It's also worth to mention noise-based dithering - where some noise pattern is added atop of the image and then rounding is performed. Usually some sort of blue noise is used for this approach.
Dithering has similar importance in digital audio. Dithered 8-bit audio sounds way better than non-dithered (harsh artifacts are replaced with tolerable white noise, and quiet details are preserved). Higher end digital equipment even applies dithering to high-bit samples, as do plug-ins in digital audio workstations.
Ulichney (who wrote the book on halftoning) came up with ordered dithering matrices that give much nicer results than Bayer's, as good error as diffusion, and parallelizable. Look up "void and cluster".
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