Determine surface homogeneity (luminance, colour, shape) by fourier transform
If I wanted to determine, how homogeneous this source of light is, I'd take a picture of it and put it through a fourier transformation, e.g. a DFT.


Why?

The ideal homogeneous area is a rectangle of some constant colour. A typical sensor has a response with well defined low and high limits. Therefore, excluding ceiling effects at those limits, if you were to shoot a gray card you could then obtain the standard deviation of its values.

A narrow standard deviation would point to a "good" level of homogeneity. A wide standard deviation would point to a "bad" level of homogeneity.

If the standard deviation seems too "coarse", too general a quantification, then the next level up (in complexity) would be to evaluate the co-occurence matrix of the image and through that, the inverse difference moment or homogeneity. The homogeneity is high in low contrast images and vice versa.

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In any case, you would have to calibrate your metric to give meaning to "good" and "bad". Good homogeneity with a web cam is entirely different than good homogeneity with a cooled ccd at the end of a very fast lens (i.e. a lens with a large diameter) for shooting a grey card under the same lighting conditions.


  How would you determine, from a fourier spectrum alone, if the source of light is homogeneous or not?


If you absolutely had to use the Discrete Fourier Transform, then you would again be looking at how is a flat or "rough" surface "expressed" in the Fourier Transform. In general, the Fourier Transform is used for the "detection" of texture, especially in the case of repeating elements. Therefore, absence of texture leaves just an average value, full presence of texture is something like the spectrum of white noise and intermediate cases would have to be judged with something like spectral flatness but adapted to the two dimensional case.

Hope this helps.