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Giorgio Bonmassar, Eric L. Schwartz, "SpaceVariant Fourier Analysis: The Exponential Chirp Transform," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no. 10, pp. 10801089, October, 1997.  
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@article{ 10.1109/34.625108, author = {Giorgio Bonmassar and Eric L. Schwartz}, title = {SpaceVariant Fourier Analysis: The Exponential Chirp Transform}, journal ={IEEE Transactions on Pattern Analysis and Machine Intelligence}, volume = {19}, number = {10}, issn = {01628828}, year = {1997}, pages = {10801089}, doi = {http://doi.ieeecomputersociety.org/10.1109/34.625108}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
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TY  JOUR JO  IEEE Transactions on Pattern Analysis and Machine Intelligence TI  SpaceVariant Fourier Analysis: The Exponential Chirp Transform IS  10 SN  01628828 SP1080 EP1089 EPD  10801089 A1  Giorgio Bonmassar, A1  Eric L. Schwartz, PY  1997 KW  Logpolar mapping KW  rotation scale and shift invariance KW  attention KW  spacevariant image processing KW  Fourier analysis KW  nonuniform sampling KW  realtime imaging KW  warped template matching. VL  19 JA  IEEE Transactions on Pattern Analysis and Machine Intelligence ER   
Abstract—Spacevariant, or foveating, vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular spacevariant map, the logpolar 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 logpolar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for spacevariant image formats, while preserving the major aspects of the shiftinvariant properties of the usual Fourier transform. We then show that a logpolar coordinate transform in frequency (similar to the MellinTransform) 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 spacevariant image processing, we expect that the fast exponential chirp transform will provide a fundamental tool for applications in this area.
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