If I have two 2D signals, and one is the shift of another. I can propose such schema for define offset via continious Fourier Transform:

$$f_2(x,y)=f_1(x-x_0,y-y_0)$$ Then $$Ff_2(s_1,s_2)=e^{-2\pi j(s_1x_0+s_2y_0)}Ff_1(s_1,s_2)$$ => $$\frac {Ff_2(s_1,s_2)}{Ff_1(s_1,s_2)}=e^{-2\pi j(s_1x_0+s_2y_0)}$$ => $$F^{-1}(\frac {Ff_2(s_1,s_2)}{Ff_1(s_1,s_2)})=F^{-1}(e^{-2\pi j(s_1x_0+s_2y_0)})=\delta(x-x_0,y-y_0)$$

But, in wiki https://en.wikipedia.org/wiki/Phase_correlation and also in many papers (e.g. 3 March,2009: An Adaptable-Multilayer Fractional Fourier Transform Approach for Image Registration) another methtod is mentioned "cross-power spectrum".

Q1: Can you describe how this formulara have been derived? Or provide a link to paper or some lecture where this formulaa have been derived? $$(\frac {Ff_1(s_1,s_2)\overline {Ff_2(s_1,s_2)}}{|Ff_1(s_1,s_2)\overline {Ff_2(s_1,s_2)}|})=e^{-2\pi j(s_1x_0+s_2y_0)}$$

Q2: What problems are exist for rather simple method which I described?



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