DFT: Properties

The DFT has a variety of important properties. These properties help us to understand the results of some image processing operations.

In this lesson we illustrate some of these properties.

Rotation

Rotating f(x,y) by the angle theta, also rotates the DFT F(u,v) by the same amount. Intuitively, looking at the DFT as a recipe of how to build the original image by combining its fundamental waves, we can see that we need to rotate all waves to obtain a rotated image.

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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)

Linear Combination

The DFT is a linear operator, the following is verified:

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 =

Translation

Translating the original image by (x0,y0), the spectrum of the DFT of the translated image will be same as the original image, but having its phase changed:

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,y), Center: |F(u,v)|, Right: Phi(u,v)

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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.

Expansion

Expanding the original image by a factor of n, filling the empty new values with zeros, results in the same DFT, but replicated as if we were looking the original DFT with a larger "window", remembering that the DFT is periodic.

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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.

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