How to calculate the Spatial frequency in Gabor filter?

Question

This question describe the gabor filter family and its application pretty well. Though, there is nothing described about the wavelength (spatial frequency) of the filter. The creation of gabor wavelets are done in the following for loop:

for v = 0 : 4
    for u = 1 : 8
        GW = GaborWavelet ( R, C, Kmax, f, u, v, Delt2 ); % Create the Gabor wavelets
          figure( 2 );
         subplot( 5, 8, v * 8 + u ),imshow ( real( GW ) ,[]); % Show the real part of Gabor wavelets

         GW_ALL( v*8+u, :) = GW(:);

    end

    figure ( 3 );
     subplot( 1, 5, v + 1 ),imshow ( abs( GW ),[]); % Show the magnitude of Gabor wavelets

end

I know that the second loop variable is the orientation with pi/8 intervals. Though, I don't know how the first loop variable is linked with the spatial frequency (wavelength) in this code and its function [pixels/cycle]. Can anyone help?

Answer 1

I found the answer finally. The GaborWavelet function is defined as follows:

function GW = GaborWavelet (R, C, Kmax, f, u, v, Delt2)
k = ( Kmax / ( f ^ v ) ) * exp( 1i * u * pi / 8 );% Wave Vector
kn2 = ( abs( k ) ) ^ 2;
GW = zeros ( R , C );
for m = -R/2 + 1 : R/2
    for n = -C/2 + 1 : C/2
        GW(m+R/2,n+C/2) = ( kn2 / Delt2 ) * exp( -0.5 * kn2 * ( m ^ 2 + n ^ 2 ) / Delt2) * ( exp( 1i * ( real( k ) * m + imag ( k ) * n ) ) - exp ( -0.5 * Delt2 ) );
    end
end

The Kmax is the maximum frequency, f is the spacing factor and v is the resolution. The spacing factor f is usually considered as sqrt(2). Based on this paper, k= 2*pi*f*exp(i*ϑ) and in the code Kmax=fmax*2*pi which is not described and is the key to find the wavelength of the filter. I also read this implementation and noticed that wavelength can easily be found using f = 1/lambda where lambda is wavelength of sinusoid.

So for example, if Kmax=pi/2 and v=0, so the k=Kmax*exp(1i*u*pi/8) and considering the above mentioned formula, lambda = 2*pi/Kmax = 4 [pixel/cycle].