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# DFT: Properties

[Laboratory]

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.

a)original image f(x,y); b)spectrum of F(u,v)
a) | b)

a)rotated f(x,y); b)rotated F(u,v)
a) | b)

### 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:

a)Image A; b)Image B; c)0.25 * A + 0.75 * B
a) | b) | c)

a)spectrum A; b)spectrum B; c)0.25 * A + 0.75 * B
a) | b) | c)

### 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.

a)f(x,y); b)|F(u,v)|; c)Phi(u,v)
a) | b) | c)

a)f(x+x0,y+0); b)|F(u,v)|; c)Phi(u,v) + 2pi(ux0/N + vy0/M)
a) | b) | c)

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.

a)original f(x,y); b)spectrum of its DFT F(u,v)
a) | b)

a)g(x,y) = f(x,y) expanded by 2; b)G(u,v) = replicated F(u,v)
a) | b)

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.

- Lab: DFT: Properties