Extract low-frequency coefficients of Fourier transformation

I'm new to Matlab and to FFT. I have applied the Fourier transformation on an image like so:

I = imread('img.jpg');
img = fftshift(I);
F = fft2(img);
magnitude = mat2gray(100*log(1+abs(fftshift(F))));    % Magnitude spectrum
phase = mat2gray( (angle(F)) );                       % Phase spectrum

Using the energy compaction property of the Discrete Fourier Transform how can I extract a 21x21 matrix of the low-frequency value coefficients of the Fourier transformation?

Thanks in advance!

Answer 1

[Wayne King]

If you shift the 2-D DFT, then the low frequency components will be in the center of the image: center along the row and column dimensions of the matrix

   X = ones(128,128);
   Xdft = fftshift(fft2(X));
   surf(abs(Xdft)); 
   view(-16,10)

Note that the DC shift is evident in the middle of the image.

The value of Xdft(65,65) is 2^14 as expected. So to extra the low-frequency components, just extract a submatrix around the center of your original matrix.

If you have a simple 2-D complex exponential

x = 0:159;
y = 0:159;
[X,Y] = meshgrid(x,y);
Z = exp(1i*pi/4*(X+Y));
Z = real(Z);
imagesc(Z);

Then note that with a matrix of 160x160, we would expect pi/4 to be shifted from the center (80,80) to (101,101) and (61,61)

   Zdft = fftshift(fft2(Z));
    imagesc(abs(Zdft))
    Zdft(101,101)
    Zdft(61,61)

Note they are complex conjugates of each other as expected.

[Andrei]

Thanks a million for the answer! It works. I'm very much ashamed for asking, but how can I extract the central submatrix of the resulted matrix?

[Andrei]

yeah, I've noticed. But how does that help me with the central matrix? (sorry I do not understand, or is this not related to my latest comment?)

[Andrei]

I've managed to figure it out, thanks for all your help!!!