How to get the differential of the objective function? Why it's Df(S)(M)=2⟨S−I,M⟩+β(2⟨∂xS−h,∂xM⟩+2⟨∂yS−v,∂yM⟩)?

@Alex I learned DFT from <Digital Image Processing> by Gonzalez, seems it omits some important theories. I will refer to some other book. Anyway, thank you:-)

@Alex Thank you! F is the DFT Matrix right? Can you give me some books or something I can find some proof of it?

Thanks! I think that's what I'm looking for! But what does "the fact that the DFT diagonalizes the gradient" mean? I've read it somewhere but don't get it. Anywhere I can find some detail?

why M=1+dx+dy? M-P inverse is only used in ||Mx-v|| (||x+dx x + dy x -v||) right? So how can M be 1+dx+dy in this example? And how to use FFT to speed up anything in M-P inverse? The paper says the trivial way to solve this equation is to use gradient decent, which is very very slow, so they come with a way to speed it up.