To generate the matrix described above:
Get the binary representation of the x coordinate and of the y coordinate within the array. Reverse the order of the bits, then interleave the two numbers (e.g. abc interleaved with def is adbecf). For example, in an 8*8 matrix, the entry at (5, 3) is calculated as follows: the x coordinate is binary 101, the y coordinate is binary 011, the two strings reversed are 101 and 110, interleaved you get 110110, which is the binary number for that entry in the array (54).
The whole array looks like:
0 32 8 40 2 34 10 42
16 48 24 58 18 50 26 58
4 36 12 44 6 38 14 46
20 52 28 60 22 54 30 62
1 33 9 41 3 35 11 43
17 49 25 57 19 51 27 59
5 37 13 45 7 39 15 47
21 53 29 61 23 55 31 63
though, of course, a
reflection of the matrix through
vertical,
horizontal, or either
diagonal would give equally
effective results, or a 90
degree, 180 degree or 270 degree
rotation.
There is a generalisation to colour pixels, and to multiple greyscales (rather than just black and white). Imagine a source file with 8 bits of greyscale information, 256 shades of grey, that is to be transformed into a 2-bits-per-pixel file, 4 different shades. In this case, 6 bits of shade are to be dropped from each pixel. Create a matrix, size 8*8 (so there's 2^6 entries). To each 8-bit value in the original file, add the corresponding entry in the matrix (overflows are to count as the maximum permissible value, in this case 255). Now simply drop the least significant 6 bits in every pixel. The result will be an ordered dither 4-shade representation of the original 256-shade picture. To perform ordered dither on a colour picture, take each of the three RGB channels of colour information and treat each seperately as a greyscale image, then recombine the colours at the end.