In this lesson we illustrate some of these properties.
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Left: original image f(x,y), Right: spectrum of F(u,v)
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Left: rotated f(x,y), Right: rotated F(u,v)
k1 f(x,y) + k2 g(x,y) <==> k1 F(u,v) + k2 G(u,v)
which is illustrated below:
0.25 +
0.75
=
0.25 +
0.75
=
Intuitively, we can see that the fundamental waves to construct the image are all the same, but they have to be positioned differently.
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Left: f(x+x0,y+0), Center: |F(u,v)|, Right: Phi(u,v) + 2pi(ux0/N + vy0/M)
Translating the original DFT by (u0,v0), we have a similar equation:
This property is used when displaying the Optical Fourier Spectrum. A translation of (N/2, M/2) in F(u,v) is equivalent of multiplying f(x,y) by (-1)**(x+y). See in the proposed exercise list.
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Left: original f(x,y), Right: spectrum of its DFT F(u,v)
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Left: g(x,y) = f(x,y) expanded by 2, Right: G(u,v) = replicated F(u,v)
The same property works for the DFT:
This property can be used for creating a large image from a replication of a small pattern. See the list of proposed exercises.