The Neugebauer equations are a set of equations used to model

color printing Color printing or colour printing is the reproduction of an image or text in color (as opposed to simpler black and white or monochrome printing). Any natural scene or color photograph can be optically and physiologically dissected into three ...

systems, developed by Hans E. J. Neugebauer. They were intended to predict the color produced by a combination of

halftone Halftone is the reprographic Reprography (a portmanteau of ''reproduction'' and ''photography'') is the reproduction of graphics through mechanical or electrical means, such as photography or xerography. Reprography is commonly used in catal ...

s printed in cyan, magenta, and yellow inks. The equations estimate the

reflectance The reflectance of the surface of a material is its effectiveness in reflecting radiant energy. It is the fraction of incident electromagnetic power that is reflected at the boundary. Reflectance is a component of the response of the electronic ...

(in

CIE XYZ The CIE 1931 color spaces are the first defined quantitative links between distributions of wavelengths in the electromagnetic visible spectrum, and physiologically perceived colors in human color vision. The mathematical relationships that defin ...

coordinates or as a function of wavelength) as a function of the reflectance of the 8 possible combinations of CMY inks (or the 16 combinations of CMYK inks), weighted by the area they take up on the paper. In wavelength form: :

R(\lambda) = \sum_^ w_i R_i(\lambda),

where ''R_i''(''λ'') is the reflectance of ink combination ''i'', and ''w_i'' is the relative proportions of the 16 colors in a uniformly colored patch. The weights are dependent on the halftone pattern and possibly subject to various forms of

dot gain Dot gain, or tonal value increase, is a phenomenon in offset lithography and some other forms of printing which causes printed material to look darker than intended. It is caused by halftone dots growing in area between the original printing film a ...

. Light can interact with the paper and ink in more complex ways. The Yule–Nielsen correction takes into account light entering through blank regions and re-emerging through ink: :

R(\lambda) = \left( \sum_^ w_i R_i(\lambda)^\frac \right)^n.

The factor ''n'' would be 2 for a perfectly diffusing Lambertian paper substrate, but can be adjusted based on empirical measurements. Further considerations of the optics, such as multiple internal reflections, can be added at the price of additional complexity. In order to achieve a desired reflectance, these equations have to be inverted to produce the actual dot areas or digital values sent to the printer, a nontrivial operation that may have multiple solutions.

References

Equations Color Printing {{mathapplied-stub

See also

References