The mathematics of halftoning

Introduction The problem of halftoning in digital printing has provided a splendid new example of a mathematical opportunity in modern technology. Digital halftoning is the technique used to display an image with a few immiscible colors discretely applied to paper. This problem involves various fields of science such as the physics of light, the biology of the human visual system, and the mathematics of analogto-digital conversion. Digital halftoning can be considered a huge optimization problem, but we shall see that by sacrificing optimality one can construct efficient algorithms which achieve very satisfactory results. Halftoning is above all an art, yet mathematics, particularly the theory of dynamical systems, helps to improve this art, while the art suggests a rich collection of mathematical problems and insights. Specifically, we address the question of how full-color images should be imitated by printers which print at positions on a lattice with colors limited to a few available choices. This question subsumes the simpler, more restrictive one of getting continuous-tone gray-scale images from black and white dots. Black and white digital halftoning was initially created for television well before digital printers! We call the available colors pixel colors and the locations on the lattice pixel locations. Purists reserve the term pixels for screens and instead use pels for printing, but we shall stick to pixels. Pixel colors are for us the colors of inks or toners. In some printers the choice of pixel colors can be augmented by superposition of these substances, but this is not allowed in highlight printers. The two images of Brian Wu (see Figure 1) illustrate halftoning for gray scale: Figure 1(b) is a coarse-grain halftone image of the fine-grain image in Figure 1(a), which is approximately a full gray-scale image. The halftone image in Figure 1 was made by a clustered dither mask, which is discussed later. It was modified to accommodate artificially large pixels. The mathematical problem of digital halftoning is to construct an algorithm whose output is a sequence of pixel colors that creates the full-color visual experience of an input image sequence. For gray-scale images, one wants discrete black and white dots to appear as continuous tones. In discussing the mathematics of printing, one cannot avoid using the language of geometry. It has long been known that our perception of color can be modeled by vector addition in a color space, the dimension of which we call the color dimension. For rendering gray-scale images with black and white dots, the color dimension is 1; for the color space of full color, it is 3.