0

I really need your help here:

Think of $f(x,y)$ as an image where the dimensions are: $NxM$.

I can't prove this property of 2D transform fourier:

if the transform of $f(x,y)$ is $F(u,v)$ so the transform of $f(x-x_0,y-y_0)$ is $F(u,v)e^{-2{\pi}i(ux_0/{N}+ vy_0/M)}$

Any help will be appreciated. thanks.

Alberto Debernardi
  • 3,046
daniel
  • 101
  • simply apply the change of variable $(x',y') =(x-x_0,y-y_0)$ in the integral – reuns Jan 22 '16 at 09:41
  • I tried but i'm stuck.. i know this should be simple... can you please write this? I will appreciate it. – daniel Jan 22 '16 at 10:00
  • do it in dimension $1$ first – reuns Jan 22 '16 at 10:02
  • I dont know what to do with the sigma..I can't just split it.. – daniel Jan 22 '16 at 12:55
  • write first your formula for the Fourier transform. and proofs of the translation property of the FT can be found everywhere on the internet. – reuns Jan 22 '16 at 12:56
  • the formula is: $F(u,v)=\frac{1}{NM}\Sigma^{N-1}{x=0}\Sigma^{M-1}{y=0}f(x,y)e^{-2\pi{i}(\frac{ux}{N}+\frac{vy}{M})} $

    after changing $x$ by $x-x_0$ and $y$ by $y-y_0$ we will get: $F'(u,v)=\frac{1}{NM}\Sigma^{N-1}{x=0}\Sigma^{M-1}{y=0}f(x-x_0,y-y_0)e^{-2\pi{i}(\frac{u(x-x_0)}{N}+\frac{v(y-y_0)}{M})}$

    and after few trival steps: $F'(u,v)=\frac{1}{NM}\Sigma^{N-1}{x=0}\Sigma^{M-1}{y=0}f(x-x_0,y-y_0)e^{-2\pi{i}(\frac{ux}{N}+\frac{vy}{M})}e^{2\pi{i}(\frac{ux_0}{N}+\frac{vy_0}{M})}$

    what can I do now? I searched this proof for a week and I was failed..

     – daniel Jan 22 '16 at 13:29
  • Study a Discrete Fourier transform course !!! in dimension $1$ :

    $$DFTf(n) = \sum_{n=0}^{N-1} f(n) e^{-2 i \pi n k/N}$$

    with $f(n+N) = f(n)$ and $a \in \mathbb{Z}$ :

    $$DFTf(n-a) = \sum_{n=0}^{N-1} f(n-a) e^{-2 i \pi n k/N} = \sum_{m=-a}^{N-1-a} f(m) e^{-2 i \pi (m+a) k/N} = \sum_{m=0}^{N-1} f(m) e^{-2 i \pi (m+a) k/N} = e^{-2 i \pi a k/N} DFTf(n)$$

     – reuns Jan 22 '16 at 14:16
  • and you are talking of the Discrete Fourier Transform (DFT) not the general Fourier transform. if you want to prove the formula for non integer shifts, you have to study the general Fourier transform or the Fourier series, the sampling and convolution theorems, etc. – reuns Jan 22 '16 at 14:16

0 Answers0