Duality between foveatization and multiscale local spectrum estimation
Abstract
In this work we demonstrate the relationship existing between two important issues in vision: multiscale local spectrum analysis, and logpolar foveatization. We show that, when applying a continuous set of selfsimilar (rotated and scaled) bandpass filters 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 logpolar warping of the spectral/spatial domain followed by a conventional invariant lowpass filtering and the corresponding inverse warping. Furthermore, the lowpass filters in the warped space and frequency domains are mirror versions of each other. Thus, filters with mirror symmetry under the log polar warping are selfdual, 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 fixed bank of steerable filters, instead of applying logpolar warpings with different centers. Using lowpass scalable filters we have implemented a realtime moving fovea. We believe that a dual finite spatial/spectral local representation of images could be a very powerful tool for many visual tasks, which could benefit from a dual explicit representation in space and spatial frequency, as well as from the rotation and scale invariance naturally achieved in both domains.
 Publication:

Human Vision and Electronic Imaging III
 Pub Date:
 July 1998
 Bibcode:
 1998SPIE.3299..306N