Summary

I used Image:Dandelion_clock.jpg to create this image. I wanted to show clearly the different behavior between the DFT and the DCT in the frequency domain.

The pictures are made of other figures. The first one on the top is just the original image: I used its gray-scale version. On the second line there is the DFT: its magnitude on the left, its histogram on the right. On the third line there is the DCT, with both magnitude and histogram.

The spectrum of the DFT has the lowest frequencies on the center of the image, while the DCT has the lowest frequencies on the top-left of the picture. It is clear how the DCT concentrates most of the energy into the lowest frequencies.

I created the single images with the following Matlab code:

% read the image
RGB = imread('Dandelion_clock.jpg');
% convert pixels to the [0 1] range
RGB = im2double(RGB);
% convert to grayscale
I = rgb2gray(RGB);

% preprocessing of I to show the spectrum poperly
[X Y] = size(I);
I2 = zeros(X,Y);
for i=1:X
    for j=1:Y
    I2(i,j)=I(i,j)*(-1)^(i+j); 
    end
end

% evaluate magnitude of the DFT
F = abs(fft2(I2));
% use log scale
F = log(1 + F);
F = log(1 + F);
% normalize
F = F/max(F(:));

% evaluate magnitude of the DCT
C = abs(dct2(I));
% use log scale
C = log(1 + C);
C = log(1 + C);
% normalize
C = C/max(C(:));

% show all the results
imshow(F), colorbar, colormap(jet);
figure, imhist(F);
figure, imshow(C), colorbar, colormap(jet);
figure, imhist(C);

First it imports the RGB image and converts it to gray-scale. Then the picture is changed to be Fourier-transformed properly. If (i,j) is the index of any pixel, multiplying all the pixels by (-1)^i+j shifts the spectrum so that the DC frequency will be in the center of the image, showing its symmetry. I didn't make anything similar with the DCT because it has no symmetry. Both pictures had a huge dynamic, so I calculated the logarithm of both, twice, in order to be able to show the transforms properly. Once all the pictures were shown on the screen, I just selected File -> Save as on Matlab to save all the pictures. I put them all together using Gimp.

Licensing

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• to remix – to adapt the work

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• attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
• share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

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