Fourier analysis and cortical architectures: The exponential chirp transform

Abstract

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† 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 provides a quasi-shift invariant realization for the applications of matched filter and phase-only filter. This work provides, for the first time, a conceptual basis for combining global spatial frequency methods with space-variant mappings in a way which is consistent with the anatomical fact that human vision, at the cortical level, takes place in log-polar coordinates. © 1997 Academic Press Limited.