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

The direct Fourier method for 3D reconstruction of a 3D volume is based on the relationship between the 3D Fourier transform of the volume and the 2D FT of a parallel-ray projection of the volume. The direct Fourier method has the potential for very fast reconstruction, but a straightforward implementation of the method leads to unsatisfactory results. This paper presents an implementation (3D-FRP) of the direct Fourier method for fully 3D PET data with incomplete oblique projections that gives results as good as (or better than) those of a much slower method (3DRP), and in the same time as a fast but less accurate method (Fourier rebinning followed by slice-by-slice reconstruction). In common with filtered backprojection (3DRP), the Fourier method (3D-FRP) is based on a discretization of an inversion formula (so it is geometrically accurate for large oblique angles), and the method involves reprojection of an initial image. The critical two steps in the method are the estimations of the samples of the 3D transform of the image from the samples of the 2D transforms of the projections on the planes through the origin of Fourier space, and vice versa for reprojection. These steps use a gridding strategy, combined with new approaches for weighting in the transform and image domains. The authors´ experimental results confirm that good image accuracy can be achieved together with a short reconstruction time