3D-FRP: Direct fourier reconstruction with Fourier reprojection for fully 3-D PET

The direct Fourier method (DIM) for three-dimensional (3-D) reconstruction of a 3-D volume is based on the relationship between the 3-D Fourier transf orm (FT) of the volume and the two-dimensional (2-D) FT of a parallel-ray p rojection of the volume. The direct Fourier method has the potential for ve ry fast reconstruction, but a straightforward implementation of the method leads to unsatisfactory results. This paper presents an implementation of t he direct Fourier method for fully 3-D positron emission tomography (PET) d ata with incomplete oblique projections (3D-FRP) that gives results as good as, or better than, those of a much slower 3-D filtered backprojection met hod (3DRP), and in the same time as a fast but less accurate method using F ourier rebinning (FORE) followed by slice-by-slice reconstruction. In commo n with 3DRP, 3D-FRP is based on a discretization of an inversion formula, s o it is geometrically accurate for large oblique angles, and both methods i nvolve reprojection of an initial image. The critical two steps in the 3D-F RP method are the estimations of the samples of the 3-D transform of the im age from the samples of the 2-D transforms of the projections on the planes through the origin of Fourier space, and vice versa for reprojection. Thes e steps use a gridding strategy, combined with new approaches for weighting in the transform and image domains. Our experimental results confirm that good image accuracy can be achieved together with a short reconstruction ti me.