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This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is saying.

What does it really mean and why is the Fourier Transform so prevalent in image processing?

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What part are you having difficulty understanding? An image as a function? 2D Fourier transform? Something else? –  Peter Taylor Jan 29 '12 at 23:48
    
The 2D Fourier Transform, yes. I am specifically interested in the application to images, though. :) –  water Jan 30 '12 at 5:08
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3 Answers 3

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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Thanks, I'll check these links out when I get home tomorrow! –  water Jan 30 '12 at 5:08
    
I recently became aware of the quaternion Fourier transform, which is apparently useful in the analysis of color images. ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=723451 –  dls Jul 30 '13 at 4:09
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The Fourier transform - any Fourier transform - splits a signal into "frequencies", and measures the amplitude and alignment of each frequency.

In the case of sound, these are audible frequencies that you can hear. But in the case of an image, things are less obvious. The mathematics is still the same, but it's harder to wrap your brain around.

The Fourier transform measures "spatial frequencies" in the image. If you imagine horizontal or vertical bars of colour repeating at different speeds, these are the "frequencies" that the Fourier transform is measuring. Much like a sound signal, an image with long, rolling, smooth colour transitions contains many low frequencies, whereas one with abrupt changes in colour possesses lots of high frequencies.

The Fourier transform thus has a couple of uses in image processing. I can think of two:


First, when you change any signal, its spectrum obviously changes as well. When you take a photograph and the camera moves, you get a blurry image. It's not at all obvious how you could try to "unblur" this image. But, when you talk about the spectrum of the image, a blur is simply a low-pass filtering operation. In principle, if you undo that filtering, you could unblur the image.

(Obviously, that's the theory. In practise, it's not that simple...)

Lots of other interesting things you could do to an image are quite complicated in terms of what happens to the individual pixels, but very simple in terms of how the spectrum changes. So using the Fourier transform to get you a spectrum is an obvious step.


Alternatively, the Fourier transform is useful for image compression. If you save the individual pixel colours less accurately, the image just looks like some God-awful computer graphics from the 1980s. But if you save the spectrum less accurately, the picture just gets slightly blurry, which is far less annoying.

By doing a sophisticated analysis of the way the human brain processes image data, you can estimate which frequencies in a given image are "the most important", and store those with high precision, while throwing away any "less important" frequencies. This is how JPEG and friends work.

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But how can you see which point in the spectrum is from which pixel in the original image? –  user8005 Jun 6 '13 at 22:20
    
Each pixel affects several frequencies, and each frequency affects several pixels. It's just that adjusting frequencies rather than adjusting pixels can be easier / more useful for certain types of problems. –  MathematicalOrchid Jun 7 '13 at 17:30
    
"Each pixel affects several frequencies, and each frequency affects several pixels." How does that make sense intuitively? –  user8005 Jun 8 '13 at 10:15
    
If you want to know which pixel is which, work with pixels. Working with frequencies is useful when you're trying to do other tasks - when you're not interested in individual pixels, but you want to manipulate the entire image in some way. For example, if you increase the amplitude of high frequencies, you make the image look sharper. (Without having to do the complicated per-pixel processing of, say, an unsharp mask.) That kind of thing. –  MathematicalOrchid Jun 8 '13 at 19:39
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I can't see where to write a comment, so I'm going to have to give you my ideas as an answer (my apologies to Stack Exchange mods).

The Fourier transform extracts how much a signal is "like" a sinusoid with particular wavelength. Each color is a different wavelength (which I guess means how fast the light photons are oscillating, or something like that). Therefore, I'm guessing that the transform means how much of the input signal is made up of different wavelengths, or colors. HTH.

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This is only true if you're talking about a single beam of light, then the Fourier spectrum translates to the familiar color spectrum (though, I am not a physicist). There is a different interpretation when we're talking about the Fourier transform of an image. –  dls Jan 30 '12 at 2:41
    
@dls: Isn't each pixel of an image a single beam of light? –  Jeff Jan 30 '12 at 5:09
    
Agreed. My point is that this application of the 1d-Fourier transform is not very prevalent in image processing. More commonly, the 2d-transform is used in the analysis of image processing (e.g. blurring), image compression (JPEG), etc. –  dls Jan 30 '12 at 5:36
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-1, this is not how the Fourier transform is used in image processing. See dls's answer. –  Rahul Jan 30 '12 at 5:58
    
@dls How about adding the word 'cosinetransform'? –  user8005 Jun 7 '13 at 19:29
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