An implementation of an efficient direct Fourier transform of polygonal areas and volumes

Abstract: Calculations of the Fourier transform of a constant quantity over an area or volume defined by polygons (connected vertices) are often useful in modeling wave scattering, or in fourier-space filtering of real-space vector-based volumes and area projections. If the system is discretized onto a regular array, Fast Fourier techniques can speed up the resulting calculations but if high spatial resolution is required the initial step of discretization can limit performance; at other times the discretized methods result in unacceptable artifacts in the resulting transform. An alternative approach is to calculate the full Fourier integral transform of a polygonal area as a sum over the vertices, which has previously been derived in the literature using the divergence theorem to reduce the problem from a 3-dimensional to line integrals over the perimeter of the polygon surface elements, and converted to a sum over the straight segments of that contour. We demonstrate a software implementation of this algorithm and show that it can provide accurate approximations of the Fourier transform of real shapes with faster convergence than a block-based (voxel) discretization.