As someone who has used inverse FFT to recover printed images: It's not as magical as this makes it out to be. It depends very heavily on the amount of effort you put in, and it takes a substantial amount of effort to get anything close to the original. Multiply by four for color images, you really have to work each channel of CMYK separately. Also, you are NOT recovering lost data, you are interpolating results from the data you have - if halftoning destroyed a detail, you can't get it back with inverse FFT.

Sorry for sounding a little too excited: As the article notes, you usually loose additional details by applying filters, which is not necessary with this method.

A common technique was applying the “grow [dark/light] areas” filter, which would eat up the space between dots. That sounds like a non-mathematical variant of this filter (and also produced varying results).

Well, it is a little exciting, honestly, at least to us newspaper guys, because it does give us something usable from essentially nothing. And it's something you genuinely could successfully apply machine learning to, and do so on your home desktop, to get result you genuinely thought were magical. (You just have to take the grain of salt with it - it's still not the original, just a very close simulacrum.

Lots and lots of math.