Space-variant Fourier analysis: the exponential chirp transform

Space-variant (or foveating) vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular space-variant map, the log-polar map, which approximates the primate visual map, and which has been applied in machine vision by a number of investigators during the past two decades. Associated with the log-polar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform. We then show that a log-polar coordinate transform in frequency provides a fast exponential chirp transform. This provides size and rotation, in addition to shift, invariant properties in the transformed space. Finally, we demonstrate the use of the fast exponential chirp algorithm on a database of images in a template matching task, and also demonstrate its uses for spatial filtering