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I have two questions related to this image;

enter image description here

Is it possible to identify any spatial domain features in the frequency domain?

My understanding: what I read around they are mostly the same as spatial domain but shows $\sin$ or $\cos$ ripples for every position in the frequency domain but not sure if it correct something like this:

enter image description here

Second question: when I have information about the images, the frequency spectrum and histogram, how do I select the compression method? Do I look for specific pattern or do I need something else? Or is it something entirely else?

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For the first question, first. In short, yes you can, somehow. It depends a lot on how you process images, and the nature or morphology of the images you analyze.

First, it is well etabished that amplitude and phase spectra of natural images are often difficult to analyze (see classical textbooks). I would emphasize three classical issues: nature of the processing, of the images, of the features.

First, the processing itself. Here, you can spot a bright cross on you FFT magnitude spectrum. It is probably due to the periodicity assumed by the FFT. This bright cross, which has a lot of energy, might obfuscate more subtle details. Therefore, some processing steps can be useful before interpreting features: filtering, windowing, cropping, etc.

Second, the morphology of the data. Fourier is great at analyzing stationnary data, either deterministic or stochastic. When the data has non-stationnary aspects, discontinuities, the power of approximation of Fourier methods decreases a lot. This does not entail that Fourier is bad, but its interpretability becomes way more complicated. However, in the Fourier plane, you can spot diagonal lines, or cones, or bumps, that concentrate spatial information. Here, with proper preprocessing, you could expect more concentrated information from the mixed sines. For instance, here, the bright line from the top-left quadrant to the bottom-right one is likely to be related to the left-to-right upward edge of the hat, and the parallel stripe patterns above.

Third, the features. Here, you are looking at visual features from the amplitude spectrum only. A lot of information also resides in the highly non-linear part of Fourier: the phase. Moreover, crafted features can be more involved, like look at at specific scales (eg wavelets or related tools) or layers in deep learning, etc.

There is a lot of ongoing works around your question. My take is that complex-valued, scale-related features can be very efficient. But the margin, you know...

Related posts:

For the second question. Early image compression (1970's, 1980's) have used by the amplitude spectrum and the phase. And the story was: for natural images, phase requires more bits than amplitude. Since then, non-complex (eg real) transforms have been more efficient (DCT-II, biorthogonal wavelets). But I do believe complex transforms will have their word again.

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