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I use discrete Fourier transform for digital image processing purposes , but I don't understand basic concept behind it. For example :

  1. What information exists in frequency domain?

  2. What is difference between spatial domain and frequency domain?

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What information exists in frequency domain?

As JasonR says in the comment, The frequency domain is a different view of the same data.

No new information is created, it just takes the "spatial" domain data (the image pixel values and their locations) and re-presents it as the coefficients of complex exponentials (sines and cosines).

What is difference between spatial domain and frequency domain?

The spatial domain is the domain of the image: the pixel values are located at particular positions in the image --- they are spatially distributed, usually in a regular grid.

The frequency domain takes this same data and finds any underlying periodicities (sine waves and cosine waves) in the spatial data, and their amplitudes and phases (spatial offsets).

For example, suppose I have the following image (Scialab, not matlab):

N = 100;
x = [1:N];
y = [1:N];
phi = 2*%pi*0.0987298374*ones(N,1)*x + 2*%pi*0.033102974*y'*ones(1,N);
im1 = sin(phi);

Which looks like (appropriately scaled to the grey scale values):

Periodic image (scaled)

Then the FFT of this is:

FFT of periodic image (scaled)

(again, with appropriate scaling).

The frequency domain version shows up the periodicities of the spatial domain as a small number of large coefficients.

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Fourier transform approximates a function to a sum of sine and cosine signals of varying frequency.

The Fourier transform is an extension of the Fourier series that results when the period of the represented function is lengthened and allowed to approach infinity.

Due to the properties of sine and cosine, it is possible to recover the amplitude of each wave in a Fourier series using an integral.

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  • $\begingroup$ One comment: for a continuous function the Fourier transform does not approximate. The function can be recovered from its Fourier transform exactly. $\endgroup$ – Dima Apr 22 '13 at 14:17
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Spatial domain in image looks like time domain in signals. Any signal (image,data...everything) can be composed of sine signals of varying frequencies (cosine signals are sine signals too, with just some lag or lead). So a definite signal can be decomposed to the sum of lots of sine signals with different frequencies. In more easy terms, any signal has a lot of components having multiple frequencies.

This is the basic underlying principle of fourier transform.

So what it basically tells is what is the strength of the signal at a definite frequency. "What frequency, What strength". So fourier transform mainly converts the signal from spatial domain to frequency domain.

So, in spatial domain, you are concerned about finding the value of the pixel at a definite location, right? Like if I go this location, what value of pixel would I find there, what color would I see there.

But in frequency domain, you can find the strength of the signal occurring at a definite frequency. Remember, any signal contains of components of multiple frequencies. So any signal can be thought of having a lot of frequencies. So in frequency domain, you get to know how strong the signal would be at this particular frequency.

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