Verify the convoliution theorem using Matlab

When using Matlab to verify “Convolution in time domain is multification in frequency domain”, simply doing fft2 is wrong.

• fft2 in defalut is doing circular convolution. (extend with mirroring). If we want to do linear convolution with fft2, we need to properly pad the signal. We can use fft2(X,M,N) to pad the signal.M=size(img,1) + size(mask,1)
• A few ways to show the results are:

%% Method1:keep fft2, use circular convolution
im_f = fft2(Y);
ker_f = fft2(G,size(Y,1),size(Y,2));
Iff = im_f.*ker_f;
Iffi = ifft2(Iff);
Yp = padarray(Y,[5,5],'circular'); 
Yd = conv2(Yp,G,'same');
% Remove the padded signal
Yd2 = Yd(4:387,4:515);
figure;imshow(abs(Iffi-Yd2)>1e-6)
%% Method2:use linear conv for fft2
% Discrete Fourier transform
M = size(Y, 1) + size(G, 1);
N = size(Y, 2) + size(G, 2);
Ye = ifft2(fft2(Y, M, N) .* fft2(G, M, N));
Ye = Ye(3:end-3, 3:end-3);  % # Adjust dimensions
figure;imshow(abs(Ye-Yb)>1e-6)