Two-Dimensional Fourier Transforms in Polar Coordinates

Abstract For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination of Hankel transforms and Fourier series—even if the function does not have circular symmetry. However, to be as useful as its Cartesian counterpart, a polar version of the Fourier operational toolset is required for the standard operations of shift, multiplication, convolution, and so on. This chapter derives the requisite polar version of the standard Fourier operations. In particular, convolution—two-dimensional, circular, and radial one-dimensional—is discussed in detail. It is shown that standard multiplication/convolution rules do apply as long as the correct definition of convolution is applied.