Fourier Analysis and Cortical Architectures: The Exponential Chirp Transform

The use of visual representations in which pixel-size and local neighborhood topology are not constant is termed space-variant vision. This is the dominant visual architecture in all higher vertebrate visual systems, and is coming 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 associated with space-variant vision presents a major algorithmic complication for image processing. In this paper we use a Lie group approach to derive a kernel which 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 Work supported by ARPA ANNT-ONR N00014-92-C-0119 and ONR MURI N6001495-I-0409 1We use the term quasi-shift invariant to refer to the nonuniform sampling nature of space-variant maps like the log-polar mapping. Thus, for example, if an object is shifted across the aperture of a space-variant system, our methods will produce invariance up to, but not including, the application of a band-pass filter that is position dependent. This will be made clear in the text.