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

TL;DR

This paper presents an efficient algorithm for computing the Fourier transform of polygonal areas and volumes directly from their vertices, offering improved accuracy and convergence over traditional discretization methods.

Contribution

The authors implement and demonstrate a software version of a vertex-based Fourier transform algorithm that outperforms voxel-based discretization in accuracy and convergence.

Findings

Faster convergence than voxel discretization.

Accurate approximation of Fourier transforms of real shapes.

Effective for modeling wave scattering and filtering.

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.