@Article{bonmassar1997:adaptive,
  author =       {Bonmassar, G. and Schwartz, E. L.},
  title =        {Fourier analysis and cortical architectures: the
                  exponential chirp transform},
  journal =      {Real-Time Imaging},
  year =         1997,
  pages =        {229--237},
  month =        {June},
  datestr =      199706,
  INSPEC =       5718766,
  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. The authors 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. They work out the special case of the
                  log-polar mapping, which is the principle
                  space-variant mapping in use; in this case, they
                  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},
  keywords =     {active vision; Fourier analysis; Fourier transforms;
                  Lie groups; real-time systems}
}