Two-Dimensional Fourier Transform (2DFT)

Two-dimensional Fourier transform also has four different forms depending on whether the 2D signal (usually spatial signal) $f(x,y)$ is periodic and whether it is discrete.

• Non-periodic, continuous signal $f(x,y)$, continuous, non-periodic spectrum $F(u,v)$

  
  

  $$F(u,v)=\int \int_{-\infty}^{\infty} f(x,y) e^{-j2\pi(ux+vy)} dx dy$$
  
  
  
  
  $$f(x,y)=\int \int_{-\infty}^{\infty} F(u,v) e^{j2\pi(ux+vy)} du dv$$
  
  
  where $u$ and $v$ are spatial frequencies in $x$ and $y$ directions, respectively, and $F(u,v)$ is the 2D spectrum of $f(x,y)$.

• Non-periodic, discrete signal $f[m,n]$, continuous, periodic spectrum $F_{UV}(u,v)$

  
  

  $$F_{UV}(u,v)=\sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} f[m,n]e^{-j2\pi(umx_o+vny_o)}$$
  
  
  
  
  $$f[m,n]=\frac{1}{UV}\int_0^U \int_0^V F(u,v) e^{j2\pi(umx_o+vny_o)} du dv$$
  
  
  where $x_o$ and $y_o$ are the spatial intervals between consecutive signal samples in the $x$ and $y$ directions, respectively, and $U=1/x_o$ and $V=1/y_o$ are sampling rates in the two directions, and they are also the periods of the spectrum $F(u,v)$.

• Periodic, continuous signal $f_{XY}(x,y)$, discrete, non-periodic spectrum $F[k,l]$

  
  

  $$F[k,l]=\frac{1}{X Y}\int_0^{X} \int_0^{Y} f_{XY}(x,y) e^{j2\pi(kxu_o+lyv_o)} dx dy$$
  
  
  
  
  $$f_{XY}(x,y)=\sum_{k=-\infty}^{\infty} \sum_{l=-\infty}^{\infty} F[k,l] e^{-j2\pi(xku_o+ylv_o)}$$
  
  
  where $X$ and $Y$ are periods of the signal in $x$ and $y$ directions, respectively, and $u_o=1/X$ and $v_o=1/Y$ are the intervals between consecutive samples in the spectrum $F[k,l]$.

• Periodic, discrete signal $f[m,n]$, discrete and periodic spectrum $F[k,l]$

  
  

  $$X[k,l]=\frac{1}{\sqrt{MN}}\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}x[m,n] e^{-j2\pi(\frac{mk}{M}+\frac{nl}{N})}$$
  
  
  
  
  $$x[m,n]=\frac{1}{\sqrt{MN}}\sum_{l=0}^{N-1}\sum_{k=0}^{M-1}X[k,l] e^{j2\pi(\frac{mk}{M}+\frac{nl}{N})}$$
  
  
  
  
  $$\;\;\;\;(0 \leq m, k \leq M-1,\;\;\;0 \leq n,l \leq N-1)$$
  
  
  where $M$ and $N$ are the numbers of samples in $x$ and $y$ directions, respectively, and $X[k,l]$ is the 2D discrete spectrum of $x[m,n]$. Both $X[k,l]$ and $x[m,n]$ can be considered as elements in two $M\times N$ matrices ${\bf x}_{M\times N}$ and ${\bf X}_{M\times N}$, respectively.