The Log-Gabor function proposed by Field [1987] is an alternative to the Gabor function to efficiently represent edges in natural images. Log-Gabor filters can be constructed with arbitrary bandwidth and the bandwidth can be optimised to produce a filter with minimal spatial extent. We develop here a log-Gabor representation, which is well suited to represent a wide range of natural images.
This framework was presented in the following paper by Sylvain Fischer, Filip Šroubek, Laurent U Perrinet, Rafael Redondo and Gabriel Cristóbal (2007). Examples and documentation is available @ https://pythonhosted.org/LogGabor/ and this package provides with a python implementation.
A log-Gabor pyramid is an oriented multiresolution scheme for the efficient coding of natural images.
To represent the edges of the image at different levels and orientations, we use a multi-scale approach constructing a set of filters of different scales and according to oriented log-Gabor filters. This is represented here by stacking images on a Golden Rectangle Perrinet (2008), that is where the aspect ratio is the golden section ϕ=1+5√2. The level represents coefficients' amplitude, hue corresponds to orientation. We present here the base image on the left and the successive levels of the pyramid in a clockwise fashion (for clarity, we stopped at level 8). Note that here we also use ϕ^2 (that is ϕ+1) as the down-scaling factor so that the pixelwise resolution of the pyramid images correspond across scales.
The Golden Laplacian Pyramid. To represent the edges of the image at different levels, we may use a simple recursive approach constructing progressively a set of images of decreasing sizes, from a base to the summit of a pyramid (see https://laurentperrinet.github.io/publication/perrinet-15-bicv for more details).