Duality of log-polar image representations in the space and spatial-frequency domains

In this paper, we study the result of applying a lowpass variant filtering using scaling-rotating kernels to both the spatial and spatial-frequency re presentations of a two-dimensional (2-D) signal (image). It is shown that i f we apply this transformation to a Fourier pair, the two resulting signals can also form a Fourier pair when the filters used in each domain maintain a dual relationship, For a large class of "self-dual" filters, a perfect s ymmetry exists, so that the lowpass scaling-rotating variant filtering (SRV F) is the same in both domains, thus commuting with the Fourier transform o perator. The lowpass SRVF of an image is often referred to as a "foveated" image, whereas its Fourier pair (the lowpass SRVF of its spectrum) can be r ealized as a local spectrum estimation around the point of attention. This lowpass SRVF is equivalent to a log-polar warping of the image representati on followed by a lowpass invariant filtering and the corresponding inverse warping. The use of the log-polar warped representation, allows us to exten d the one-dimensional (1-D) scale transform to higher dimensions, in partic ular to images, for which we have defined a scale-rotation invariant repres entation. We also present an efficient implementation using steerable filte rs to compute both the foveated image and the local spectrum.