Abstract: The cone-beam reconstruction theory has been proposed by Kirillov in
1961, Tuy in 1983, Feldkamp in 1984, Smith in 1985, Pierre Grangeat in 1990.
The Fourier slice theorem is proposed by Bracewell 1956, which leads to the
Fourier image reconstruction method for parallel-beam geometry. The Fourier
slice theorem is extended to fan-beam geometry by Zhao in 1993 and 1995. By
combining the above mentioned cone-beam image reconstruction theory and the
above mentioned Fourier slice theory of fan-beam geometry, the Fourier slice
theorem in cone-beam geometry is proposed by Zhao 1995 in short conference
publication. This article offers the details of the derivation and
implementation of this Fourier slice theorem for cone-beam geometry. Especially
the problem of the reconstruction from Fourier domain has been overcome, which
is that the value of in the origin of Fourier space is 0/0. The 0/0 type of
limit is proper handled. As examples, the implementation results for the single
circle and two perpendicular circle source orbits are shown. In the cone-beam
reconstruction if a interpolation process is considered, the number of the
calculations for the generalized Fourier slice theorem algorithm is
O(N^4), which is close to the filtered back-projection
method, here N is the image size of 1-dimension. However the interpolation
process can be avoid, in that case the number of the calculations is
O(N^5).
