Lie groups, space-variant Fourier analysis and the exponential chirp transform

The use of visual representations in which retinal neurons receptive fields are not constant over the visual field is universal in the visual systems of higher vertebrates, and is coming to play an important role in active vision applications. The breaking of translation symmetry that is unavoidably associated with non-uniform sampling presents a major algorithmic complication for image processing. In this paper we use a Lie group approach to derive a kernel which provides a quasi-shift (i.e. approximate shift) invariant template matching capability, under normal convolution in the distorted (range) coordinates of the non-uniform mapping. We work out the special case of the log-polar mapping, which is of great interest in vision; in this case, we call the associated linear integral transform the "exponential chirp transform" (ECT). The method is, however, general for other forms of mapping, or warp, function.