Applying 2D Gabor Wavelet on Image - matlab

Question

EDIT: This is the code I am using that generates an edge detected looking image:
cookiesImage = rgb2gray(imread('Cookies.png'));
width = 45;
height = 45;
KMAX = pi / 2;
f = sqrt(2);
delta = pi / 3;
output = zeros(size(cookiesImage, 1), size(cookiesImage, 2), 8);
for i = 0 : 7
wavelets = GaborWavelet(width, height, KMAX, f, i, 2, delta);
figure(1);
subplot(1, 8, i + 1), imshow(real(wavelets), []);
output(:, :, i + 1) = imfilter(cookiesImage, wavelets, 'symmetric');
end
display = sum(abs(output).^2, 3).^0.5;
display = display./max(display(:));
figure(2); imshow(display);
function GWKernel = GaborWavelet (width, height, KMAX, f, u , v, delta)
delta2 = delta * delta;
kv = KMAX / (f^v);
thetaU = (u * pi) / 8;
kuv = kv * exp (1i * thetaU);
kuv2 = abs(kuv)^2;
GWKernel = zeros (height, width);
for y = -height/ 2 + 1 : height / 2
for x = -width / 2 + 1 : width / 2
GWKernel(y + height / 2, x + width / 2) = (kuv2 / delta2) * exp(-0.5 * kuv2 * (x * x + y * y) / delta2) * (exp(1i * (real(kuv) * y + imag (kuv) * x )) - exp (-0.5 * delta2));
end
end
This is the function that I am using for the Wavelets and this is how I am trying to apply them but all I am getting is an edge detected looking image, rather than one as in this link.

Answer 1

Running your code, I see the following wavelets being generated:
These look a lot like rotated second derivatives. This is only the real (even) component of the Gabor filter kernels. The imaginary (odd) counterparts look like first derivatives.
This is why you feel like your result is like an edge-detected image. It sort of is.
Try increasing the size of the filter (not the footprint widthxheight, but the delta that determines the size of the envelope). This will make it so that you see a larger portion of the sinusoid waves that form the Gabor kernel.
Next, the result image you show is the sum of the square magnitude of the individual Gabor filters. Try displaying the real or imaginary components of one of the filter results, you'll see it looks more like you'd expect:
imshow(real(output(:,:,3)),[])
I'm not familiar with this parametrization of the Gabor kernel, but note that it has a Gaussian envelope. Therefore, the footprint (width, height) of the kernel can be adjusted to the size of this Gaussian (which seems to use delta as the sigma). I typically recommend using a kernel footprint of 2*ceil(3*sigma)+1 for the Gaussian kernel. The same applies here:
width = 2*ceil(3*delta)+1;
height = width;
This will speed up computations, as you see in the image above your kernels have lots of near-zero values in them, it is possible to crop them to a smaller size without affecting the output.
The GaborWavelet function can also be simplified a lot using vectorization:
function GWKernel = GaborWavelet (width, height, KMAX, f, u , v, delta)
delta2 = delta * delta;
kv = KMAX / (f^v);
thetaU = (u * pi) / 8;
kuv = kv * exp (1i * thetaU);
kuv2 = abs(kuv)^2;
x = -width/2 + 1 : width/2;
[x,y] = meshgrid(x,x);
GWKernel = (kuv2 / delta2) * exp(-0.5 * kuv2 * (x .* x + y .* y) / delta2) ...
.* (exp(1i * (real(kuv) * y + imag (kuv) * x )) - exp (-0.5 * delta2));