# Log Gabor : an oriented multiresolution scheme for images inspired by biology

LogGabor is a collection of tools for representing edges at multiple scales.

Figure 1:

**The log-Gabor transform compared to other multiresolution schemes.***a.*Schematic contours of the log-Gabor filters implented in Fischer (2007) in the Fourier domain with 5 scales and 8 orientations (only the contours at 78% of the filter maximum are drawn).*b.*The real part of the corresponding filters is drawn in the spatial domain. The two first scales are drawn at the bottom magnified by a factor of 4 for a better visualization. The different scales are arranged in lines and the orientations in columns. The low-pass filter is drawn in the upper-left part.*c.*The corresponding imaginary parts of the filters are shown in the same arrangement. Note that the low-pass filter does not have imaginary part. Insets (b) and (c) show the final filters built through all the processes described in Section II.*d.*In the proposed scheme the elongation of log-Gabor wavelets increases with the number of orientations nt . Here the real parts (left column) and imaginary parts (right column) are drawn for the 3, 4, 6, 8, 10, 12 and 16 orientation schemes.*e.*As a comparison orthogonal wavelet filters ’Db4’ are shown. Horizontal, vertical and diagonal wavelets are arranged on columns (low-pass on top).*f.*As a second comparison, steerable pyramid filters [30] are shown. The arrangement over scales and orientations is the same as for the log-Gabor scheme.Figure 2:

*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`*. Using simple down-scaling and up-scaling operators we may approximate well a Laplacian operator. This is represented here by stacking images on a*, that is where the aspect ratio is the golden section $\phi \eqdef \frac{1+\sqrt{5}}{2}$. 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 $\phi^2$ (that is $\phi+1$) as the down-scaling factor so that the resolution of the pyramid images correspond across scales. Note at last that coefficient are very kurtotic: most are near zero, the distribution of coefficients has`Golden Rectangle``long tails`*.*