Why fourier transform is not possible on color images?

Question

Why fourier transform(dft) is not possible on color images? Why only on gray scale images Fourier transforms is possible?

Solution 1

The Fourier Transform works with the intensity of a signal in time or space and converts to intensities of a signal in frequency. There's no single number that can represent the intensity of a color without converting to grayscale.

You can split your image into 3 different images, one for each of the Red, Green, and Blue components. Each of those can be processed with a Fourier Transform independently.

Solution 2

When you apply a Fourier Transform (FT) (or some related transform, e.g. DCT) you are looking at spatial frequencies in the image.

Intuitively, that means that FT re-organizes the spatial information that you have in your image in the form of a matrix which corresponds to coeficients of 2D sinusoids that, if you sum up, you would get the original image.

As Mark Ransom noted, you can apply DFT to each color channel separately. In fact, that basically the approach of the JPEG enconding process, which uses a very similar transform (Discrete Consine Transform - DCT).

This site is relevant to the subject. Note that you can click on the figures to see a larger version.

Solution 3

The usual Fourier trasnform is defined using complex exponentials. If we allow the quaternions in the exponent, instead of simply i=sqrt{-1}, we get a tool called the quaternion Fourier transform. It apparently has some use in the Fourier analysis of color images.

The problem of defining the Fourier transform of a colour image

Solution 4

It is possible if you expand your notion of a Fourier transform (search for multidimensional Fourier transform image or spatiochromatic image) but not as common as their traditional counterparts.

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Comments

[Alceu Costa]

Note that an image is a 2D signal on space and not on time.

[Mark Ransom]

@AlceuCosta, interesting. I've so long thought of the output of the FFT as "frequencies" and of frequencies being a property of time that I never noticed the discrepancy.

[Jonas Heidelberg]

you often say "spatial frequency" when talking about images, to make clear there is no time involved...