Analytical calculation of volumes-of-intersection for iterative, fully 3-D PET reconstruction.

Use of iterative algorithms to reconstruct three-dimensional (3-D) positron emission tomography (PET) data requires the computation of the system probability matrix. The pure geometrical contribution can easily be approximated by the length-of-intersection (LOI) between lines-of-response (LOR) and individual voxels. However, more accurate geometrical projectors are desirable. Therefore, we have developed a fast method for the analytical calculation of the 3-D shape and volume of volumes-of-intersection (VOI). This method provides an alternative robust projector with a uniformly continuous sampling of the image space. The enhanced calculation effort is facilitated by using several speedup techniques. Exploiting intrinsic symmetry relations and the sparseness of the system matrix allows to create an efficiently compressed matrix which can be precomputed and completely stored in memory. In addition, a new voxel addressing scheme has been implemented. This scheme avoids time-consuming symmetry transformations of voxel addresses by using an octant-wise symmetrically ordered field of voxels. The above methods have been applied for a fully 3-D, iterative reconstruction of 3-D sinograms recorded with a Siemens/CTI ECAT HR+ PET scanner. A comparison of the performance of the reconstruction using LOI weighting and VOI weighting is presented.