Discrete Fourier Transform (DFT)

Forward DFT

$$F(u,v)=\sum _{x=0}^{m-1}\sum _{y=0}^{n-1}f(x,y)\exp \left[-j2\pi \left(\frac{ux}{m}+\frac{vy}{n}\right)\right]$$

Inverse DFT

$$f(x,y)=\frac{1}{mn}\sum _{u=0}^{m-1}\sum _{v=0}^{n-1}F(u,v)\exp \left[j2\pi \left(\frac{ux}{m}+\frac{vy}{n}\right)\right]$$ $$x,u=0,1,2,\dots ,m-1;y,v=0,1,2,\dots ,n-1$$

Properties of DFT

Property	Spatial Domain	Frequency Domain
Periodicity	NIL	$F(u,v)=F(u+m,v)=F(u,v+n)=F(u+m,v+n)$
Translation	$f(x-x_0,y-y_0)$	$F(u,v)e^{-j2\pi (u{x}_0+v{y}_0)}$
Convolution	$f\ast h$	$F\cdot H$
Linearity & Scaling	$a{f}_1+b{f}_2$	$a{F}_1+b{F}_2$

Convolution

In Spatial Domain, we slide and sum. In Frequency Domain, we multiply spectra. Convolution in useful in filtering.