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There's a new image format available, called BPG (Better Portable Graphics). You can see an example of how it differs from JPEG here: http://bellard.org/bpg/lena.html

The curious thing to note about that page is that it actually loads the bpg versions as bpg files from the server and decompresses them with javascript. You could probably cut down on the server bandwidth usage a lot if you changed the site to use BPG where sensible.

In addition to being just plain more efficient, it also supports lossless compression and transparency. In my limited tests, the lossless mode compresses significantly better than PNG. I also suspect the highest quality lossy modes mostly just leave out noise.

Compression is slow but decompression is fast and most computers actually possess hardware that is capable in decoding it in hardware. (It's based on HEVC algorithms.)

For more info and sources or windows binaries, see here:
http://bellard.org/bpg/