Ordered Dithering

Classical Algorithm - Developed Algorithms

We can express the patterns in compact form as the order of dots added:

     8  3  4          and        1  7  4

     6  1  2                         5  8  3

     7  5  9                         6  2  9

Then we can simply use the value in the array as a threshold.  If the value of the pixel (scaled into the 0-9 range) is less than the number in the corresponding cell of the matrix, plot that pixel black, otherwise, plot it white.  This process is called ordered dither.  As before, clustered patterns should be used for devices which blur dots.  In fact, the clustered pattern ordered dither is the process used by most newspapers, and the term halftoning refers to this method if not otherwise qualified.

Bayer has shown that for matrices of orders which are powers of two there is an optimal pattern of dispersed dots which results in the pattern noise being as high-frequency as possible.  The pattern for a 2x2 and 4x4 matrices are as follows:

        1  3        1  9  3 11        These patterns (and their rotations

        4  2       13  5 15  7        and reflections) are optimal for a

                    4 12  2 10        dispersed-pattern ordered dither.

                   16  8 14  6

Bayer's method is in very common use and is easily identified by the cross-hatch pattern artifacts it produces in the resulting display.  This artifact is the major drawback of the technique which is otherwise very fast and powerful.  Ordered dithering also performs very badly on images which have already been dithered to some extent.  As stated earlier, dithering should be the last stage in producing a physical display from a digitally stored image.  The dithered image should never be stored itself. 

Exemplifying: