there were no Gibbs artefacts visible. The spatial resolution was similar to the resolution with TOR-CD in all cases. The resampling step removed most of the Gibbs artefacts and reduced the noise level but also degraded the spatial resolution substantially. We conclude that the blob model can be considered just as a special case of a TOR-based reconstruction. The latter approach provides a more natural description of the detection process and allows for modifications that are not readily representable within the blob framework.