Spiral Fourier transformation in optics and image processing

The “spiral Fourier transformation” (sFT) is an addition to the family of Fourier transformations. Its basis functions are complex exponentials with phase contour lines that are, in general, spirals in Cartesian coordinates and thus straight lines in polar coordinates. This makes the sFT useful for the analysis and processing of 2D signals with circular, elliptical, and spiral geometry. For those cases, the sFT yields a modal spectrum that consists of significantly fewer modes, so that computing times and storage requirements are reduced as compared with the conventional Fourier transformation in Cartesian and polar coordinates. We discuss the basic mathematical properties of the sFT, give several examples, and describe its potential use in optics and in digital image processing.