Some Important Details

The gradient below has been halftoned using a "blue noise" technique that generally illustrates the state-of-the-art in dispersed dot halftoning. This is far from being either random, or maximally "smooth", in any mathematical sense.

Random Dot

Truly Random "White Noise" Halftone

The halftone shown below was generated with simple random numbers. Note that a truly random dither looks pretty bad. For a halftone that looks good, it is necessary to "scatter" the dots so that they are widely separated from each other at every density.

Random Dot White Noise

A "Mathematically Perfect" Solution

In a mathematical proof published by Bayer in the 1970's, the dither pattern below was shown to be the smoothest possible at all levels of gray. By "smoothest", we mean that dots are as far apart as they can be for every density.

This "smoothest possible" solution is certainly not very acceptable, as it is filled with patterns that offend the eye.

Bayer Dither

Poor Compromise

Below we show a poor compromise, troubled by the infamous "broken checkerboard" or "worm" problem in some middle densities.

The fundamental problem is that at exactly 50% gray, the smoothest possible dither is a checkerboard, and use of the smoothest possible dither makes problems such as broken checkerboards unavoidable somewhere in midtones.

Nasty visual effects can only be controlled by avoiding the smoothest possible dither, and there is no ideal "nearly smoothest" solution. As with so much in image processing, the final design of good halftoning must be a matter of art rather than science.

Poor Compromise