Duality between foveatization and multiscale local spectrum estimation

In this work we demonstrate the relationship existing between two important issues in vision: multi-scale local spectrum analysis, and log-polar foveatization. We show that, when applying a continuous set of self-similar (rotated and scaled) band-pass lters to estimate the local spectrum at a given point of attention of the image, the inverse Fourier transform of this local spectrum is a log-polar foveated version of the original image at that position. Both the local spectrum and its associated foveated image can be obtained through log-polar warping of the spectral/spatial domain followed by a conventional invariant low-pass ltering and the corresponding inverse warping. Furthermore, the low-pass lters in the warped space and frequency domains are mirror versions of each other. Thus, lters with mirror symmetry under the log-polar warping are self-dual, and make the foveatization process commute with the Fourier transform. Nevertheless, in order to implement a fovea that can be easily moved across the image, it is preferable to use a xed bank of steerable lters, instead of applying log-polar warpings with diierent centers. Using low-pass scalable lters we have implemented a real-time moving fovea. We believe that a dual nite spatial/spectral local representation of images could be a very powerful tool for many visual tasks, which could beneet from a dual explicit representation in space and spatial frequency, as well as from the rotation and scale invariance naturally achieved in both domains.