There's a series of videos showing the algorithm: https://www.youtube.com/watch?v=LFXN9PiOGtY

I finally understood DCT after watching https://www.youtube.com/watch?v=Q2aEzeMDHMA

And someone created a "JPEG to PNG" tool that "finds the smoothest possible picture that encodes to the given JPEG file": https://github.com/victorvde/jpeg2png . I tried it with some very terribly compressed images, and the outputs were a lot better.