The use of visual representations in which pixel-size and local neighbor-hood topology are not constant is termed space-variant vision. This is the dominant visual 
architecture in all higher vertebrate visual systems, and iscoming to play an important role in real-time active vision applications in the form of log-polar, foveating 
pyramid, and related approaches to machine vision.

The breaking of translation symmetry that is unavoidably associatedwith space-variant vision presents a major algorithmic complication for image processing. In this paper 
we use a Lie group approach to derive a kernelwhich provides a generalization of the Fourier Transform that provides a quasi-shift invariant 1 template matching 
capability in the distorted (range)coordinates of the space-variant mapping. We work out the special case of the log-polar mapping, which is the principle space-variant 
mapping in use;in this case, we call the associated integral transform the "exponential chirp transform" (ECT). The method is, however, general for other forms of 
mapping, or warp, function.

Examples from the two-dimensional (image processing) log-polar transformation are presented along with the demonstration that the ECT preserves the foveating aspect of 
the space domain mapping, and therefore pro-vides a quasi-shift-invariant realization for the applications of matched filter and phase-only filter. This work provides, 
for the first time, a conceptual ba-sis for combining global spatial frequency methods with space-variant mappings in a way which is consistent with the anatomical fact 
that humanvision, at the cortical level, takes place in log-polar coordinates.