How to visualize and understand Gabor filter using MATLAB

Question

I am reading a paper, in which the author defines gabor kernels as :

$$ \psi_{\mu,\nu}(x,y)=\frac{\lVert \mathbf{k}_{\mu,\nu} \rVert}{\sigma^{2}}\exp \left(-\frac{\lVert \mathbf{k}_{\mu,\nu}\rVert^{2}\lVert \mathbf{z}\rVert^{2}}{2\sigma^{2}}\right)\times\left[\exp\left(i\mathbf{k}_{\mu,\nu}^{T}\mathbf{z}\right)-\exp\left(-\frac{\sigma^{2}}{2}\right)\right] $$ where $\mu$ and $\nu$ define the orientation and scale of tghe Gabor kernels, respectively, $\mathbf{z}=(x,y)^{T}$, and the wave vector is defined as

$$ \mathbf{k}_{\mu,\nu} = \left(k_{\nu}\cos{\phi_{\mu}},k_{\nu}\sin{\phi_{\mu}}\right)^{T} \quad\text{with}\quad k_{\nu} = \frac{k_{\rm max}}{f^{\nu}}, k_{\rm max}=\frac{\pi}2,f=\sqrt{2},\quad\text{and}\quad\phi_{\mu}=\frac{\pi\mu}{8}$$

Now, I want to plot the graph of this function using matlab to visually see and understand the Gabor filter. I have very basic knowledge in using matlab.

Specifically, I am interested in viewing these Gabor Kernels at single scale $(\nu=0)$ and four orientations $(\mu\in\{0,2,4,6\})$ with $\sigma=1$

Right now I am trying to understand Gabor filters. Any help will be greatly appreciated.

Thanks in advance.

Answer 1

The following is my implementation, of the above problem :

In my_gabor_filter.m, I have the following code :

function psi = my_gabor_filter(x,y,mu,nu,sigma) 
    phi = pi*mu/8;
    f = sqrt(2);
    k_max = pi/2;
    k_nu = k_max/(f^nu);

    % This is the wave vector
    k_vec = [k_nu*cos(phi),k_nu*sin(phi)]';
    z_vec = [x,y]';

    k_vec_norm = norm(k_vec);
    z_vec_norm = norm(z_vec);

    exp1 = (k_vec_norm/(sigma^2));
    exp2 = (exp(-((k_vec_norm^2)*(z_vec_norm^2)/(2*sigma^2))));
    exp3 = (exp(complex(0,k_vec'*z_vec))-exp(-(sigma^2/2)));

    psi = exp1*exp2*exp3;
end

and, in plot_gabor_filter.m I did the following :

function plot_gabor_filter()
    mu = 0;
    nu = 0;
    sigma = 1;

    [X,Y] = meshgrid(-2:0.1:2,-2:0.1:2);

    Z = zeros(41,41);

    for i = 1:1:41
        for j = 1:1:41
            Z(i,j) = imag(my_gabor_filter(X(i,j),Y(i,j),mu,nu,sigma));
        end
    end

    surf(X,Y,Z,'EdgeColor','black')

    xlabel('x axis');
    ylabel('y axis');
    zlabel('g(x,y)');
end

and when I run plot_gabor_filter, I get the following output, which helped me in visualizing the gabor function :

Answer 2

I checked out the code. I think in the exp3 line, while using the dot product there is a problem. The correct format for dot product of a and b is: a.*b - you have written it as a*b.

Also while taking dot product you have arranged it as 2x1 with a 1x2 vector. Try it in the other orders, i.e.: a 1x2 with a 2x1 or a 2x1 with a 2x1 or a 1x2 with a 1x2 vector product. I'm not very sure about which order is correct.

Make sure that your title reflects the content of the question to get better visibility in these websites. Your title is about concepts but the question is about correcting the MATLAB code.