Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates

摘要

For functions that are best described with spherical coordinates, the three-dimensional Fourier transform cannbe written in spherical coordinates as a combination of spherical Hankel transforms and spherical harmonicnseries. However, to be as useful as its Cartesian counterpart, a spherical version of the Fourier operationalntoolset is required for the standard operations of shift, multiplication, convolution, etc. This paper derives thenspherical version of the standard Fourier operation toolset. In particular, convolution in various forms is discussednin detail as this has important consequences for filtering. It is shown that standard multiplication andnconvolution rules do apply as long as the correct definition of convolution is applied. © 2010 Optical Society ofnAmerica