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Based on this link

Is it possible to consider a model to introduce FFT2 as a parameter to determine how random the image is? (randomness test like Entropy)

consider Lena's image and its FFT2:

enter image description here

If we do the FFT2 calculations mentioned in the above link, we have: enter image description here

Compared to the output of FFT2 of the white noise image: enter image description here

For example, Is it correct to say that as the image moves away from the random state, the output plot of the 2D Fourier transform also focuses the data on the left instead of the middle of the plot?

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  • $\begingroup$ OK, so there's more happening than just the FFT2 (as you allude to). The question is: what is your measure of randomness? Would a histogram like your first one, but flipped around the vertical access (so the peak was at 1 instead of 0) be "more random" than the first plot? More random than the second plot? Would a histogram that is uniform (same value at from 0 to 1) be "more random" ? What do you mean by "random" ? $\endgroup$
    – Peter K. ♦
    Commented Oct 15, 2022 at 17:31
  • $\begingroup$ @Peter K. I have considered the image of white noise as a random image, and to answer your question, yes, the image of white noise is more random than the image of Lena. $\endgroup$
    – user64854
    Commented Oct 16, 2022 at 1:39
  • $\begingroup$ Analyzing the degree of patternless encrypted data is called the randomness test. For example, in image encryption, an encrypted image with an entropy close to 8 is closer to the random state than an image with an entropy close to 7. The hypothesis I propose is that if we accept that the output of FFT2 calculations is in the form of a normal graph similar to a white noise image, then our data is random, and whatever we move away from this state, we have an image in which a pattern can be found. Can it be shown that this hypothesis is correct? $\endgroup$
    – user64854
    Commented Oct 16, 2022 at 1:39
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    $\begingroup$ That will not work. The distribution (Gaussian) of the samples has very little to do with the randomness of the samples. You have to answer the question what is your measure of randomness?. You keep referring to Entropy. Why not use that directly and not both with FFTs? $\endgroup$
    – Peter K. ♦
    Commented Oct 16, 2022 at 20:25

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A couple of points:

  1. In my answer to the thread you refer to I claim that square root of the absolute value of FFT of white noise is not distributed normally, but real and imaginary pars of FFT are.

  2. Normal distributions in FFT do not necessarily make image unrecognizable. I've corrupted the Lena image so that its FFT looks similar to that of white noise, but you can still recognize the image:

enter image description here

The reader is invited to independently verify the above claim, but I've included my histograms of FFT components and derived information:

enter image description here

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