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Space-Variant Fourier Analysis: The Exponential Chirp Transform
October 1997 (vol. 19 no. 10)
pp. 1080-1089

Abstract—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 (similar to the Mellin-Transform) 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. Given the general lack of algorithms in space-variant image processing, we expect that the fast exponential chirp transform will provide a fundamental tool for applications in this area.

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Index Terms:
Logpolar mapping, rotation scale and shift invariance, attention, space-variant image processing, Fourier analysis, nonuniform sampling, real-time imaging, warped template matching.
Citation:
Giorgio Bonmassar, Eric L. Schwartz, "Space-Variant Fourier Analysis: The Exponential Chirp Transform," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 10, pp. 1080-1089, Oct. 1997, doi:10.1109/34.625108
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