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I understand the 1D Fourier transform but I don't have any experience with the 2D version so there are a couple of things I'm wondering about. If you take the Fourier transform of a 2D image you end up with another 2D image. What do the intensities of the following sets of pixels represent:

  1. center pixel
  2. center vertical row of pixels
  3. four corner pixels

Am I correct in thinking that the intensity of the center pixel represents the mean value of all the pixels in the original image?

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    $\begingroup$ This appears to be a pure math question (or, depending on taste, it belongs on the signal processing site). $\endgroup$ – knzhou Oct 20 '16 at 18:16
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You don't define or elaborate regarding the original image you want to transform, so although you are correct in your question:

Am I correct in thinking that the intensity of the center pixel represents the mean value of all the pixels in the original image?

at least according to this Web page, Intuitive Explanation of Fourier Theory.

It seems to me, as an almost total beginner, but from reading a little on the subject, and having a past interest in data compression on analogue music files, that the answers to your question 2 and 3 are completely dependent on the input you use.

enter image description here

For example if the input image is a sinusoidal grating, as shown below, the resultant Fourier image will have a bright spot at the center, the DC term, with two flanking peaks on either side, whose distance from the center will vary with the spatial frequency of the sinusoid.

I have seen this answer on other related pages, such as 2D Image Transformation, where it states that:

In most implementations the Fourier image is shifted in such a way that the DC-value (i.e. the image mean) F(0,0) is displayed in the center of the image. The further away from the center an image point is, the higher is its corresponding frequency.

But as far as your questions:

What do the intensities of the following sets of pixels represent: 2. center vertical row of pixels? 3. four corner pixels?

It really would seem to depend on your original image as to what exactly is represented by those points on the output file.

What I would suggest is that that your read the two link pages, and please bear in mind that my "answer" to this question was based my interest in the similarities between MP3 encoding, which reduces file sizes by "throwing away" sounds that would be inaudible, and 2D compression.

From reading the links, it it obvious that 2D sampling is a more sophisticated version of this technique, so I hope you receive a similiarly more sophisticated answer than this one, that we both can learn from.

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  • $\begingroup$ What the pixels represent is independent of the image. It is the pixel intensities themselves that are dependent on the image under consideration. $\endgroup$ – ManUtdBloke Oct 26 '16 at 12:03

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