A novel extension of the parallel‐beam projection‐slice theorem to divergent fan‐beam and cone‐beam projections

Abstract

The general goal of this paper is to extend the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections without rebinning the divergent fan-beam and cone-beam projections into parallel-beam projections directly. The basic idea is to establish a novel link between the local Fourier transform of the projection data and the Fourier transform of the image object. Analogous to the two- and three-dimensional parallel-beam cases, the measured projection data are backprojected along the projection direction and then a local Fourier transform is taken for the backprojected data array. However, due to the loss of the shift invariance of the image object in a single view of the divergent-beam projections, the measured projection data is weighted by a distance dependent weight w(r) before the local Fourier transform is performed. The variable r in the weighting function w(r) is the distance from the backprojected point to the x-ray source position. It is shown that a special choice of the weighting function, w(r)=1/r, will facilitate the calculations and a simple relation can be established between the Fourier transform of the image function and the local Fourier transform of the 1/r-weighted backprojection data array. Unlike the parallel-beam cases, a one-to-one correspondence does not exist for a local Fourier transform of the backprojected data array and a single line in the two-dimensional (2D) case or a single slice in the 3D case of the Fourier transform of the image function. However, the Fourier space of the image object can be built up after the local Fourier transforms of the 1/r-weighted backprojection data arrays are shifted and then summed in a laboratory frame. Thus the established relations Eq. (27) and Eq. (29) between the Fourier space of the image object and the Fourier transforms of the backprojected data arrays can be viewed as a generalized projection-slice theorem for divergent fan-beam and cone-beam projections. Once the Fourier space of the image function is built up, an inverse Fourier transform could be performed to reconstruct tomographic images from the divergent beam projections. Due to the linearity of the Fourier transform, an image reconstruction step can be performed either when the complete Fourier space is available or in parallel with the building of the Fourier space. Numerical simulations are performed to verify the generalized projection-slice theorem by using a disc phantom in the fan-beam case.