Abstract: For parallel beam geometry the Fourier reconstruction works via the
Fourier slice theorem (or central slice theorem, projection slice theorem). For
fan beam situation, Fourier slice can be extended to a generalized Fourier
slice theorem (GFST) for fan-beam image reconstruction. We have briefly
introduced this method in a conference. This paper reintroduces the GFST method
for fan beam geometry in details. The GFST method can be described as
following: the Fourier plane is filled by adding up the contributions from all
fanbeam projections individually; thereby the values in the Fourier plane are
directly calculated for Cartesian coordinates such avoiding the interpolation
from polar to Cartesian coordinates in the Fourier domain; inverse fast Fourier
transform is applied to the image in Fourier plane and leads to a reconstructed
image in spacial domain. The reconstructed image is compared between the result
of the GFST method and the result from the filtered backprojection (FBP)
method. The major differences of the GFST and the FBP methods are: (1) The
interpolation process are at different data sets. The interpolation of the GFST
method is at projection data. The interpolation of the FBP method is at
filtered projection data. (2) The filtering process are done in different
places. The filtering process of the GFST is at Fourier domain. The filtering
process of the FBP method is the ramp filter which is done at projections. The
resolution of ramp filter is variable with different location but the filter in
the Fourier domain lead to resolution invariable with location. One advantage
of the GFST method over the FBP method is in short scan situation, an exact
solution can be obtained with the GFST method, but it can not be obtained with
the FBP method. The calculation of both the GFST and the FBP methods are at
O(N^3), where N is the number of pixel in one dimension.
