Fourier Properties

 

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Jan 24, 2001

Discrete Fourier Transforms

 

The discrete Fourier transform pair for a sampled space array f(x, y) of dimensions M x N to a frequency array F(w, z):

                   

 

Properties of two-dimensional Fourier transform (assume M=N):

Commutivity:
            [af(x,y) + bg(x,y)] ó aF(w,z) + bG(w,z)
Translation:
            f(x, y) exp[j2π(wox + zoy)/N] F(w - wo, z - zo)
            f(x - xo, y - yo) F(w, z) exp[-j2π(wxo + zyo)/N]
  Periodicity:
            F(w,z) = F(w+N, z) = F(w, z+N) = F(w+N, z+N)
Rotation:
            Let x = r cos θ, y = r sin θ, w = ρ cos φ, z = ρ sin φ
            Then     f(r, θ + θo) ó F(ρ, φ + θo)
Scaling:
            a f(x, y) ó a F(w, z)
            f(ax, by) ó (1/|ab|)F(w/a, z/b)

The Fourier transform of a constant valued image  is a delta function located at (0,0) and multiplied by the constant value.

f(x,y) ó Aδ(m,n)

The comb function preserves its form under a Fourier transform:

       

Convolution


A basic reason for the importance of the Fourier transform is that it allows the mathematical complexity of a convolution integral to be replaced by the product of Fourier transforms. 

 

Do Homework Assignment 2.

 

Last modified on January 29, 2001