[ska]

JPEG isn't crude, agreed. But your comments seem off to me.

The core trick isn't DCT per se, it is transform coding, Which both DCY and wavelets are, followed by (usually) quantization and then entropy coding.

Typical wavelets used are orthonormal transforms, no loss there. The "lack of texture and softened edges" are a choice of the model used, not a consequence of the transform. True also of DCT and blocking artifacts. This should be obvious since both approaches allow for lossless encoders.

Typically wavelets will beat or match JPEG in any situation both in terms of psnr and the like, and perceptually (though this latter is much more controversial and poorly defined -- and to be fair I am not up to date on the literature here but I would be surprised of that has changed in the last decade).

The real reason for the huge popularity of DCT in still compression at first and video codecs later is that it is cheap to implement in hardware. And once there becomes very cheap to use.

JPEG 2000 is needlessly complex, at its core a wavelet codec is also very simple and elegant.