I know this is an old quesiton, but posting here in case someone finds it useful. Here are three different ways of getting the 2D DFT of an image. What is asked for is shown in method 2, by the matrix called Fvec, which can be applied to a vectorized form of the input image.
%2d dft transforms
%gen image
m = 10; n = 20;
x = rand(m,n);
%2d dft, method 1: apply to cols at a time, and then to rows
F1D_m = dftmtx(m);
F1D_n = dftmtx(n);
X1 = F1D_m * x * F1D_n;
%2d dft, method 2: vectorize input, gen matrix by linear transformation ideas
%refer to sec 2.6 of Linear Algebra, Gilbert Strang
x = reshape(x,[],1);
Fvec = zeros(m*n,m*n);
for i=1:m*n
inpvec = zeros(m*n,1);
inpvec(i) = 1;
inpvec = reshape(inpvec,m,n);
%finding out one col of the transformation. since i/p and o/p basis are
%canonical, nothing further is required
opvec = fft2(inpvec);
opvec = reshape(opvec,[],1);
Fvec(:,i) = opvec;
end
X2 = Fvec * x; %input,output are vectors
X2 = reshape(X2,m,n); %convert back to image
x = reshape(x,m,n); %convert back to image
%2d dft, method 3: directly use the 2d fft
X3 = fft2(x);
