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.
