Interpolation-based reconstruction methods for tomographic imaging in 3D positron emission tomography

Positron Emission Tomography (PET) is considered a key diagnostic tool in neuroscience, by means of which valuable insight into the metabolism function in vivo may be gained. Due to the underlying physical nature of PET, 3D imaging techniques in terms of a 3D measuring mode are intrinsically demanded to assure satisfying resolutions of the reconstructed images. However, incorporating additional cross-plane measurements, which are specific for the 3D measuring mode, usually imposes an excessive amount of projection data and significantly complicates the reconstruction procedure. For this reason, interpolation-based reconstruction methods deserve a thorough investigation, whose crucial parts are the interpolating processes in the 3D frequency domain. The benefit of such approaches is apparently short reconstruction duration, which can, however, only be achieved at the expense of accepting the inaccuracies associated with the interpolating process. In the present paper, two distinct approaches to the realization of the interpolating procedure are proposed and analyzed. The first one refers to a direct approach based on linear averaging (inverse distance weighting), and the second one refers to an indirect approach based on two-dimensional convolution (gridding method). In particular, attention is paid to two aspects of the gridding method. The first aspect is the choice of the two-dimensional convolution function applied, and the second one is the correct discretization of the underlying continuous convolution. In this respect, the geometrical structure named the Voronoi diagram and its computational construction are considered. At the end, results of performed simulation studies are presented and discussed