2D Discrete Fourier Transform and Inverse DFT in matlab - matlab

Question

I am currently implementing 2D DFT and IDFT for images in matlab without using built-in library. I successfully output a spectrum image after DFT but I fail to get back the original image after IDFT.
Here's my code for DFT
input = im2double(img_input);
[M, N] = size(input);
Wm = zeros(M, M);
Wn = zeros(N, N);
for x = 1:M-1
for y = 1:N-1
input(x, y) = input(x, y) * (-1)^(x + y);
end
end
for u = 0:M-1
for x = 0:M-1
Wm(u+1, x+1) = exp(-li * pi * 2 * u * x/ M);
end
end
for v = 0:N-1
for y = 0:N-1
Wn(v+1, y+1) = exp(-li * pi * 2 * v * y / N);
end
end
F = Wm * input * Wn / 200;
output = im2uint8(log(1 + abs(F)));
IDFT:
[M, N] = size(input);
Wm = zeros(M, M);
Wn = zeros(N, N);
for x = 0:M-1
for u = 0:M-1
Wm(x+1, u+1) = exp(2 * pi * 1i * u * x/ M);
end
end
for y = 0:N-1
for v = 0:N-1
Wn(y+1, v+1) = exp(2 * pi * 1i * v * y / N);
end
end
f = Wm * input * Wn;
for x = 1:M-1
for y = 1:N-1
f(x, y) = f(x, y) * (-1)^(x + y);
end
end
output = im2uint8(abs(f));
I multiply the input by (-1)^(x+y) in order to shift the coordinate origin. I have no idea why I can't get back the original image after performing IDFT on the DFTed image.
Original Image
Image after DFT
Image after IDFT

Answer 1

Your code works fine. To get output of the second function to be identical to img_input of the first function, I had to make the following changes:
1st function:
F = Wm * input * Wn; % Don't divide by 200 here.
output = im2uint8(log(1 + abs(F))); % Skip this line altogether
2nd function: Make sure F from the first function is used as input here.
f = Wm * input * Wn / N / M; % Divide by N*M, proper normalization
Note that the normalization is usually put into the IDFT, but you can also put it into the DFT if you prefer. The normalization by 200 is not correct though.