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It would need an extra HTTP request, yes. In practice for the PNG solution, one would be better off with sending the DPI in the request headers, so that the "correct" image is the only one sent.

For JPEG 2000, the network characteristics are important, but I don't think it would be too bad on a mobile network. Low DPI devices might get a bit "too much", but it wouldn't be a problem - they can just throw it out (or incorporate more detail).

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Headers do seem the best place for a non vendor specific standard to be set. I'd love to see something like

viewport-dpi - the dpi viewport-max - the maximum possible pixel dimension and possibly viewport-current - the pixel dimensions at the time of the request

Decent responsive design should deal with the differing viewport sizes, but it might be nice to get a hint before delivering your page what direction to weight that response in.

What do you do when using a projector, and your DPI is something like 3? (serious question, I don't know how this is traditionally handled)

Something like device-pixel-ratio would be more useful than DPI. Raw DPI is meaningless unless you take distance into account, as is very clear in the case of the projector.

Would this work better over WebP?

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