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Consider this small picture of a sunflower, and its histogram:

Picture of a sunflower Histogram for the sunflower picture

What would the Fourier transform of the first picture look like? Is there any relationship between the histogram and the Fourier transform?

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This post is a copy from the top voted answer to the question What does the Fourier Transform mean in the context of images? The answer is math.sx user due to dls.

In sound processing, the Fourier Transform has a physically intuitive meaning. A sound or function $f : [0,b] \rightarrow [-1,1]$ can be represented as a trigonometric series. And each term of the series corresponds to a frequency which you perceive. Many effects (like filtering, reverb, etc.) have an interpretation in the frequency domain which is useful for analysis. Unfortunately, the physical interpretation isn't as simple when we start talking about images. But on the analytic usefulness transfers over.

If we're talking about grey-scale, then an image is simply a function $f : [0,1]^2 \rightarrow [0,1]$. It takes a point in the square $[0,1]\times[0,1]$ and produces a value between 0 and 1, the intensity. The Fourier Transform just says we can represent this function in the frequency domain using a countable basis of trigonometric functions. Say you want to blur an image; this corresponds to a low-pass filter in the frequency domain. The following link shows many examples of Fourier transforms of images, gives an explanation of the physical interpretation (which I don't claim to understand entirely) and shows examples of basic image processing. This website shows more illustrative examples.

Another very important use of (variants of) the 2d-Fourier transform is image compression. The Wikipedia page for the JPEG codec lists the basis functions used to represent images.

Finally, note that if you're talking about an RGB image you can represent the image using the Fourier transform on each color component.

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  • $\begingroup$ But how can you see which point in the spectrum is from which pixel in the original image? $\endgroup$ – user8005 Jun 6 '13 at 22:19
  • $\begingroup$ seems better to me to just link to the other question rather than copying stackexchange answers, unless the answer is somehow different/adds something new/a combination etc $\endgroup$ – vzn Jun 10 '13 at 3:02
  • $\begingroup$ @vzn It's a CW, you can edit it to your likings if you think it's in the community's best interest. $\endgroup$ – Pål GD Jun 11 '13 at 8:52
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    $\begingroup$ There's some really good information in this answer, but I don't see how the connection was made between Fourier transforms and histograms. $\endgroup$ – Galen Apr 22 at 1:26

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